# Patent

## Title
Elliptic curve point octupling for weighted projective coordinates

## Abstract

Systems and methods for elliptic curve octupling for weighted projective coordinates are described. In one aspect, a weighted projective point P on an elliptic curve is identified. 8P is computed from P independent of repeated doubling operations using fewer field multiplications.

## Background
TECHNICAL FIELD
The systems and methods of this specification relate to elliptic curve cryptography.
BACKGROUND
Cryptographic systems are concerned with confidentiality, authenticity, integrity, and non-repudiation of data sent from a first party to a second party. Modern cryptographic schemes based on the discrete logarithm problem in a finite abelian group are designed to address these concerns. One such finite abelian group, which is becoming increasingly popular, is a group of points on an elliptic curve (EC) over a finite field with group operation provided by simple algebraic formulae. This is because such group operations are relatively simple to realize in hardware or software. However, to fully realize implementation efficiencies of using such group, system designers need to pay close attention to underlying implementations of associated field arithmetic.
For example, in scenarios where field inversions are significantly more expensive than multiplication, it is typically more efficient to utilize weighted projective coordinates so that point addition can be performed using field multiplications, as described by Blake et al, “Elliptic Curves in Cryptography”, Cambridge University Press, 1999, page 59-60, and defer field inversions (usually only one such inversion operation is implemented at the end of a long sequence of multiplications). However, the computational cost of eliminating inversions is that an increased number of multiplications must generally be calculated. An efficient method from multiplying two elements in a finite group G is essential to performing efficient exponentiation.
Exponentiation is commonly used in public-key cryptography to calculate a scalar multiple n of points P on an elliptic curve, where n is a very large integer (e.g., a random number or private key), and wherein P is a weighted projective coordinate. An unsophisticated way to compute nP is to do n−1 multiplications in the group G. For cryptographic applications, the order of the group G typically exceeds 2160 elements, and may exceed 22024 elements. Such operations are computationally intensive, and most choices of n are large enough that it becomes infeasible to calculate nP using n−1 successive multiplications by P. However, there are a number of techniques that can be used to reduce the computational costs of exponentiation.
For instance, repeated square-and-multiply algorithms (i.e., binary exponentiation) and windowing methods such as described by Blake et al, “Elliptic Curves in Cryptography”, Cambridge University Press, 1999, pages 63-72, can reduce the computational costs of exponentiation. More particularly, repeated square-and-multiply algorithms divide the exponent n into smaller sums of powers of two (2), which respectively take less processing resources to compute. For instance, given a projective point P with coordinates (x, y, z) on an elliptic curve over a finite field, n can be divided into pieces of size 23 (i.e., using a window of 3) to calculate scalar multiples of P (23P, or 8P) with multiple point doubling iterations. To accomplish this, existing systems typically input P=(x, y, z) into the square-and-multiply algorithm to generate 2P. Next, the coordinates for 2P (output from the first doubling operation) are input as (x, y, z) into the same square-and-multiply algorithm to obtain 4P. Finally, this iterative process is repeated one more time to input the coordinates for 4P (output from the second doubling operation) as (x, y, z) into the same square-and-multiply algorithm to obtain 8P. This process to obtain 8P involves a total of 30 field multiplications.
SUMMARY
Systems and methods for elliptic curve octupling for weighted projective coordinates are described. In one aspect, a weighted projective point P on an elliptic curve is identified. 8P is computed from P with direct formulas using 26 multiplications independent of repeated doubling operations.



BRIEF DESCRIPTION OF THE DRAWINGS
In the Figures, the left-most digit of a component reference number identifies the particular Figure in which the component first appears.
FIG. 1 illustrates an exemplary system for elliptic curve point octupling for weighted projective coordinates.
FIG. 2 shows an exemplary procedure for elliptic curve point octupling for weighted projective coordinates.
FIG. 3 illustrates an example of a suitable computing environment for fully or partially implementing the systems and methods for elliptic curve point octupling for weighted projective coordinates.



DETAILED DESCRIPTION
Overview
The systems and methods for elliptic curve point octupling for weighted projective coordinates substantially optimize weighted projective point exponentiation by calculating 8P without repeated doubling operations. In one implementation, this is accomplished via a straight in-line algorithm in 26 field multiplications. Use of 26 field multiplications to calculate 8P is in contrast to conventional techniques, which require 30 field multiplications and repeated/iterative doublings to compute 8P (i.e., 2P to 4P to 8P). By reducing the number of field multiplications to compute 8P from 30 to 26, the systems and methods provide substantial processing performance improvements over conventional 8P binary exponentiation implementations.
These and other aspects of the systems and methods for elliptic curve point octupling for weighted projective coordinates are now described in greater detail.
An Exemplary System
Although not required, the systems and methods for elliptic curve point octupling for weighted projective coordinates are described in the general context of computer-executable instructions (program modules) being executed by a computing device such as a personal computer. Program modules generally include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. While the systems and methods are described in the foregoing context, acts and operations described hereinafter may also be implemented in hardware.
FIG. 1 illustrates an exemplary system 100 for elliptic curve point octupling for weighted projective coordinates. System 100 includes a first computing device 102 coupled over a communications network 103 to a second computing device 104. Communications network 103 may include any combination of a local area network (LAN) and a general wide area network (WAN) communication environments, such as those which are commonplace in offices, enterprise-wide computer networks, intranets, and the Internet. First and second computing devices 102 and 104 respectively represent any type of computing device such as a personal computer, a laptop, a server, handheld or mobile computing device (e.g., a cellular phone, personal digital assistant), and/or so on.
Computing device 102 includes program module(s) 106 and program data 108. Program modules 106 include, for example, signing/encrypting module 110 to respectively encrypt or sign original data 112 using elliptic curve (EC) point octupling exponentiation. Independent of the below described algorithms for elliptic curve point octupling, the particular cryptographic protocol implemented by signing/encrypting module 110 is arbitrary and a function of the particular cryptographic algorithm selected for implementation. To perform elliptic curve point octupling, signing/encrypting module 110 operates on a set of points on elliptic curve 114. Elliptic curve 114 conforms to the following equation: y2=x3+axz4+bz6 in field characteristic not equal to 2 or 3. The characteristic of the underlying field is defined to be the smallest non-zero natural number, p, such that p times any element of the field is zero.
A point P on elliptic curve 114 is represented in weighted projective form as P=(x: y: z). In one implementation, and if the cryptographic protocol implemented is a signature scheme, the point P may be a part of the public information for the cryptosystem. In another implementation, and if the cryptographic protocol implemented is a key agreement scheme (e.g., Diffie-Hellman key agreement) or an encryption scheme (e.g., ElGamal encryption), the point P may be a part of the public or private information for the cryptosystem.
In one implementation, signing/encrypting module 110 implements elliptic curve point octupling exponentiation using a direct (non-inline) algorithm. TABLE 1 shows an exemplary direct elliptic curve point octupling algorithm for signing/encrypting module 110 to determine 8P (see, component 116 of FIG. 1) from a point P on elliptic curve 114.







TABLE 1





EXEMPLARY DIRECT ELLIPTIC CURVE OCTUPLING


ALGORITHM TO COMPUTE 8P OF A POINT P


ON AN ELLIPTIC CURVE



















  
    
      
        
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              3
              ⁢
              
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                              (
                              
                                
                                  
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                                          2
                                        
                                      
                                      +
                                      
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                                        4
                                      
                                    
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                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
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                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
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                                  2
                                
                              
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                          2
                        
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                  ⁢
                  
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                    ⁢
                    
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                  )
                
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                              12
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                                    3
                                    ⁢
                                    
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                        8
                        ⁢
                        
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                          4
                        
                      
                    
                    )
                  
                  2
                
              
              )
            
            2
          
          +
          
            256
            ⁢
            
              
            
            ⁢
            a
            ⁢
            
              
            
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
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                                      3
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                    +
                                    
                                      az
                                      4
                                    
                                  
                                  )
                                
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                        -
                        
                          8
                          ⁢
                          
                            y
                            4
                          
                        
                      
                      )
                    
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                  ⁢
                  
                    y
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
                )
              
              2
            
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          -
          
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            ⁢
            
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                              ⁢
                              
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                              4
                            
                          
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                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    8
                    ⁢
                    
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                  )
                
                2
              
              +
              
                16
                ⁢
                
                  ay
                  4
                
                ⁢
                
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                  4
                
              
            
            )
          
          ⁢
          
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              ⁢
              
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                          3
                          ⁢
                          
                            x
                            2
                          
                        
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                          4
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
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                      2
                    
                  
                
                )
              
              ⁢
              
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                    (
                    
                      
                        3
                        ⁢
                        
                          x
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                      +
                      
                        az
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                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
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                                2
                              
                            
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                              4
                            
                          
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                      )
                    
                    -
                    
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                      ⁢
                      
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                        4
                      
                    
                  
                  )
                
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              -
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                2
              
            
            )
          
          -
        
      
    
    
      
        
          
            
            ⁢
            
              8
              ⁢
              
                
              
              ⁢
              
                
                  (
                  
                    
                      
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                            3
                            ⁢
                            
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                          +
                          
                            az
                            4
                          
                        
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                      ⁢
                      
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                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                4
              
            
            )
          
          4
        
      
    
  
   






 






  
    
      
        
          t
          :=
          
          ⁢
          
            8
            ⁢
            
              (
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    12
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                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
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                          2
                        
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                          ⁢
                          
                            xy
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                            3
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                    ⁢
                    
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                      4
                    
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                -
                
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                  ⁢
                  
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                    4
                  
                
              
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                              ⁢
                              
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                  ⁢
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
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                            3
                            ⁢
                            
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                            12
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                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
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                                  4
                                
                              
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                    -
                    
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                      ⁢
                      
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                        4
                      
                    
                  
                  )
                
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          )
        
      
    
    
      
        
          
          ⁢
          
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
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                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
              
              )
            
            ⁢
            yz
          
        
      
    
  
   













Table 1 gives the formula for the quantity 8P=(r: s: t), where P is the point given in weighted projective coordinates (x: y: z) on the elliptic curve y2=x3+axz4+bz6. In contrast to conventional binary exponentiation techniques to generate 8P from a point P on an EC, which iteratively invoke the doubling algorithm to input output (e.g, 2P or 4P) from a previous point doubling operation, the direct elliptic curve point octupling algorithm of TABLE 1 calculates the coordinates (r, s, t) of 8P of a point P on an elliptic curve 114 independent of any iterative invocation of point doubling operations.
Such a direct point octupling algorithm can be used as the basic operation for an exponentiation routine using window size 3 (where the window size indicates which power of 2 is used to decompose the exponent). For example, 587*P can be calculated as 8*(8*(8*P+P)+P)+3*P instead of as 2*(2*(2*(2*(2*(2*(2*2*2*P+P)))+P))+P)+P.
In one implementation, signing/encrypting module 110 calculates 8P (116) of a point P on elliptic curve 114 using a straight line algorithm of TABLE 2 for elliptic curve point octupling exponentiation. For purposes of discussion, a straight line formula is a process for obtaining the desired result by executing each statement in the formula in succession, much like operations are implemented by a computer, with no loops or branches to the code. Program instructions of the straight line program of TABLE 2 are bolded and proceeded by a forward angle bracket. In each instruction there is the following symbol “:=”, which means definition. The program instructions are shown in program description language (PDL). For purposes of comparison, an execution trace (non-bolded) of the corresponding program instructions from the direct algorithm of TABLE 2 is presented adjacent to each program instruction of the straight line algorithm of TABLE 2.







TABLE 2





EXEMPLARY STRAIGHT LINE PROGRAM FOR ELLIPTIC


CURVE POINT OCTUPLING















>x2 := x{circumflex over ( )}2;







x2 := x2







>y2 := y{circumflex over ( )}2;







y2 := y2







>y4 := y2{circumflex over ( )}2;







y4 := y4







>xy2 := x*y2;







xy2 := xy2







>yz := y*z;







yz := yz







>z2 := z{circumflex over ( )}2;







z2 := z2







>z4 := z2{circumflex over ( )}2;







z4 := z4







>az4 := a*z4;







az4 := az4







>ay4z4 := az4*y4;







ay4z4 := ay4z4







>T1 := 3*x2 + az4;







T1 := 3x2 + az4







>T12 := T1{circumflex over ( )}2;







T12 := (3x2 + az4)2







>T2 := 12*xy2 − T12;







T2 := 12xy2 − (3x2 + az4)2







>T3 := T12 − 8*xy2;







T3 := (3x2 + az4)2 − 8xy2







>T32 := T3{circumflex over ( )}2;







T32 := ((3x2 + az4)2 − 8xy2)2







>T4 := T1*T2 − 8*y4;







T4 := (3x2 + az4)(12xy2 − (3x2 + az4)2) − 8y4







>T42 := T4{circumflex over ( )}2;







T42 := ((3x2 + az4)(12xy2 − (3x2 + az4)2) − 8y4)2







>T44 := T42{circumflex over ( )}2;







T44 := ((3x2 + az4)(12xy2 − (3x2 + az4)2) − 8y4)4







>T5 := 3*T32 + 16*zy4z4;







T5 := 3((3x2 + az4)2 − 8xy2)2 + 16ay4z4







>T52 := T5{circumflex over ( )}2;







T52 := (3((3x2 + az4)2 − 8xy2)2 + 16ay4z4)2







>T52 := T44*256*ay4z4;







T6 := 256a((3x2 + az4)(12xy2 − (3x2 + az4)2) − 8y4)4y4z4







>T7 := T3*T42;







T7 := ((3x2 + az4)2 − 8xy2)((3x2 + az4)(12xy2 − (3x2 + az4)2) − 8y4)2







>T8 := T52 − 8*T7;


 






  
    
      
        
          
            T
            ⁢
            
              
            
            ⁢
            8
          
          :=
          
          ⁢
          
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            -
            
              8
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      xy
                      2
                    
                  
                
                )
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
              
              )
            
            2
          
        
      
    
  
   






 


>T82 := 3*T8{circumflex over ( )}2;


 






  
    
      
        
          
            T
            ⁢
            
              
            
            ⁢
            82
          
          :=
          
          ⁢
          
            3
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                    +
                                    
                                      az
                                      4
                                    
                                  
                                  )
                                
                                2
                              
                              -
                              
                                8
                                ⁢
                                
                                  xy
                                  2
                                
                              
                            
                            )
                          
                          2
                        
                      
                      +
                      
                        16
                        ⁢
                        
                          ay
                          4
                        
                        ⁢
                        
                          z
                          4
                        
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            )
          
          2
        
      
    
  
   






 


>T9 := T82 + T6;


 






  
    
      
        
          
            T
            ⁢
            
              
            
            ⁢
            9
          
          :=
          
          ⁢
          
            3
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                    +
                                    
                                      az
                                      4
                                    
                                  
                                  )
                                
                                2
                              
                              -
                              
                                8
                                ⁢
                                
                                  xy
                                  2
                                
                              
                            
                            )
                          
                          2
                        
                      
                      +
                      
                        16
                        ⁢
                        
                          ay
                          4
                        
                        ⁢
                        
                          z
                          4
                        
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
              )
            
            2
          
          +
        
      
    
    
      
        
          
          ⁢
          
            256
            ⁢
            
              
            
            ⁢
            a
            ⁢
            
              
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              4
            
            ⁢
            
              y
              4
            
            ⁢
            
              z
              4
            
          
        
      
    
  
   






 


>T92 := (T9){circumflex over ( )}2;


 






  
    
      
        
          
            T
            ⁢
            
              
            
            ⁢
            92
          
          :=
          
          ⁢
          
            (
            
              3
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
              )
            
            2
          
          +
        
      
    
    
      
        
          
            
            ⁢
            
              256
              ⁢
              
                
              
              ⁢
              a
              ⁢
              
                
              
              ⁢
              
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                4
              
              ⁢
              
                y
                4
              
              ⁢
              
                z
                4
              
            
            )
          
          2
        
      
    
  
   






 


>S10 := T52 − 8*T7;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            10
          
          :=
          
          ⁢
          
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            -
            
              8
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      xy
                      2
                    
                  
                
                )
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
              
              )
            
            2
          
        
      
    
  
   






 


>S11 := 12*T7 − T52;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            11
          
          :=
          
          ⁢
          
            12
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  y
                  4
                
              
              )
            
            2
          
          -
          
            
              (
              
                
                  3
                  ⁢
                  
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                    2
                  
                
                +
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
              
              )
            
            2
          
        
      
    
  
   






 


>S12 := T5*S11;


 






  
    
      
        
          S12
          :=
          
          ⁢
          
            
              (
              
                
                  3
                  ⁢
                  
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                    2
                  
                
                +
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
              
              )
            
            ⁢
            
              (
              
                12
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                    
                      8
                      ⁢
                      
                        
                          (
                          
                            x
                            ⁢
                            y
                          
                          )
                        
                        2
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            -
          
        
      
    
    
      
        
          
          ⁢
          
            
              (
              
                
                  3
                  ⁢
                  
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                    2
                  
                
                +
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
              
              )
            
            2
          
          )
        
      
    
  
   






 


>S13 := S12 − 8*T44;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            13
          
          :=
          
          ⁢
          
            
              (
              
                
                  3
                  ⁢
                  
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                    2
                  
                
                +
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
              
              )
            
            ⁢
            
              (
              
                12
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                    
                      8
                      ⁢
                      
                        xy
                        2
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      az
                      4
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
              
              )
            
            2
          
          -
          
            (
            
              
                3
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                  2
                
              
              +
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              4
            
          
        
      
    
  
   






 


>S132 := S13{circumflex over ( )}2;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            132
          
          :=
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              ⁢
              
                (
                
                  12
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            -
            
              (
              
                3
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
                )
              
              4
            
            )
          
          2
        
      
    
  
   






 


>S14 := S10*S132;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            14
          
          :=
          
          ⁢
          
            (
            
              
                
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              
                                (
                                
                                  
                                    
                                      (
                                      
                                        
                                          3
                                          ⁢
                                          
                                            x
                                            2
                                          
                                        
                                        +
                                        
                                          az
                                          4
                                        
                                      
                                      )
                                    
                                    2
                                  
                                  -
                                  
                                    8
                                    ⁢
                                    
                                      xy
                                      2
                                    
                                  
                                
                                )
                              
                              2
                            
                          
                          +
                          
                            16
                            ⁢
                            
                              ay
                              4
                            
                            ⁢
                            
                              z
                              4
                            
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
              
                
                  
                    8
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                
                                  12
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                                -
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              )
                            
                          
                          -
                          
                            8
                            ⁢
                            
                              y
                              4
                            
                          
                        
                        )
                      
                      2
                    
                  
                
              
            
            )
          
        
      
    
    
      
        
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              ⁢
              
                (
                
                  12
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            -
            
              (
              
                3
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
                )
              
              4
            
            )
          
          2
        
      
    
  
   






 


>r := T92 − 8*S14;


 






  
    
      
        
          r
          :=
          
          ⁢
          
            (
            
              3
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                    
                      8
                      ⁢
                      
                        xy
                        2
                      
                    
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              
                            
                            ⁢
                            
                              az
                              4
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              12
                              ⁢
                              
                                xy
                                2
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        8
                        ⁢
                        
                          y
                          4
                        
                      
                    
                    )
                  
                  2
                
              
              )
            
            2
          
          +
        
      
    
    
      
        
          
            
              
              ⁢
              
                256
                ⁢
                
                  
                
                ⁢
                a
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              12
                              ⁢
                              
                                xy
                                2
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        8
                        ⁢
                        
                          y
                          4
                        
                      
                    
                    )
                  
                  4
                
                ⁢
                
                  y
                  4
                
                ⁢
                
                  z
                  4
                
              
              )
            
            2
          
          -
        
      
    
    
      
        
          
          ⁢
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                    +
                                    
                                      az
                                      4
                                    
                                  
                                  )
                                
                                2
                              
                              -
                              
                                8
                                ⁢
                                
                                  xy
                                  2
                                
                              
                            
                            )
                          
                          2
                        
                      
                      +
                      
                        16
                        ⁢
                        
                          ay
                          4
                        
                        ⁢
                        
                          z
                          4
                        
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
              
              )
            
            2
          
          )
        
      
    
    
      
        
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              ⁢
              
                (
                
                  12
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            -
            
              (
              
                3
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                          ⁢
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                          )
                        
                        2
                      
                      +
                      
                        16
                        ⁢
                        
                          ay
                          4
                        
                        ⁢
                        
                          z
                          4
                        
                      
                    
                    )
                  
                  2
                
                )
              
              -
              
                8
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              12
                              ⁢
                              
                                xy
                                2
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        8
                        ⁢
                        
                          y
                          4
                        
                      
                    
                    )
                  
                  4
                
              
            
            )
          
          2
        
      
    
  
   






 


>S15 := 12*S14 − T92;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            15
          
          :=
          
          ⁢
          
            12
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                    +
                                    
                                      az
                                      4
                                    
                                  
                                  )
                                
                                2
                              
                              -
                              
                                8
                                ⁢
                                
                                  xy
                                  2
                                
                              
                            
                            )
                          
                          2
                        
                      
                      +
                      
                        16
                        ⁢
                        
                          ay
                          4
                        
                        ⁢
                        
                          z
                          4
                        
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
                )
              
              ⁢
              
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            az
                            4
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
            
            )
          
          ⁢
          
            (
            
              (
              
                3
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    8
                    ⁢
                    
                      xy
                      2
                    
                  
                  )
                
                2
              
              +
              
                16
                ⁢
                
                  ay
                  4
                
                ⁢
                
                  z
                  4
                
              
            
            )
          
          ⁢
          
            (
            
              12
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      xy
                      2
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
              -
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                2
              
            
            )
          
          -
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              12
                              ⁢
                              
                                xy
                                2
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        8
                        ⁢
                        
                          y
                          4
                        
                      
                    
                    )
                  
                  4
                
              
              )
            
            2
          
          -
          
            (
            
              3
              ⁢
              
                (
                
                  (
                  
                    3
                    ⁢
                    
                      (
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                          ⁢
                          
                            az
                            4
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                  2
                
                +
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
              
              )
            
            2
          
          -
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      x
                      2
                    
                  
                  +
                  
                    az
                    4
                  
                
                )
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
              )
            
            2
          
          +
          
            256
            ⁢
            
              
            
            ⁢
            a
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        x
                        2
                      
                    
                    +
                    
                      az
                      4
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      12
                      ⁢
                      
                        xy
                        2
                      
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                4
              
              ⁢
              
                y
                4
              
              ⁢
              
                z
                4
              
            
            )
          
          2
        
      
    
  
   






 


>S134 := S132{circumflex over ( )}2;


 






  
    
      
        
          
            S
            ⁢
            
              
            
            ⁢
            134
          
          :=
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              ⁢
              
                (
                
                  12
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            -
            
              (
              
                3
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
                )
              
              4
            
            )
          
          4
        
      
    
  
   






 


>s := T9*S15 − 8*S134;


 






  
    
      
        
          s
          :=
          
          ⁢
          
            (
            
              3
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            az
                            4
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
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>t := 8*S13*T4*yz;


 






  
    
      
        
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As shown by TABLE 2, the straight line algorithm systematically generates building blocks, or pieces of the equations presented in TABLE 1. For purposes of exemplary illustration, such building blocks are represented with respective portions of “other data” 118 For example, the definition “xy2:=x*y2” sets xy2 equal to x*y2. Notice that later in the sequence, the term T2 leverages the definition of xy2 as T2:=12*xy2−T12, without having to recalculate x*y2, etc. In this manner, the number of field calculations needed to compute (r, s, t) of 8P of a point on an elliptic curve 114 is reduced to twenty six (26). That is, signing/encrypting module 110 generates 12 squares and implements 14 multiplications for a total of 26 field multiplications to determine coordinates (r, s, t) of 8P (116) for point P on an elliptic curve 114. This is a significant reduction in the number of field multiplications represented by existing techniques.
Signing/encrypting module 110 cryptographically processes original data 112 using any public key-based cryptographic algorithm that has been modified to implement elliptic curve octupling exponentiation, to generate cryptographically processed data. Cryptographically processed data is shown as a respective portion of “other data” 118. Although such cryptographic processing has been described with respect to RSA, Diffie-Hellman key exchange, and ElGamal encryption, signing/encrypting module 110 can employ other algorithms modified to utilize elliptic curve octupling exponentiation, to cryptographically process original data. In other words, the particular public key-based cryptographic algorithm implemented by signing/encrypting module 110, independent of the algorithm for calculating 8P of points on an elliptic curve 114 as shown above in TABLES 1 and 2, is arbitrary because it is a function of the particular cryptographic algorithm selected for implementation.
For example, respective implementations of signing/encrypting module 110 implement one or more of the following cryptographic protocols: identity-based cryptography (e.g., plain, blind, proxy, ring, undeniable, etc.), encryption protocols (e.g., authenticated, broadcast, encryption with keyword search, etc.), batch signatures, key agreement (plain, authenticated, group, etc.), trust authorities and public key certification, hierarchical cryptosystems, threshold cryptosystems and signatures, chameleon hash and signatures, authentication, applications and systems, access control, key agreement, non-interactive key distribution, credentials (e.g., anonymous, hidden, self-blindable, etc.), secret handshakes, provably secure signatures, short signatures, aggregate, ring, and verifiably encrypted signatures, blind and partially blind signatures, proxy signatures, undeniable signatures, sign-cryption, multi-signatures and threshold signatures, limited-verifier and designated-verifier signatures, threshold cryptosystems, hierarchical and role-based cryptosystems, chameleon hash and signatures, verifiable random functions, strongly insulated encryption, intrusion-resilient encryption, certificate-less PKC, al, traitor tracing, and/or so on.
In one implementation, device 102 communicates cryptographically processed data to device 104 for verifying/decrypting. For purposes of exemplary illustration, cryptographically processed data received by device 104 is shown as signed/encrypted data 120. Computing device 104 includes verifying/decrypting module 122 to verify or decrypt the received cryptographically processed data. In one scenario, verifying/decrypting module 118 verifies as cryptographically processed data when the cryptographically processed data has been cryptographically signed. In this scenario, verifying/decrypting module 118 utilizes one or more implementations of the elliptic curve point octupling exponentiation described above to verify the cryptographically processed data. In another scenario, verifying/decrypting module 116 decrypts cryptographically processed data that has been encrypted. In this scenario, verifying/decrypting module 118 utilizes one or more implementations of the elliptic curve point octupling exponentiation described above to decrypt the cryptographically processed data.
Exemplary Elliptic Curve Point Octupling
FIG. 2 shows an exemplary procedure 200 to for elliptic curve point octupling for weighted projective coordinates. For purposes of discussion and exemplary illustration, the operations of procedure 200 are described with respect to components of FIG. 1. To this end, the left-most digit of a component reference number identifies the particular figure in which the component first appears. At block 202, signing/encrypting module 110 (FIG. 1) identifies a point P of an elliptic curve 114 over a finite field Fq. Point P has weighted projective coordinates of (x, y, z). At block 202, signing/encrypting module 110 computes 8P of P independent of repeated doubling operations. 8P has coordinates (r, s, t). In one implementation, 8P is computed using a direct elliptic curve point octupling algorithm, such as shown above in TABLE 1. In another implementation, 8P is computed using a straight-line elliptic curve point octupling algorithm with 26 field multiplications, such as shown above in TABLE 2. At block 206, results of the elliptic curve point octupling algorithm are utilized to encrypt or cryptographically sign original data, or decrypt or verify cryptographically processed data.
An Exemplary Operating Environment
FIG. 3 illustrates an example of a suitable computing environment in which elliptic curve octupling for weighted projective coordinates may be fully or partially implemented. Exemplary computing environment 300 is only one example of a suitable computing environment for the exemplary system of FIG. 1 and exemplary operations of FIG. 2, and is not intended to suggest any limitation as to the scope of use or functionality of systems and methods the described herein. Neither should computing environment 300 be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in computing environment 300.
The methods and systems described herein are operational with numerous other general purpose or special purpose computing system, environments or configurations. Examples of well-known computing systems, environments, and/or configurations that may be suitable for use include, but are not limited to, personal computers, server computers, multiprocessor systems, microprocessor-based systems, network PCs, minicomputers, mainframe computers, distributed computing environments that include any of the above systems or devices, and so on. Compact or subset versions of the framework may also be implemented in clients of limited resources, such as handheld computers, or other computing devices. The invention is practiced in a distributed computing environment where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.
With reference to FIG. 3, an exemplary system elliptic curve octupling for weighted projective coordinates includes a general purpose computing device in the form of a computer 310 implementing, for example, system 100 of FIG. 1. The following described aspects of computer 310 are exemplary implementations of computing devices 102 and/or 104 of FIG. 1. Components of computer 310 may include, but are not limited to, processing unit(s) 320, a system memory 330, and a system bus 321 that couples various system components including the system memory to the processing unit 320. The system bus 321 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. By way of example and not limitation, such architectures may include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus.
A computer 310 typically includes a variety of computer-readable media. Computer-readable media can be any available media that can be accessed by computer 310 and includes both volatile and nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer-readable media may comprise computer storage media and communication media. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by computer 310.
Communication media typically embodies computer-readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism, and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example and not limitation, communication media includes wired media such as a wired network or a direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of the any of the above should also be included within the scope of computer-readable media.
System memory 330 includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM) 331 and random access memory (RAM) 332. A basic input/output system 333 (BIOS), containing the basic routines that help to transfer information between elements within computer 310, such as during start-up, is typically stored in ROM 331. RAM 332 typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit 320. By way of example and not limitation, FIG. 3 illustrates operating system 334, application programs 335, other program modules 336, and program data 337.
The computer 310 may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only, FIG. 3 illustrates a hard disk drive 341 that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive 351 that reads from or writes to a removable, nonvolatile magnetic disk 352, and an optical disk drive 355 that reads from or writes to a removable, nonvolatile optical disk 356 such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive 341 is typically connected to the system bus 321 through a non-removable memory interface such as interface 340, and magnetic disk drive 351 and optical disk drive 355 are typically connected to the system bus 321 by a removable memory interface, such as interface 350.
The drives and their associated computer storage media discussed above and illustrated in FIG. 3, provide storage of computer-readable instructions, data structures, program modules and other data for the computer 310. In FIG. 3, for example, hard disk drive 341 is illustrated as storing operating system 344, application programs 345, other program modules 346, and program data 347. Note that these components can either be the same as or different from operating system 334, application programs 335, other program modules 336, and program data 337. Application programs 335 includes, for example program modules of computing devices 102 or 104 of FIG. 1. Program data 337 includes, for example, program data of computing devices 102 or 104 of FIG. 1. Operating system 344, application programs 345, other program modules 346, and program data 347 are given different numbers here to illustrate that they are at least different copies.
A user may enter commands and information into the computer 310 through input devices such as a keyboard 362 and pointing device 361, commonly referred to as a mouse, trackball or touch pad. Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit 320 through a user input interface 360 that is coupled to the system bus 321, but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB).
A monitor 391 or other type of display device is also connected to the system bus 321 via an interface, such as a video interface 390. In addition to the monitor, computers may also include other peripheral output devices such as printer 396 and audio device(s) 397, which may be connected through an output peripheral interface 395.
The computer 310 operates in a networked environment using logical connections to one or more remote computers, such as a remote computer 380. In one implementation, remote computer 380 represents computing device 102 or networked computer 104 of FIG. 1. The remote computer 380 may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and as a function of its particular implementation, may include many or all of the elements described above relative to the computer 310, although only a memory storage device 381 has been illustrated in FIG. 3. The logical connections depicted in FIG. 3 include a local area network (LAN) 381 and a wide area network (WAN) 373, but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.
When used in a LAN networking environment, the computer 310 is connected to the LAN 371 through a network interface or adapter 370. When used in a WAN networking environment, the computer 310 typically includes a modem 372 or other means for establishing communications over the WAN 373, such as the Internet. The modem 372, which may be internal or external, may be connected to the system bus 321 via the user input interface 360, or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer 310, or portions thereof, may be stored in the remote memory storage device. By way of example and not limitation, FIG. 3 illustrates remote application programs 385 as residing on memory device 381. The network connections shown are exemplary and other means of establishing a communications link between the computers may be used.
Conclusion
Although the systems and methods for elliptic curve octupling for weighted projective coordinates have been described in language specific to structural features and/or methodological operations or actions, it is understood that the implementations defined in the appended claims are not necessarily limited to the specific features or actions described. For example, although signing/encryption module 110 (FIG. 1) and verifying/decrypting module 122 (FIG. 1) are shown on different respective computing devices (i.e., devices 102 and 104), in another implementation, logic associated with these program modules can be implemented on a single computing device 102. Accordingly, the specific features and operations of system 100 are disclosed as exemplary forms of implementing the claimed subject matter.

## Claims

1. A method implemented by a single instruction multiple data (SIMD) computing system, the method comprising:
identifying by the SIMD computing system, coordinates x, y, and z of a weighted projective point P on an elliptic curve;
computing by the SIMD, coordinates r, s, and t for 8P of P independent of repeated doubling operations;
storing the coordinates r, s, and t to a computer readable storage medium associated with the computing system for subsequent use by the SIMD computing system;
wherein computing 8P with a straight-line elliptic curve point octupling algorithm results in 26 field multiplications to calculate 8P;
receiving original data;
determining 8P for respective points on the elliptic curve with a direct or an inline direct elliptic curve point octupling algorithm; and
encrypting or signing the original data based upon a result of the direct or the inline direct elliptic curve point octupling algorithm.

2. A method as recited in claim 1, wherein the straight-line elliptic curve point octupling algorithm comprises the following, wherein the symbol := represents a define operation:


x2:=x^2;



y2:=y^2;



y4:=y2^2;



xy2:=x*y2;



yz:=y*z;



z2:=z^2;



z4:=z2^2;



az4:=a*z4;



ay4z4:=az4*y4;



T1:=3*x2+az4;



T12:=T1^2;



T2:=12*xy2−T12;



T3:=T12−8*xy2;



T32:=T3^2;



T4:=T1*T2−8*y4;



T42:=T4^2;



T44:=T42^2;



T5:=3*T32+16*ay4z4;



T52:=T5^2;



T6:=T44*256*ay4z4;



T7:=T3*T42;



T8:=T52−8*T7;



T82:=3*T8^2;



T9:=T82+T6;



T92:=(T9)^2;



S10:=T52−8*T7;



S11:=12*T7−T52;



S12:=T5*S11;



S13:=S12−8*T44;



S132:=S13^2;



S14:=S10*S132;



r:=T92−8*S14;



S15:=12*S14−T92;



S134:=S132^2;



s:=T9*S15−8*S134; and



t:=8*S13*T4*yz.

3. A method as recited in claim 1, wherein computing 8P further comprises computing 8P using a direct elliptic curve point octupling algorithm.

4. A method as recited in claim 3, wherein symbol := represents a define
operation, and wherein the direct elliptic curve point octupling algorithm comprises:




  
    
      
        
          
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                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                  )
                
                2
              
              -
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                2
              
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      x
                      2
                    
                  
                  +
                  
                    az
                    4
                  
                
                )
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                4
              
              )
            
            2
          
          -
          
            (
            
              3
              ⁢
              
                (
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                  
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              16
              ⁢
              
                ay
                4
              
              ⁢
              
                z
                4
              
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                          ⁢
                          
                            8
                            ⁢
                            
                              y
                              4
                            
                          
                          )
                        
                        2
                      
                      )
                    
                    2
                  
                  +
                  
                    256
                    ⁢
                    
                      
                        a
                        ⁡
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    12
                                    ⁢
                                    
                                      xy
                                      2
                                    
                                  
                                  -
                                  
                                    
                                      (
                                      
                                        
                                          3
                                          ⁢
                                          
                                            x
                                            2
                                          
                                        
                                        +
                                        
                                          az
                                          4
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                                )
                              
                            
                            -
                            
                              8
                              ⁢
                              
                                y
                                4
                              
                            
                          
                          )
                        
                      
                      4
                    
                    ⁢
                    
                      y
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
        
      
    
    
      
        
          
          ⁢
          
            8
            ⁢
            
              (
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    12
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                        
                          8
                          ⁢
                          
                            xy
                            2
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
            -
            
              (
              
                3
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                                          
                  ⁢
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                  )
                
                4
              
              )
            
            4
          
          ;
          and
        
      
    
  




  
    
      
        
          
            t
            ⁢
            
              :
            
          
          =
          
          ⁢
          
            8
            ⁢
            
              (
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    12
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                      
                      ⁢
                      
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  xy
                  2
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
          
          -
          
            (
            
              3
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    2
                  
                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                            
                      ⁢
                      
                        8
                        ⁢
                        
                          xy
                          2
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                                  
                ⁢
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
                )
              
              4
            
            )
          
          ⁢
          
            (
            
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        x
                        2
                      
                    
                    +
                    
                      az
                      4
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      12
                      ⁢
                      
                        xy
                        2
                      
                    
                    -
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      2
                    
                  
                  )
                
              
              -
              
                8
                ⁢
                
                  y
                  4
                
              
            
            )
          
          ⁢
          
            yz
            .

5. A method as recited in claim 1, the method further comprising:
receiving original data;
determining 8P for respective points on the elliptic curve with a direct or inline direct elliptic curve point octupling algorithm; and
decrypting or verifying cryptographically processed original data based on results of the direct or inline direct elliptic curve point octupling algorithm.

6. A computer-readable storage medium encoded with instructions that, when executed by a processor of a device, cause the device to perform acts comprising:
identifying coordinates x, y, z of a weighted projective point P on an elliptic curve;
determining coordinates (r, s, t) associated with 8P of P independent of repeated doubling operations storing the coordinates (r, s, t) to a computer readable storage medium associated with the computing system for subsequent use by the computing system;
computing of 8P with a straight-line elliptic curve point octupling algorithm resulting in 26 field multiplications to calculate 8P;
determining 8P for respective points on the elliptic curve with a direct or an inline direct elliptic curve point octupling algorithm; and
decrypting or verifying cryptographically processed data based on results of the direct or the inline direct elliptic curve point octupling algorithm.

7. A computer-readable storage medium as recited in claim 6, wherein symbol := represents a define operation, and wherein the straight-line elliptic curve point octupling algorithm that determines r, s, t comprises:


x2:=x^2;



y2:=y^2;



y4:=y2^2;



xy2:=x*y2;



yz:=y*z;



z2:=z^2;



z4:=z2^2;



az4:=a*z4;



ay4z4:=az4*y4;



T1:=3*x2+az4;



T12:=T1^2;



T2:=12*xy231 T12;



T3:=T12−8*xy2;



T32:=T3^2;



T4:=T1*T2−8*y4;



T42:=T4^2;



T44:=T42^2;



T5:=3*T32+16*ay4z4;



T52:=T5^2;



T6:=T44*256*ay4z4;



T7:=T3*T42;



T8:=T52−8*T7;



T82:=3*T8^2;



T9:=T82+T6;



T92:=(T9)^2;



S10:=T52−8*T7;



S11:=12*T7−T52;



S12:=T5*S11;



S13:=S12−8*T44;



S132:=S13^2;



S14:=S10*S132;



r:=T92−8*S14;



S15:=12*S14−T92;



S134:=S132^2;



s:=T9*S15−8*S134; and



t:=8*S13*T4*yz.

8. A computer-readable storage medium as recited in claim 6, the acts further comprising computing 8P using a direct elliptic curve point octupling algorithm.

9. A computer-readable storage medium as recited in claim 8, wherein symbol := represents a define operation, and wherein the direct elliptic curve point octupling algorithm that determines r, s, t comprises:



  
    
      
        
          
            r
            ⁢
            
              :
            
          
          =
          
          ⁢
          
            (
            
              3
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                
                ⁢
                
                  
                
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                  )
                
                2
              
              )
            
            2
          
          +
          
            256
            ⁢
            
              a
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                    )
                  
                  4
                
                ⁢
                
                  y
                  4
                
                ⁢
                
                  z
                  4
                
              
              )
            
            2
          
          -
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                    +
                                    
                                      az
                                      4
                                    
                                  
                                  )
                                
                                2
                              
                              -
                              
                                8
                                ⁢
                                
                                  xy
                                  2
                                
                              
                            
                            )
                          
                          2
                        
                      
                      +
                      
                        16
                        ⁢
                        
                          ay
                          4
                        
                        ⁢
                        
                          z
                          4
                        
                      
                    
                    )
                  
                  2
                
                -
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
                -
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  8
                  ⁢
                  
                    y
                    4
                  
                
                )
              
              2
            
            )
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              ⁢
              
                (
                
                  12
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  xy
                  2
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
          
          -
        
      
    
    
      
        
          
            
            ⁢
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        x
                        2
                      
                    
                    +
                    
                      az
                      4
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      12
                      ⁢
                      
                        xy
                        2
                      
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                               
                      ⁢
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                4
              
              )
            
            2
          
          ;
        
      
    
  




  
    
      
        
          
            s
            ⁢
            
              :
            
          
          =
          
          ⁢
          
            (
            
              3
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      (
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                        
                        ⁢
                        
                          az
                          4
                        
                        )
                      
                      2
                    
                    -
                    
                      8
                      ⁢
                      
                        xy
                        2
                      
                    
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              12
                              ⁢
                              
                                xy
                                2
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        8
                        ⁢
                        
                          y
                          4
                        
                      
                    
                    )
                  
                  2
                
              
              )
            
            2
          
          +
        
      
    
    
      
        
          
          ⁢
          
            256
            ⁢
            
              
                a
                ⁡
                
                  (
                  
                    
                      
                        (
                        
                          
                            3
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            az
                            4
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            12
                            ⁢
                            
                              xy
                              2
                            
                          
                          -
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
              
              4
            
            ⁢
            
              y
              4
            
            ⁢
            
              z
              4
            
          
          )
        
      
    
    
      
        
          
          ⁢
          
            (
            
              12
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        
                                          x
                                          2
                                        
                                      
                                      +
                                      
                                        az
                                        4
                                      
                                    
                                    )
                                  
                                  2
                                
                                -
                                
                                  8
                                  ⁢
                                  
                                    xy
                                    2
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        +
                        
                          16
                          ⁢
                          
                            ay
                            4
                          
                          ⁢
                          
                            z
                            4
                          
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  xy
                  2
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
          
          )
        
      
    
    
      
        
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              ⁢
              
                (
                
                  12
                  ⁢
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
                ⁢
                
                  xy
                  2
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              2
            
          
          -
        
      
    
    
      
        
          
            
            ⁢
            
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      x
                      2
                    
                  
                  +
                  
                    az
                    4
                  
                
                )
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                4
              
              )
            
            2
          
          -
          
            (
            
              3
              ⁢
              
                (
                
                  (
                  
                    3
                    ⁢
                    
                      (
                      
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                        -
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      8
                      ⁢
                      
                        xy
                        2
                      
                    
                    )
                  
                  2
                
                +
                
                  16
                  ⁢
                  
                    ay
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
              
              )
            
            2
          
          -
          
            8
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                -
                
                  8
                  ⁢
                  
                    xy
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      x
                      2
                    
                  
                  +
                  
                    az
                    4
                  
                
                )
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
              )
            
            2
          
          +
          
            256
            ⁢
            
              a
              (
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        x
                        2
                      
                    
                    +
                    
                      az
                      4
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      12
                      ⁢
                      
                        xy
                        2
                      
                    
                    -
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                          ⁢
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          )
                        
                        -
                        
                          8
                          ⁢
                          
                            y
                            4
                          
                        
                      
                      )
                    
                    4
                  
                  ⁢
                  
                    y
                    4
                  
                  ⁢
                  
                    z
                    4
                  
                
                )
              
              2
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                
              
            
          
        
      
    
    
      
        
          
            
            ⁢
            
              16
              ⁢
              
                ay
                4
              
              ⁢
              
                z
                4
              
            
            )
          
          ⁢
          
            (
            
              12
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      xy
                      2
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        12
                        ⁢
                        
                          xy
                          2
                        
                      
                      -
                    
                  
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        2
                      
                      )
                    
                    -
                    
                      8
                      ⁢
                      
                        y
                        4
                      
                    
                  
                  )
                
                2
              
              -
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                2
              
            
            )
          
          -
        
      
    
    
      
        
          
            
                      
              ⁢
              
                8
                ⁢
                
                  
                    (
                    
                      
                        
                          (
                          
                            
                              3
                              ⁢
                              
                                x
                                2
                              
                            
                            +
                            
                              az
                              4
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              12
                              ⁢
                              
                                xy
                                2
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        8
                        ⁢
                        
                          y
                          4
                        
                      
                    
                    )
                  
                  4
                
              
              )
            
            4
          
          ;
          and
        
      
    
  




  
    
      
        
          
            t
            ⁢
            
              :
            
          
          =
          
          ⁢
          
            8
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  
                                    x
                                    2
                                  
                                
                                +
                                
                                  az
                                  4
                                
                              
                              )
                            
                            2
                          
                          -
                          
                            8
                            ⁢
                            
                              xy
                              2
                            
                          
                        
                        )
                      
                      2
                    
                  
                  +
                  
                    16
                    ⁢
                    
                      ay
                      4
                    
                    ⁢
                    
                      z
                      4
                    
                  
                
                )
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            (
            
              12
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    2
                  
                  -
                  
                    8
                    ⁢
                    
                      xy
                      2
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    
                      (
                      
                        
                          3
                          ⁢
                          
                            x
                            2
                          
                        
                        +
                        
                          az
                          4
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                  ⁢
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                  )
                
                2
              
              -
              
                
                  (
                  
                    
                      3
                      ⁢
                      
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    3
                                    ⁢
                                    
                                      x
                                      2
                                    
                                  
                                  +
                                  
                                    az
                                    4
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              8
                              ⁢
                              
                                xy
                                2
                              
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
                        4
                      
                      ⁢
                      
                        z
                        4
                      
                    
                  
                  )
                
                2
              
            
            )
          
          -
          
            8
            ⁢
            
              (
              
                (
                
                  
                    3
                    ⁢
                    
                      x
                      2
                    
                  
                  +
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        az
                        4
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          12
                          ⁢
                          
                            xy
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  x
                                  2
                                
                              
                              +
                              
                                az
                                4
                              
                            
                            )
                          
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    8
                    ⁢
                    
                      y
                      4
                    
                  
                
                )
              
              4
            
            )
          
          ⁢
          
            (
            
              
                (
                
                  
                    3
                    ⁢
                    
                      x
                      2
                    
                  
                  +
                  
                    az
                    4
                  
                
                )
              
              ⁢
              
                (
                
                  
                    12
                    ⁢
                    
                      xy
                      2
                    
                  
                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                ⁢
                
                  
                    (
                    
                      
                        3
                        ⁢
                        
                          x
                          2
                        
                      
                      +
                      
                        az
                        4
                      
                    
                    )
                  
                  2
                
                )
              
              -
              
                8
                ⁢
                
                  y
                  4
                
              
            
            )
          
          ⁢
          
            yz
            .

10. The computer-readable storage medium recited in claim 6, the acts further comprising:
determining 8P for respective points on the elliptic curve with a direct or inline direct elliptic curve point octupling algorithm; and
encrypting or signing original data based on results of the direct or inline direct elliptic curve point octupling algorithm.

11. A computing device comprising:
a processor configured to support at least in part Single Input Multiple Data processing; and
a memory coupled to the processor, the memory comprising instructions executable by the processor for, that when executed by a processor of the computing device, cause the computing device to perform acts comprising:
identifying coordinates x, y, z corresponding to a weighted projective point P on an elliptic curve; and
computing coordinates (r, s, t) corresponding to 8P of P independent of repeated doubling operations;
storing the coordinates (r, s, t ) to a computer readable storage medium associated with the computing system for subsequent use by the computing system;
computing 8P with a straight-line elliptic curve point octupling algorithm resulting in 26 field multiplications to calculate 8P;
determining 8P for respective points on the elliptic curve with a direct or inline direct elliptic curve point octupling algorithm; and
encrypting or signing original data based on results of the direct or the inline direct elliptic curve point octupling algorithm; or
decrypting or verifying cryptographically processed data based on results of the direct or the inline direct elliptic curve point octupling algorithm.

12. A computing device as recited in claim 11, wherein the straight-line elliptic curve point octupling algorithm determines r, s, t as follows where the symbol:= represents a define operation:


x2:=x^2;



y2:=y^2;



y4:=y2^2;



xy2:=x*y2;



yz:=y*z;



z2:=z^2;



z4:=z2^2;



az4:=a*z4;



ay4z4:=az4*y4;



T1:=3*x2+az4;



T12:=T1^2;



T2:=12*xy2−T12;



T3:=T12−8*xy2;



T32:=T3^2;



T4:=T1*T2−8*y4;



T42:=T4^2;



T44:=T42^2;



T5:=3*T32+16*ay4z4;



T52:=T5^2;



T6:=T44*256*ay4z4;



T7:=T3*T42;



T8:=T52−8*T7;



T82:=3*T8^2;



T9:=T82+T6;



T92:=(T9)^2;



S10:=T52−8*T7;



S11:=12*T7−T52;



S12:=T5*S11;



S13:=S12−8*T44;



S132:=S13^2;



S14:=S10*S132;



r:=T92−8*S14;



S15:=12*S14−T92;



S134:=S132^2;



s:=T9*S15−8*S134; and



t:=8*S13*T4*yz.

13. A computing device as recited in claim 11, wherein the acts further comprise computing 8P using a direct elliptic curve point octupling algorithm.

14. A computing device as recited in claim 13, wherein Symbol := represents a define operation, and wherein determining r, s, t with the direct elliptic curve point octupling algorithm further comprises:



  
    
      
        
          
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                      ⁢
                      
                        
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                              3
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                              az
                              4
                            
                          
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                        2
                      
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                    -
                    
                      8
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                        4
                      
                    
                  
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                      3
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                          (
                          
                            
                              
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                                    3
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                                    az
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                              2
                            
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                              8
                              ⁢
                              
                                xy
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                          )
                        
                        2
                      
                    
                    +
                    
                      16
                      ⁢
                      
                        ay
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                      ⁢
                      
                        z
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                  )
                
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            )
          
          -
        
      
    
    
      
        
          
            
              
              ⁢
              
                8
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                    (
                    
                      
                        
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                              12
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                                    3
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                              2
                            
                          
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                      -
                      
                        8
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                  4
                
              
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            4
          
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          and
        
      
    
  




  
    
      
        
          
            t
            ⁢
            
              :
            
          
          =
          
          ⁢
          
            8
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                            8
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                )
              
            
          
        
      
    
    
      
        
          
          ⁢
          
            (
            
              12
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                      (
                      
                        
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                                3
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                          2
                        
                      
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                  -
                
              
            
          
        
      
    
    
      
        
          
            
              
                
                  
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                  -
                  
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            )
          
          ⁢
          
            yz
            .