Abstract:
A cryptographic system (CS) comprised of generators ( 502 ), ( 504 ), ( 510 ), an encryption device (ED), and a decryption device (DD). The generator ( 502 ) generates a data sequence (DS) including payload data. The generator ( 504 ) generates an encryption sequence (ES) including random numbers. The ED ( 506 ) is configured to perform a CGFC arithmetic process. As such, the ED is comprised of a mapping device (MD) and an encryptor. The MD is configured to map the DS and ES from Galois field GF[p k ] to Galois extension field GF[p k+1 ]. The encryptor is configured to generate an encrypted data sequence (EDS) by combining the DS and ES utilizing a Galois field multiplication operation in Galois extension field GF[p k+1 ]. The generator ( 510 ) is configured to generate a decryption sequence (DS). The DD ( 508 ) is configured to generate a decrypted data sequence by performing an inverse of the CGFC arithmetic process utilizing the EDS and DS.

Description:
BACKGROUND OF THE INVENTION 
     1. Statement of the Technical Field 
     The inventive arrangements relate to efficient implementations of Galois field multiplication in cryptographic systems. More particularly, the inventive arrangements relate to an efficient and invertible closed Galois field combination (CGFC) process for combining two or more input sequences in a cryptographic system. 
     2. Description of the Related Art 
     Sequence generators are used in applications ranging from signal processing, error correction codes, pseudo-random number generation and cryptographic systems. The underlying mechanism in each of these applications is a sequence that is finite in nature, with the smaller the repetition period of the sequence, the easier it is for an outsider to determine or predict the future values of the sequence. To increase the security of a sequence, most applications increase the length of the generating polynomial, use some type of nonlinear combination technique with multiple orthogonal sequences or otherwise increase the length of the sequence. 
     These techniques often result in significantly increasing implementation complexity, which translates into increased resource consumption and processing time. Also, for each sequence period, the generating algorithm is typically fixed and tied to a specific irreducible polynomial over a Galois field. It is also desirable to implement combination techniques such that a priori knowledge of one or more of the input sequences yields an efficient method to detect and correct sequence errors that may occur during storage or transmission. 
     Static examples of modification of a number sequence exist in communication systems. One such example is the Reed Solomon error correction code and derivatives thereof. The Reed Solomon error correction code computes an output value based on Galois extension field multiplications using a primitive element and a fixed irreducible polynomial. 
     In view of the forgoing, there remains a need for a method to combine a data stream with a random number sequence that is more computationally efficient than conventional methods. There also remains a need for a method to efficiently increase the repetition period of a random number sequence. There further remains a need for a method to combine a data stream with a random number sequence that is more resistant to errors than conventional methods. 
     SUMMARY OF THE INVENTION 
     The invention concerns a method for encrypting and decrypting a data sequence. The method involves generating a data sequence comprising payload data and an encryption sequence comprising random numbers. The random numbers can be pseudo-random numbers or pseudo-chaotic random numbers. The method also involves mapping the data sequence and the encryption sequence from a Galois field GF[p k ] to a Galois extension field GF[p k+1 ]. More particularly, the data sequence is mapped to the Galois extension field GF[p k+1 ] by performing an Affine transformation of the data sequence. Similarly, the encryption sequence is mapped to the Galois extension field GF[p k+1 ] by performing an Affine transformation of the encryption sequence. 
     The method further involves generating an encrypted data sequence by performing a CGFC arithmetic process. The CGFC arithmetic process generally involves combining the data sequence with the encryption sequence utilizing a Galois field multiplication operation in the Galois extension field GF[p k+1 ]. More particularly, the CGFC arithmetic process involves combining the data sequence and the encryption sequence by selectively performing a plurality of Galois field multiplication operations using a distinct subset of digits from each of sequences. 
     The method also includes the step of generating a decryption sequence expressed in a Galois field GF[p k ]. Notably, the decryption sequence is a mathematical inversion of the encryption sequence. This inversion may be created through a calculation of the Extended Euclid Algorithm or similar process. The method further includes the step of generating a decrypted data sequence by performing an inverse of the CGFC arithmetic process utilizing the encrypted data sequence and the decryption sequence. 
     The invention also concerns a closed Galois field cryptographic (CGFC) system. The CGFC system is comprised of a first generator, a second generator, and an encryption device. The first generator is configured to generate a data sequence comprising payload data. The second generator is configured to generate an encryption sequence comprising pseudo random numbers or pseudo-chaotic random numbers. The encryption device is coupled to the first and second generators. The encryption device is configured to perform a CGFC arithmetic process. In this regard, it should be understood that the encryption device is comprised of a mapping device and an encryptor. The mapping device is configured to map the data sequence and the encryption sequence from a Galois field GF[p k ] to a Galois extension field GF[p k+1 ]. The encryptor is configured to generate an encrypted data sequence by combining the data sequence with the encryption sequence utilizing a Galois field multiplication operation in the Galois extension field GF[p k+1 ]. 
     The CGFC system is further comprised of a third generator and a decryption device. The third generator is configured to generate a decryption sequence expressed in a Galois field GF[p k ]. The decryption sequence is a mathematical inversion of the encryption sequence. The inversion may be created through a calculation of the Extended Euclid Algorithm or similar process. The decryption device is coupled to the third generator. The decryption device is configured to generate a decrypted data sequence by performing an inverse of the CGFC arithmetic process utilizing the encrypted data sequence and the decryption sequence. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments will be described with reference to the following drawing figures, in which like numerals represent like items throughout the figures, and in which: 
         FIG. 1  is a flow chart of a closed Galois field combination (CGFC) method that is useful for understanding the invention. 
         FIG. 2  is a diagram of a CGFC arithmetic process that is useful for understanding the invention. 
         FIG. 3  is a diagram of a CGFC arithmetic process for encoding data that is useful for understanding the invention. 
         FIG. 4  is a diagram of a CGFC arithmetic process for decoding data that is useful for understanding the invention. 
         FIG. 5  is a block diagram of a closed Galois field cryptographic system that is useful for understanding the invention. 
     
    
    
     DETAILED DESCRIPTION 
     The invention will now be described more fully hereinafter with reference to accompanying drawings, in which illustrative embodiments of the invention are shown. This invention, may however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. For example, the present invention can be embodied as a method, a data processing system or a computer program product. Accordingly, the present invention can take the form as an entirely hardware embodiment, an entirely software embodiment or a hardware/software embodiment. 
     In many communications system applications, it is desirable to have an efficient means of combining two or more number sequences. Such communications system applications include a random number generation application, an error coding application, an encryption application, a modulation application and the like. In encryption applications, the combining process is performed for masking an input data stream by some invertible operation using a number stream produced to deny determination of its generating characteristics. The term “invertible” as used herein refers to the existence of a method for undoing the masking operation and obtaining the original data stream. It is also desirable to have an efficient means of combining two (2) independent randomly generated number streams. This combining process can provide a non-linear random number stream. Non-linearly generated random number streams make it very difficult to identify generating structures. As such, some embodiments of the present invention provide a method and apparatus for efficiently combining two (2) inputs via multiplication in a Galois extension field. Such an embodiment satisfies efficiency in both time and hardware design. Such an embodiment further provides a simple nearly symmetric inverse operator. 
     Other embodiments of the present invention provide a method for efficiently masking data in digital hardware while making unintended decoding difficult. In this regard, it should be appreciated that such a method can be used in a variety of communications system applications. Such a method can be combined with other nonlinear or mixed-radix processing techniques to induce chosen statistical characteristics. 
     Referring now to  FIG. 1 , there is provided a flow chart of a closed Galois field combination (CGFC) method  100  for combining two (2) data inputs. Prior to describing the CGFC method  100 , a brief discussion of Galois fields is provided to assist a reader in understanding the present invention. It should be understood that the following discussion is a general discussion of the fundamental properties for a Galois field. For simplicity purposes, a complete or precise definition of a Galois field is not provided. 
     As should be understood, a Galois field GF[p k ] is a cyclic finite structure created on a discrete collection of elements {0, 1, 2, . . . , p k −1}. For example, a Galois field GF[2 3 ] is a cyclic finite structure containing elements {0, 1, 2, 3, 4, 5, 6, 7}. The Galois field GF[p k ] has certain analytical properties that allow arithmetic operations to be performed within the Galois field GF[p k ]. Such arithmetic operations include an addition operation, a subtraction operation, and a multiplication operation. The Galois field GF[p k ] is often referred to as a closed structure since results of said arithmetic operations equal another element in the same Galois field. The arithmetic operations may be viewed as modular arithmetic. For example, if the Galois field GF[p k ] is selected as a Galois field GF[7 1 ], then the modular arithmetic for selected elements can be defined as follows: 2−4=−1·7+5=5 mod 7; 5+6=11=1·7+4=4 mod 7; and 5·6=30=4·7+2=2 mod 7. 
     An arithmetic operation of division in the Galois field GF[p k ] requires a definition of unique multiplicative inverses. More generally, division by a number in Galois field GF[p k ] is more properly equivalent to a multiplication by said number&#39;s inverse. This notional division operation is only possible if a multiplicative inverse exists. For example, if the Galois field GF[p k ] is selected as a Galois field GF[11], then the modular arithmetic for selected elements can be defined as follows: 4·3=12=1 mod 11. In such a scenario, four (4) is the multiplicative inverse of three (3) in Galois field GF[11]. Similarly, three (3) is the multiplicative inverse of four (4) in Galois field GF[11]. 
     As should be understood, the notional arithmetic operation of division in the Galois field GF[p k ] is only valid for elements that result in a nonzero value when reduced modulo p. For example, if the Galois field GF[p k ] is selected as a Galois field GF[2 3 ], then the modular arithmetic for elements zero (0) and X is defined as follows: 0·X=0≡0 mod 8. X is any element of Galois field GF[2 3 ]. In such a scenario, there does not exist a multiplicative inverse of zero (0) within the Galois field GF[2 3 ]. As such, a multiplication of element zero (0) times another element of Galois field GF[2 3 ] can not be inverted to obtain one (1) of the two (2) elements multiplied together. Notably, there also does not exist a multiplicative inverse for the even numbered elements {0, 2, 4, 6} within the Galois field GF[2 3 ]. More generally, there does not exist a multiplicative inverse for any even numbered elements in a Galois field GF[2 k ]. Even more generally, in a Galois field GF[p k ] it is true that only numbers that are not divisible by the prime characteristic p will have unique inverses. 
     Referring again to  FIG. 1 , the present invention provides a CGFC method  100  for bypassing the above-described shortcomings of Galois field multiplication. The present invention also provides a CGFC method  100  which ensures the existence of a multiplicative inverse and subsequently an invertible cryptographic operation. Generally, the method involves performing an Affine transformation on any plurality of k-bit p-adic data streams. The resulting data streams can be efficiently and invertibly combined by a means of Galois field multiplication in a Galois field GF[p k+1 ]. The Affine transformation yields additional degrees of freedom which expands the variability of output values and thereby makes reverse engineering of the k-bit data streams more difficult. As will be appreciated by those skilled in the art, Affine transformations generally involve a linear transformation along with a nonzero additive constant. In an exemplary embodiment of binary systems (p=2), the affine transformation is a linear multiplication by two (2) along with an addition of one (1) to the end of a digital word or removing a one (1) from the end of a digital word. 
     The following example is provided in order to further illustrate an Affine transformation. The scope of the present invention, however, is not to be considered limited in any way thereby. 
     EXAMPLE 1 
     Let p equal two (2) and k equal three (3). A simple Affine transformation from a Galois field GF[8] to a Galois field GF[16] is performed by appending a one (1) behind a least significant bit (LSB) of each three (3) bit representation of numbers in the Galois field GF[8]. The Affine transformation yields 0→1, 1→3, 2→5, 3→7, 4→9, 5→11, 6→13, and 7→15. Each number in the extended Galois field has a unique multiplicative inverse. Explicitly, 1 −1  mod 16≡1,3 −1  mod 16≡11, 5 −1  mod 16=13, 7 −1  mod 16≡7, 9 −1  mod 16≡9, 1 −1  mod 16≡3, 13 −1  mod 16≡5, and 15 −1  mod 16≡15. The Affine transformation from the Galois field GF[16] to the Galois field GF[8] is performed as (y−1)/2, where y is the result of the Galois field GF[16] multiplication. The Affine transformation from the Galois field GF[2 k ] to the Galois field GF[2 k+1 ] always yields an odd value. The result of the Galois field GF[2 k+1 ] multiplication y is always odd. y−1 is always even. The Affine transformation from the Galois field GF[2 k+1 ] to the Galois field GF[2 k ] is simply accomplished by dropping the least significant bit (LSB) from y and keeping the k most significant bits. 
     Referring again to  FIG. 1 , the CGFC method  100  begins at step  102  and continues with step  104 . In step  104 , two data inputs X 1  and X 2  are received at a closed Galois field combination system. The closed Galois field combination system can be, but is not limited to, a closed Galois field encryption system such as that described below in relation to  FIG. 5 . Thereafter, step  106  is performed where a dimension of a Galois field GF[p k] is increased by a value p. After step  106 , step  108  is performed where a nonzero element from a Galois field GF[p] is added to each of the data inputs X 1  and X 2 . It should be noted that steps  106  and  108  collectively provide an Affine transformation between an original number range to a number range of a Galois extension field. The Affine transformation can be selected as a constant function or a function of time. Affine transformations are well known to persons skilled in the art, and therefore will not be described in detail herein. However, it should be understood that an Affine transformation may be considered a bijective mapping. Bijective, injective, and surjective mappings are also well known to persons skilled in the art, and therefore will not be described herein. 
     In step  110 , the two data inputs are combined via a Galois field multiplication operation in a Galois extension field GF[p k+1 ]. The Galois field multiplication operation is performed to guarantee invertible reconstruction of the data inputs X 1  and X 2 . This invertible operation is desirable such that a first data input X 1  can be obtained via a pre-defined mathematical process and a priori knowledge of a second data input X 2 . 
     After step  110 , the CGFC method  100  continues with a step  112 . In step  112 , a known digit is subtracted from a result of said Galois field multiplication operation. In step  114 , a dimension of a Galois field GF[p k+1 ] is decreased by a value p. Collectively, steps  112 ,  114  perform an Affine transformation that is the inverse of the original Affine transformation performed in steps  106 ,  108 . Subsequently, step  116  is performed where the CGFC method  100  ends. 
     Referring again to step  110  of  FIG. 1 , a brief discussion of the rationale and mechanics of multiplication in a Galois field is provided to assist a reader in understanding the present invention. It should be noted that the scope of the present invention is not to be considered limited in any way thereby. 
     As should be understood, multiplication of two (2) numbers in a Galois field results in another number within the same Galois field. The multiplication process may not be invertible, even given knowledge of one of the original numbers since the other number may not have a multiplicative inverse. For example, the multiplication of the numbers two (2) and four (4) in a Galois field GF[8] yields a number zero (2×4=8≡0 modulo 8). If the output value of zero (0) and an input value is known, then the other input value can not be reconstructed since the numbers two (2) or four (4) do not have a multiplicative inverse in a Galois field GF[8]. More generally, the multiplication process is non-invertible for some values in a Galois field GF[8]. 
     The solution to guaranteed invertible multiplication is to restrict the numbers being multiplied to those which have multiplicative inverses in a given Galois field. This restriction places additional requirements on the random number generator to either produce acceptable numbers or else to map outputs to acceptable numbers. One method of implementing a structure such that all elements have unique inverses is to increase the dimension of the original Galois field, mapping the initial number set to a distinct invertible subset of the extended Galois field. The Galois field having an increased dimension is referred to herein as a Galois extension field. It can be shown that there exists an injective mapping between the elements in the original Galois field and those in the Galois extension field that possess multiplicative inverses. This invertability allows a unique multiplication of numbers in the original Galois field through the injective mapping, implementing an effective bijective mapping. 
     EXAMPLE 2 
     As an example, compare the direct (non-invertible) multiplication of elements in a Galois field GF[3 2 ] with the (invertible) multiplications of the same elements when projected upon a subset of the Galois extension field GF[3 3 ]. The elements in the Galois field GF[3 2 ] are provided in the following Table 1. The elements in the Galois field GF[3 3 ] are provided in the following Table 2. It should be noted that the numbers in Tables 1 and 2 are represented in 2-digit 3-adic representation. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 BASIC TIMES TABLE FOR 3 2 , USING DIRECT 
               
               
                 MULTIPLICATION 
               
             
          
           
               
                   
                 00 
                 01 
                 02 
                 10 
                 11 
                 12 
                 20 
                 21 
                 22 
               
               
                   
                   
               
             
          
           
               
                 00 
                 00 
                 00 
                 00 
                 00 
                 00 
                 00 
                 00 
                 00 
                 00 
               
               
                 01 
                 00 
                 01 
                 02 
                 10 
                 11 
                 12 
                 20 
                 21 
                 22 
               
               
                 02 
                 00 
                 02 
                 11 
                 20 
                 22 
                 01 
                 10 
                 12 
                 21 
               
               
                 10 
                 00 
                 10 
                 20 
                 00 
                 10 
                 20 
                 00 
                 10 
                 20 
               
               
                 11 
                 00 
                 11 
                 22 
                 10 
                 21 
                 02 
                 20 
                 01 
                 12 
               
               
                 12 
                 00 
                 12 
                 01 
                 20 
                 02 
                 21 
                 10 
                 22 
                 11 
               
               
                 20 
                 00 
                 20 
                 10 
                 00 
                 20 
                 10 
                 00 
                 20 
                 10 
               
               
                 21 
                 00 
                 21 
                 12 
                 10 
                 01 
                 22 
                 20 
                 11 
                 02 
               
               
                 22 
                 00 
                 22 
                 21 
                 20 
                 12 
                 11 
                 10 
                 02 
                 01 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 TIMES TABLE FOR 3 2  USING GALOIS EXTENSION 
               
               
                 FIELD WITH NONZERO ELEMENTS 1 
               
             
          
           
               
                   
                 00 
                 01 
                 02 
                 10 
                 11 
                 12 
                 20 
                 21 
                 22 
               
               
                   
                   
               
             
          
           
               
                 00 
                 00 
                 01 
                 02 
                 10 
                 11 
                 12 
                 20 
                 21 
                 22 
               
               
                 01 
                 01 
                 12 
                 00 
                 11 
                 22 
                 10 
                 21 
                 02 
                 20 
               
               
                 02 
                 02 
                 00 
                 21 
                 12 
                 10 
                 01 
                 22 
                 20 
                 11 
               
               
                 10 
                 10 
                 11 
                 12 
                 20 
                 21 
                 22 
                 00 
                 01 
                 02 
               
               
                 11 
                 11 
                 22 
                 10 
                 21 
                 02 
                 20 
                 01 
                 12 
                 00 
               
               
                 12 
                 12 
                 10 
                 01 
                 22 
                 20 
                 11 
                 02 
                 00 
                 21 
               
               
                 20 
                 20 
                 21 
                 22 
                 00 
                 01 
                 02 
                 10 
                 11 
                 12 
               
               
                 21 
                 21 
                 02 
                 20 
                 01 
                 12 
                 00 
                 11 
                 22 
                 10 
               
               
                 22 
                 22 
                 20 
                 11 
                 02 
                 00 
                 21 
                 12 
                 10 
                 01 
               
               
                   
               
             
          
         
       
     
     It should be noted that the multiplication results shown in the first row and column of Table 1 are always zero (00). This result implies that the values are non-invertible. Further, the statistical distribution of the output sequence, given two uniformly random input sequences, will be skewed such that 21/81 outputs are equal to zero (00) thereby eliminating the desired uniform distributions. Table 2 displays a perfectly uniformly distributed set of outputs whenever the inputs are also uniformly distributed. Given an input value and an output value, we can reconstruct the other input uniquely. This invertible property is most easily seen from the fact that each output value occurs only once in each row or column of Table 2. 
     It should be noted that Galois field multiplication is more computationally efficient than the equivalent multiplication in a weighted number system since the number of output values is restricted to a finite set. The phrase “weighted number system” as used herein refers to a number system other than a residue number system. 
     It should also be noted that Galois field multiplication in binary number systems such as a Galois field GF[2 k ] are useful in practice due to the structure of digital logic. Computational savings for Galois field multiplication in digital hardware as compared to conventional multiplications exceed fifty percent (50%). Algorithms for effecting the Galois field multiplication operation are well known to those skilled in the art, and therefore will not be described herein. In general, the multiplication result in a Galois field may be obtained more efficiently since only partial information of the intermediate arithmetic steps is required to determine the unique output value. 
     It should further be noted that combining two (2) numbers via Galois field multiplication results in the masking of the two (2) numbers whenever there is no knowledge of the same. This property is desirable in numerous encryption applications. This property is also comparable to combination techniques that combine two (2) numbers through a bit-wise XOR (exclusive or) logical operation except that each digit in the individual input sequence values has a larger effect on the other digits in the output value. The bit-wise XOR logical operation can be viewed as a special case of Galois extension field multiplication for a Galois field GF[2 1 ], where a number maps to its own multiplicative inverse in the Galois extension field GF[2 2 ] using Affine transformations f(x)=2x+1 and f −1 (x)=(x−1)/2. 
     Referring now to  FIG. 2 , the above described CGFC method  100  can be further defined by a CGFC arithmetic process  200 . The CGFC arithmetic process  200  begins with data inputs X 1  and X 2 . The data input X 1  can be payload data or a random number of a random number sequence. The data input X 2  can be a pseudo-random number of a pseudo-random number sequence or a pseudo-chaotic random number of a pseudo-chaotic random number sequence. The phrase “payload data” as used herein refers to data that is a numerical equivalent to a written symbol of a natural language, such as a letter or a punctuation mark. For example, if a number three (3) is a numerical equivalent to a letter C, then the data input X 1  has a value of three (3). The data input can also be the numerical equivalent for data of any type using a predefined coding process. 
     The data inputs X 1 , X 2  are expressed as a member of the Galois field GF[p k ] of dimension p k . According to an embodiment of the invention, the data inputs X 1 , X 2  are expressed in a Galois field with a power of two (2) dimension, i.e., a Galois field GF[2 k ] representation. In such a scenario, the manipulation of elements of the Galois field GF[2 k ] in binary arithmetic is trivial. 
     The data inputs X 1 , X 2  are expressed in a p-adic number representation or a mixed-radix number system representation. The p-adic number representation and mixed-radix number systems are well known to persons skilled in the art, and therefore will not be described in detail herein. However, it should be appreciated that a p-adic representation is an extension of the natural numbers based on the divisibility of integers by powers of a number p. As two common examples, the commonly used decimal system uses a 10-adic representation and a binary system uses a 2-adic representation. This p-adic representation can also be viewed as the natural representation of a weighted number system, where digits in the p-adic representation correspond to the powers of a number p. The phrase “natural number” as used herein refers to the set of positive integers greater than zero (0). In practice p is greater then and equal to two (p&gt;2) since the value of p equal to one (p=1) contains no capacity for information. The extension of the field of natural numbers results in a p-adic number sequence. The p-adic number sequence is comprised of k elements (x 0 , x 1 , . . . , x k−1 ) of residue x k  modulo p. 
     It is notable that the p-adic number representation of a weighted number system GF[p k ] is isomorphic to the k-length composite GF[p]·GF[p]· . . . ·GF[p]. Any operator that is bijective on the weighted number system may also be viewed as bijective on the k-length composite. Operating on a p-adic word as a weighted number system induces a correlation between elements, while operating on the k independent fields of size p remains independent at the output. The first case is much more difficult to decode if one or more of the data inputs is unknown. 
     As shown in  FIG. 2 , the initial multiplication operations  202 ,  204  and addition operations  206 ,  208  are performed using the data inputs X 1 , X 2 . These operations  202 ,  204 ,  206 ,  208  are performed to increase a size of the finite or Galois field GF[p k ] from p k  to p k+1  via an Affine transformation. p is taken to be an arbitrary prime number chosen based on the p-adic representation of the data inputs. Still, the invention is not limited in this regard. The products of the multiplication operations  202 ,  204  are herein after referred to as Y 1  and Y 2 . The products Y 1 , Y 2  are defined by mathematical Equations (1-2).
 
 Y=X   1   ·p   1   (1)
 
 Y   2   =X   2   ·p   1   (2)
 
where Y 1  and Y 2  are products obtained via a multiplication operation  202 ,  204 .
 
     Subsequently, the addition operations  206 ,  208  are performed to add non-zero constants C 1 , C 2  from a finite or Galois field GF[p 1 ] to each product Y 1 , Y 2 . The non-zero constants C 1 , C 2  have a value of 1, 2, . . . , or p−1. These addition operations  206 ,  208  complete the input Affine transformations and assure the existence of inverses of the data inputs X 1 , X 2 . It should be appreciated that if p=2 and the arithmetic operations  206 ,  208  are performed in binary arithmetic then the Affine transformation is a simple appending of a binary one (1) to the right of a least significant bit (LSB) of the data input X 1 , X 2  thus implementing 2X+1 as one (1) is the only nonzero element in a Galois field GF[2]. 
     The results of the addition operations  206 ,  208  are hereinafter referred to as Z 1  and Z 2 . The results Z 1 , Z 2  are defined by mathematical Equations (3-4).
 
 Z   1   =Y   1   +C   1 ( X   1   ·p   1 )+ C   1   (3)
 
 Z   2   =Y   2   +C   2 ( X   2   ·p   1 )+ C   2   (4)
 
where Z 1  and Z 2  are results of addition operations. Y 1  and Y 2  are products obtained via multiplication operations  202 ,  204 . C 1  and C 2  are nonzero constants selected from a finite or Galois field GF[p]. Equivalently, C 1  and C 2  may be chosen as unrestricted constants selected from a finite or Galois field GF[p−1], and additively increased by the value one (1). X 1  and X 2  are data inputs expressed as RNS residue values, p-adic number system representations, or mixed-radix number system representations. p is a prime number constant defining a finite field size of a Galois field GF[p].
 
     Thereafter, a Galois field multiplication  210  is performed using the results Z 1 , Z 2 . This Galois field multiplication  210  is performed to combine the data inputs X 1 , X 2  in an expanded Galois field GF[p k+1 ]. The result of the Galois field multiplication  210  is defined by mathematical Equation (5).
 
 RV   1   =Z   1   ·Z   2 (mod  p   k+1 )=[( X   1   ·p   1 )+ C   1 ]·[( X   2   ·p   1 )+ C   2 ](mod  p   1 )  (5)
 
where RV 1  is a result of the Galois field multiplication  210 . p k+1  is a modulus having a value defined by the size of the extended Galois field GF[p k+1 ]. It should be understood that when two elements from a Galois field GF[p k+1 ] are multiplied the product is another element within that Galois field GF[p k+1 ]. In effect, a finite structure size is maintained while a multiplication operation is being performed using the results Z 1 , Z 2 . For example, if p k+1  is selected to have a value of sixteen (16) and the product Z 1 , Z 2  results in a value of forty-five (45), then the result of the arithmetic operation forty-five modulo sixteen (45 modulo 16) equals thirteen (13). The value thirteen (13) is an element in the finitely many invertible elements {1, 3, . . . , 15} of the Galois extension field GF[16].
 
     As shown in  FIG. 2 , the nonzero constants C 1 , C 2  are combined together via Galois field multiplication  212 . This Galois field GF[p 1 ] multiplication  212  is performed to combine the nonzero constants C 1 , C 2  in Galois field GF[p 1 ]. The result of the Galois field multiplication  212  can be defined by mathematical Equation (6).
 
 RV   2   =C   1   ·C   2 (mod  p )  (6)
 
where RV 2  is a value representing the nonzero constants C 1 , C 2 . p is a modulus having a prime number value defining a finite field size of a Galois field GF[p]. In the case of p equal to two (p=2) and C 1  times C 2  equals one (C 1 =C 2 =C 1 ·C 2 =1), there is no need to perform this multiplication  212  in a Galois extension field since multiplicative inverses will always exist for both data inputs X 1 , X 2 .
 
     As shown in  FIG. 2 , (RV 1 −RV 2 )·p −1  is computed to perform an Affine transformation  214  from a Galois field GF[p k+1 ] to a Galois field GF[p k ]. The Affine transformation  214  includes subtracting RV 2  from RV 1  to obtain a result Q. This subtraction operation is performed in GF[p k+1 ]. The result Q can be defined by a mathematical Equation (7).
 
 Q=RV   1   −RV   2   (7)
 
     Thereafter, the difference Q is combined with a multiplicative inverse of p via a multiplication operation  216  to obtain a combined output (CO). The combined output can be defined by mathematical Equation (8).
 
 CO=Q·p   −1   (8)
 
where Q is a result obtained from subtracting RV 2  from RV 1 . p −1  is the multiplicative inverse of the prime p, considered in a weighted number system.
 
     The overall CGFC arithmetic process  200  can be generally defined by a mathematical Equation (9). 
                   CO   =           [       (       p   ·     X   1       +     C   1       )     ·     (       p   ·     X   2       +     C   2       )       ]     ⁢     (     mod   ⁢           ⁢     p     k   +   1         )       -         C   1     ·     C   2       ⁢           ⁢     (     mod   ⁢           ⁢   p     )         p             (   9   )               
where p is a prime number constant defining a finite field size of a Galois field GF[p m ]. X 1  and X 2  are data inputs expressed as values in a p-adic number system representation or a mixed-radix number system representation. C 1  and C 2  are nonzero values selected from a Galois field GF[p]. p k+1  is a number value defining a finite field size of a Galois field GF[p k+1 ]. It should be noted that for p equal to two (p=2) all products in the extended Galois field GF[2 k+1 ] are odd. Therefore, the operation defined in mathematical Equation (8) can be performed simply in binary arithmetic by truncating the k+1 bit result of the modulo 2 k+1  multiply RV 1  to the k most significant bits (MSBs). More generally, the least significant bit (LSB) is removed from the result of subtracting RV 2  from RV 1 .
 
     Referring again to  FIG. 2 , a method of extending the repetition period of a random number sequence can be easily constructed. Given two (2) random number sequences as inputs X 1  and X 2  that come from pseudo-random number generators with mutually prime repetition periods, the output numerical sequence will be a nonlinear combination that yields another pseudo-random number sequence with a repetition period equal to the product of the input repetition periods. The method described in this paragraph extends trivially to the combination of larger numbers of random number generators since multiplication inside a Galois field is associative. In this context, the “associative” property refers to the ability to reorder inputs to an arithmetic process in any combination, resulting in the same output. Note that the CGFC arithmetic process  200  (described above in relation to  FIG. 2 ) as a whole is associative. 
     It should be understood that the CGFC arithmetic process  200  can be modified in accordance with an encoding process. In such a scenario, the data input X 1  can be a number from a numerical sequence representing data. X 2  can be a number from a coding sequence. The coding sequence can be, but is not limited to, a pseudo-random number sequence or a pseudo-chaotic random number sequence. An example of such a modified CGFC arithmetic process  200  is provided in  FIG. 3 . 
     Referring now to  FIG. 3 , a block diagram of a CGFC arithmetic process  300  for encoding data is provided. It should be understood that the CGFC arithmetic process  300  may be used in conjunction with other processes to create stream ciphers or other coding algorithms. Such structures include, but are not limited to, feedback loops, scrambling, and mixed-radix operations. 
     As shown in  FIG. 3 , each value of a data sequence  302  is combined with a value of a code sequence  304  through a Galois extension field multiplication  314  (described above in relation to  FIG. 1  and  FIG. 2 ). It should be noted that the data sequence  302  and/or the code sequence  304  can be selected to have a known mathematical relationship. The data sequence  302  and the code sequence  304  can also be comprised of two or more digits expressed in a weighted number system. In such a scenario, the data sequence  302  and the code sequence  304  can be combined by selectively performing a Galois extension field multiplication  314  using a distinct subset of digits from each sequence  302 ,  304 . 
     It should be noted that the dynamic range of the code sequence  304  values is assumed to be greater than or equal to that of the data sequence  302  values. As used herein, the phrase “dynamic range” refers to the number of possible values for a number or a measurement. The dynamic range is equivalent to a cardinality of a set of possible input values when the code generator does not operate on a subcycle. 
     It should also be noted that the code sequence  304  may be formed from a truncated or random selection of values generated by one or more random number generators. Random number generators are well known to persons skilled in the art, and therefore will not be described in detail herein. The coded data output sequence  320  of the encoding process  300  represents a block of encoding or encryption cipher on the data sequence  302  values using the coded sequence  304  values. 
     It should further be noted that a nearly symmetric method for decoding the coded data output sequence  320  exists. A decoding process can be performed to obtain the data sequence  302  from the combined output coded data sequence  320 . Such an inverse CGFC arithmetic process is described below in  FIG. 4 . 
     Referring now to  FIG. 4 , a first input sequence  402  is the coded data output sequence  320  developed in the encoding process  300  (described above in relation to  FIG. 3 .). A second input sequence  404  is a synchronized replica of the code sequence  304  utilized in the encoding process  300 . An “inverse” operation  414  on this code sequence  404  referring to a bijective mapping from the original code sequence  304  values to the unique multiplicative inverse of the post-Affine transformation code sequence values inside the same Galois extension field that is used for the combination step. This inverse operation is calculated on a value by value basis. The inverse operation may be implemented using any combination of arithmetic processes, digital logic, lookup tables or software. Efficient methods to locate a multiplicative inverse inside residue number systems are well known to those skilled in the art, and therefore will not be described in detail herein. However, it should be noted that one common method for determining the multiplicative inverse inside a Galois field is via an Extended Euclidean algorithm. The Extended Euclidean algorithm is well known to persons skilled in the art, and will not be described in detail herein. 
     Given the code sequence  404 , the decoding process  400  follows a modified combination technique to the CGFC arithmetic process  200  (described above in relation to  FIG. 2 ). In this regard, it should be appreciated that the decoding process  400  includes multiplications operations  406 ,  408  and addition operations  410 ,  412 . The decoding process  400  also includes a Galois extension field multiplication operation  416  and a Galois field multiplication operation  418 . The decoding process further includes an addition operation  420  and a multiplication operation  422 . The output sequence  424  of the decoding process  400  is identical to original data sequence  302  of  FIG. 3 . 
     Table 3 presents all of the encoded and decoded values possible from the data and code sequences if the Galois field GF[8] follows the flows shown in  FIG. 3  and  FIG. 4 . It should be appreciated that the table heading “Data Sequence from a First Data Source” refers to values of the Data Sequence  302  of  FIG. 3 . The table heading “Code Sequence from a Second Data Source” refers to values of the Code Sequence  304  of  FIG. 3 . The table heading “First Data Source Affine Transformation” refers to the value of Z 1  produced by the multiplication and addition operations  306 ,  310  of  FIG. 3 . The table heading “Second Data Source Affine Transformation” refers to the value of Z 2  produced by the multiplication and addition operations  308 ,  312  of  FIG. 3 . The table heading “GF[16] Result” refers to the intermediate result RV 1  of  FIG. 3 . The table heading “GF[8] Result” refers to the intermediate result CO of  FIG. 3 . The table heading “1/Data Source 2 in GF[16]” refers to the intermediate result Z 4  of  FIG. 4 . The table heading “GF[8] Decoded Data” refers to the decoded data sequence  424  of  FIG. 4 . It should be noted that in this binary (p=2) example, the arbitrary nonzero value that is appended to each of the values is always one (1). It should also be noted that the decoding step requires multiplication by the multiplicative inverse of the random sequence rather than the sequence value itself. 
                                                                                   TABLE 3               Data   Code   First   Second                       Sequence   Sequence   Data   Data           1/Data       from a First   from a   Source   Source           Source   GF[8]       Data   Second Data   Affine   Affine   GF[16]   GF[8]   2 in   Decoded       Source   Source   Trans.   Trans.   Result   Result   GF[16]   Data                                0   0   1   1   1   0   1   0       0   1   1   3   3   1   11   0       0   2   1   5   5   2   13   0       0   3   1   7   7   3   7   0       0   4   1   9   9   4   9   0       0   5   1   11   11   5   3   0       0   6   1   13   13   6   5   0       0   7   1   15   15   7   15   0       1   0   3   1   3   1   1   1       1   1   3   3   9   4   11   1       1   2   3   5   15   7   13   1       1   3   3   7   5   2   7   1       1   4   3   9   11   5   9   1       1   5   3   11   1   0   3   1       1   6   3   13   7   3   5   1       1   7   3   15   13   6   15   1       2   0   5   1   5   2   1   2       2   1   5   3   15   7   11   2       2   2   5   5   9   4   13   2       2   3   5   7   3   1   7   2       2   4   5   9   13   6   9   2       2   5   5   11   7   3   3   2       2   6   5   13   1   0   5   2       2   7   5   15   11   5   15   2       3   0   7   1   7   3   1   3       3   1   7   3   5   2   11   3       3   2   7   5   3   1   13   3       3   3   7   7   1   0   7   3       3   4   7   9   15   7   9   3       3   5   7   11   13   6   3   3       3   6   7   13   11   5   5   3       3   7   7   15   9   4   15   3       4   0   9   1   9   4   1   4       4   1   9   3   11   5   11   4       4   2   9   5   13   6   13   4       4   3   9   7   15   7   7   4       4   4   9   9   1   0   9   4       4   5   9   11   3   1   3   4       4   6   9   13   5   2   5   4       4   7   9   15   7   3   15   4       5   0   11   1   11   5   1   5       5   1   11   3   1   0   11   5       5   2   11   5   7   3   13   5       5   3   11   7   13   6   7   5       5   4   11   9   3   1   9   5       5   5   11   11   9   4   3   5       5   6   11   13   15   7   5   5       5   7   11   15   5   2   15   5       6   0   13   1   13   6   1   6       6   1   13   3   7   3   11   6       6   2   13   5   1   0   13   6       6   3   13   7   11   5   7   6       6   4   13   9   5   2   9   6       6   5   13   11   15   7   3   6       6   6   13   13   9   4   5   6       6   7   13   15   3   1   15   6       7   0   15   1   15   7   1   7       7   1   15   3   13   6   11   7       7   2   15   5   11   5   13   7       7   3   15   7   9   4   7   7       7   4   15   9   7   3   9   7       7   5   15   11   5   2   3   7       7   6   15   13   3   1   5   7       7   7   15   15   1   0   15   7                    
Closed Galois Field Cryptographic System
 
     It should be understood that conventional cryptographic systems typically require an asymmetric encryption or decryption processing technique implemented in digital hardware. The phrase “asymmetric processing” as used herein refers to a significant difference in time, processing structure, or computational resources required to perform associated operations. This asymmetric encryption or decryption processing technique can limit many ciphers functionality and applicability. For example, an RSA algorithm involves selecting public and private keys for encrypting data. This selection typically makes decrypting a message by an intended user far easier than an encryption step by a messenger. The digital hardware typically includes a relatively large number of digital logic gates. As such, the conventional cryptographic systems have a relatively high power consumption and relatively long data processing time. Encryption methods that permit symmetric processing permit design re-use of hardware as well as more flexibility in function and applicability. 
     Embodiments of a closed Galois field (CGF) cryptographic system (e.g., shown in  FIG. 5 ) provides a symmetric encryption and decryption processing technique implemented in digital hardware. The phrase “symmetric processing” as used herein refers to an ability to both encrypt and decrypt a message using a substantially similar processing structure, time and resources. In effect, the embodiment of the CGF cryptographic system has a relatively short data processing time as compared to conventional cryptographic systems. The embodiments of the CGF cryptographic system also have relatively low power consumption as compared to conventional cryptographic systems due to the efficient combination means. 
     Referring now to  FIG. 5 , there is illustrated one embodiment of a closed Galois field (CGF) cryptographic system  500  which could be used to implement the inventive arrangements. The CGF cryptographic system  500  is comprised of a message generator  502  and random number generators (RNGs)  504 ,  510 . The CGF cryptographic system  500  is also comprised of a closed Galois field (CGF) encryption system  506  and a closed Galois field (CGF) decryption system  508 . The message generator  502  is comprised of hardware and software configured to generate payload data. The message generator  502  is also comprised of hardware and software configured to perform operations for expressing payload data in a block format. The message generator  502  is further comprised of hardware and software configured to communicate payload data to the CGF encryption system  506 . 
     The RNG  504  is comprised of hardware and software configured to generate an encryption sequence. The encryption sequence can be comprised of linearly or nonlinearly generated pseudo-random numbers or pseudo-chaotic random numbers. The RNG  504  is also comprised of hardware and software configured to communicate a random number sequence to the CGF encryption system  506 . 
     The CGF encryption system  506  is comprised of hardware and software configured to receive payload data from the message generator  502  and an encryption sequence from the RNG  504 . The CGF encryption system  506  is also comprised of hardware and software configured to implement and perform a CGFC method  100  (described above in relation to  FIG. 1 ) and a CGFC arithmetic process  300  (described above in relation to  FIG. 3 ) for encrypting payload data using the random number sequence. The CGF encryption system  506  is further comprised of hardware and software configured to communicate a combined output to the CGF decryption system  508 . 
     The RNG  510  is comprised of hardware and software configured to generate a decryption sequence. The decryption sequence can be comprised of linearly or nonlinearly generated pseudo-random numbers or pseudo-chaotic random numbers. The decryption sequence can be an inverted sequence based on the encryption sequence generated by the RNG  504 . The RNG  510  is also comprised of hardware and software configured to perform data block formatting. The RNG  510  is further comprised of hardware and software configured to communicate a decryption sequence to the CGF decryption system  508 . 
     The CGF decryption system  508  is comprised of hardware and software configured to receive a combined output from the CGF encryption system  506  and a decryption sequence from the RNG  510 . The CGF decryption system  508  is also comprised of hardware and software configured to implement and perform an inverse CGFC arithmetic process  400  (described above in relation to  FIG. 4 ) for decrypting payload data. The CGF decryption system  508  is further comprised of hardware and software configured to communicate message data to an external device (not shown). 
     In light of the foregoing description of the invention, it should be recognized that the present invention can be realized in hardware, software, or a combination of hardware and software. Any kind of computer system, or other apparatus adapted for carrying out the methods described herein, is suited. A typical combination of hardware and software could be a general purpose computer processor, with a computer program that, when being loaded and executed, controls the computer processor such that it carries out the methods described herein. Of course, an application specific integrated circuit (ASIC), and/or an FPGA could also be used to achieve a similar result. 
     The present invention can also be embedded in a computer program product, which comprises all the features enabling the implementation of the methods described herein, and which, when loaded in a computer system, is able to carry out these methods. Computer program or application in the present context means any expression, in any language, code or notation, of a set of instructions intended to cause a system having an information processing capability to perform a particular function either directly or after either or both of the following a) conversion to another language, code or notation; b) reproduction in a different material form. Additionally, the description above is intended by way of example only and is not intended to limit the present invention in any way, except as set forth in the following claims. 
     The invention described and claimed herein is not to be limited in scope by the preferred embodiments herein disclosed, since these embodiments are intended as illustrations of several aspects of the invention. Any equivalent embodiments are intended to be within the scope of this invention. Indeed, various modifications of the invention in addition to those shown and described herein will become apparent to those skilled in the art from the foregoing description. Such modifications are also intended to fall within the scope of the appended claims. 
     The Abstract of the Disclosure is provided to comply with 37 C.F.R. §1.72(b), requiring an abstract that will allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the following claims.