Patent Publication Number: US-7218735-B2

Title: Cryptography method on elliptic curves

Description:
This disclosure is based upon French Application No. 00/05006, filed on Apr. 18, 2000 and International Application No. PCT/FR01/01195, filed Apr. 18, 2001, the contents of which are incorporated herein by reference. 
   BACKGROUND OF THE INVENTION 
   The present invention relates to a cryptography method on elliptic curve. Such a method is based on the use of a public key algorithm and can be applied to the generation of probabilistic digital signals of a message and/or to a key exchange protocol and/or to a message enciphering algorithm. 
   An algorithm for generating and verifying digital signatures consists in calculating one or more integers, in general a pair, known as the signature, and associated with a given message in order to certify the identity of a signature and the integrity of the signed message. The signature is said to be probabilistic when the algorithm uses a random variable in the generation of the signature, this random variable being secret and regenerated at each new signature. Thus the same message transmitted by the same user may have several distinct signatures. 
   Key exchange protocol and enciphering algorithms also use a secret random variable k generated at each new application of the algorithm. 
   Public key cryptography algorithms on elliptic curves are being used more and more. Such an algorithm is based on the use of points P(x,y) on a curve E satisfying the equation:
 
 y   2   +xy=x   3   +ax   2   +b,  with  a  and  b  two elements of a finite field.
 
   Addition or subtraction operations are performed on the points P of the curve E. The operation consisting in adding k times the same point P is called the scalar multiplication of P by k, and corresponds to a point C on the elliptic curve defined by C(x′,y′)=k·P(x,y). 
   An example of such an algorithm can be illustrated by the ECDSA (from the English Elliptic Curve Digital Standard Algorithm), which is an algorithm for generating and verifying probabilistic digital signatures. 
   The parameters of the ECDSA are:
         E, an elliptic curve defined on the set Z p , the number of points on the curve E being divisible by a large prime number N, in general N&gt;2 160 ,   P(x,y), a given point on the elliptic curve E.       

   The secret key d is a randomly fixed number between 0 and N−1, and the public key Q is related to d by the scalar multiplication equation Q(x 1 ,y 1 )=d·P(x,y). 
   Let m be the message to be sent. The ECDSA signature of m is the pair of integers (r,s) included in the range [1, N−1] and defined as follows:
         let k be a random number chosen in the range [1, N−1], k being a random variable regenerated at each signature;   calculation of the point C obtained by the scalar multiplication C(x′,y′)=k·P(x,y);   r=x′ mod N;   s=k −1 (h(m)+d·r) mod N;       

   with h(m) the result of the application of a hash function h, which is a pseudo-random cryptographic function, to the initial message m. 
   The verification of the signature is performed, using public parameters (E, P, N, Q), as follows: 
   Intermediate calculations are carried out:
         w=s −1  mod N;   u 1 =h(m)·w mod N;   u 2 =r·w mod N;   An addition and scalar multiplication operation is performed by calculating the point on the curve E corresponding to u 1 P+u 2 Q=(x 0 ,y 0 )       

   It is checked whether v=x 0  mod N Υ r. 
   If this equality is true, the signature is authentic. 
   The generation of the signature (r,s) was performed with the secret key d and a secret random number k different for each signature, and its verification with the parameters of the public key. Thus anyone can authenticate a card and its bearer without holding its secret key. 
   The cost of execution of such a signature algorithm on elliptic curve is directly related to the complexity and speed of the scalar modification operation for defining the point C=k·P. 
   Improvements to the cryptography method on elliptic curves have been developed in order to facilitate and accelerate this scalar multiplication operation. In particular, the article by J. A. Solinas “An Improved Algorithm for Arithmetic on a Family of Elliptic Curves”, which appeared in Proceedings of Crypto&#39;97, Springer Verlag, describes one possible improvement. 
   In order to accelerate the method for calculating a scalar multiplication in the context of an algorithm on elliptic curve E, it has thus been envisaged working on a particular family of elliptic curves, known as abnormal binary elliptic curves or Koblitz curves, on which a particular operator is available, known as a Frobenius operator, making it possible to calculate the scalar multiplication operations more quickly. 
   The Koblitz curves are defined on the mathematical set GF(2 n ) by the equation:
 
 y   2   +xy=x   3   +ax   2 +1 with  aε{ 0,1]
 
   The Frobenius operator T is defined as:
 
τ[ P ( x,y )]=( x   2   ,y   2 ) with the equation τ 2 +2=(−1) 1−a τ
 
   Applying the operator τ to a given point P on the elliptic curve E constitutes a quick operation since the work is done in the mathematical set GF(2 n ), n being the size of the finite field, for example n=163. 
   In order to facilitate the calculation of the scalar multiplication C(x 1 ,y 1 )=k·P(x,y), the integer k is decomposed so as to amount to addition and subtraction operations. In this way the non-adjacent form of the integer k is defined by the NAF (from the English Non-Adjacent Form), which consists in writing an integer k in the form of a sum:
 
 k=Σ ( i= 0 to 1−1) e   i 2 i  with  e   i ε{−1, 0, 1} and 1≅ n. 
 
   In the case of a Koblitz elliptic curve, the NAF can be expressed by means of the Frobenius operator:
 
 k =Σ( i− 0 to 1)  e   i τ i .
 
   Thus the operation of scalar multiplication of P by k amounts to applying the Frobenius operator to the point P, which is easy and rapid. 
   In addition, the calculation of the scalar multiplication k·P can be accelerated further by the precalculation and storage of a few pairs (k i , P i =k i ·P), these pairs advantageously being able to be stored in the memory of the device implementing the signature algorithm. In fact P forms part of the public parameters of the key of the signature algorithm. 
   For a random variable k of 163 bits, it is thus possible, by storing 42 scalar multiplication pairs (k i ,P i ), to reduce the number of addition/subtraction operations to 19 instead of 52 without any precalculation. 
   The object of the present invention is a cryptography method on elliptic curve which makes it possible to reduce the number of additions of the scalar multiplication still further. 

   DESCRIPTION OF THE INVENTION 
   The invention relates more particularly to a cryptography method for generating probabilistic digital signatures and/or for a key exchange protocol and/or for an enciphering algorithm, the said method being based on the use of a public key algorithm on an abnormal binary elliptic curve (E) (Koblitz curve) on which a point P(x,y) is selected, pairs (k i ,P i ) being stored with P i  the point corresponding to the scalar multiplication of the point P by k i , the said method comprising steps consisting in generating a random variable k and calculating a point C corresponding to the scalar multiplication of P by k (C=k·P), characterised in that the generation of the said random variable k and the calculation of the point C are performed simultaneously. 
   According to one application, the cryptographic algorithm for generating a probabilistic digital signature is the ECDSA (from the English Elliptic Curve Digital Standard Algorithm). 
   According to another application, the key exchange protocol cryptographic algorithm is the ECDH (from the English Elliptic Curve Diffie-Hellmann). 
   According to one characteristic, the method is based on the use of a Koblitz curve defined on the mathematical set GF(2 n ) on which a so-called Frobenius operator τ[P(x,y)]=(x 2 ,y 2 ) is available, the method being characterised in that it includes the following steps:
         initialising the random variable k=0 and the point C=0,   performing a loop for j ranging from 1 to n iter , the said loop consisting in:
           generating the following random variables at each new iteration:
               a, between 0 and n, with n the size of the finite field on which the curve is defined,   u {−1,1},   i between 0 and t, with t the number of pairs (k i ,P i ) stored,   
               calculating the point C j =C j−1 +u·τ a ·P i ,   generating the random variable k j =k j−1 +u·k i ·τ a      
           converting k into an integer at the end of the loop,   simultaneously presenting the random variable k and the point C=k·P.       

   According to one characteristic, the number t of pairs (k i ,P i ) stored is between 35 and 45. 
   According to another characteristic, the number of iterations of the loop (n iter ) is fixed at between 10 and 12. 
   According to another characteristic, the size of the mathematical field n on which the Koblitz curve is defined is equal to 163. 
   The invention also relates to a secure device of the smart card type, or a calculation device of the computer type provided with enciphering software, having an electrical component able to implement the signature method according to the invention. 
   The method according to the invention has the advantage of reducing the time taken for calculating the scalar product of P and k, which constitutes an essential step in the use of a cryptography method on elliptic curve, firstly by generating the random variable k simultaneously with the calculation of the scalar product k·P and secondly by reducing the number of addition operations by the precalculation of pairs k i , P i =k i ·P. 
   The particularities and advantages of the invention will emerge more clearly from a reading of the following description made with reference to the ECDSA algorithm and given by way of illustrative and non-limiting example. The method according to the invention can in fact also be applied to a key exchange protocol or to an enciphering algorithm for example. 
   Let E be a Koblitz elliptic curve defined on the set GF(2 n ), with n=163 the size of the mathematical field on which the work is being carried out, and let P(x,y) be a given point on this curve. 
   The Frobenius operator τ[P(x,y)]=(x 2 ,y 2 ) is then available and constitutes a quick operation given the field GF(2 n ) on which the work is being carried out. 
   Firstly a certain number of pairs (k i , P i =k·P) are calculated, which are stored in the component implementing the signature method (a smart card microcontroller for example) The number of pairs is fixed at t between 35 and 45, which constitutes a compromise between the memory space occupied and the required acceleration for the signature generation calculation method. 
   The method according to the present invention consists in accelerating the method for generating a probabilistic signature by using precalculated and stored pairs (k i ,P i ) generating the random variable k at the same time as calculating the point C=k·P. 
   Firstly the values of C and k are initialised to 0. 
   Then a loop on j of n iter  iterations is implemented, which performs the following operations:
         generating the following random variables at each new iteration of j:
           r, between 0 and n,   u {−1,1},   i between 0 and t,   
           calculation of C j =C j−1 +u·τ r ·P i      calculation of k j =k j−1 +u·k i ·τ r          

   Then there is obtained, at the output from the loop, the random variable k, which is converted into an integer, and the point C corresponding to the scalar multiplication of P by k. 
   The signature (r,s) is then generated according to the conventional procedure of the ECDSA, or of another algorithm using Koblitz elliptic curves, with the values of k and C defined according to the method of the invention. 
   The generation of k simultaneously with the calculation of the point C accelerates the signature generation method, in particular by reducing the number of additions necessary for calculating the scalar multiplication of P by k. The number of additions for calculating the point C is in fact n iter  −1. 
   According to the degree of security and the performance required, n iter  is fixed at between 10 and 12 iterations. 
   Thus, with k an integer of 163 bits and by storing approximately 40 pairs (k i ,P i ), it is possible to calculate the scalar multiplication k·P by performing only 9 to 11 addition operations.