CA2129203A1 - Public key cryptography utilizing elliptic curves - Google Patents

Abstract

An elliptic curve encryption system represents coordinates of a point on the curve as a vector of binary digits in a normal basis representation. A key is generated from multiple additions of one or more points in a finite field. Inverses of values are computed using a finite field multiplier and successive exponentiations.

Claims (3)

1. A method of computing an inverse of a number x with a finite field multiplier operating in the finite field GF(2M) and having elements A2i (oi<? that constitute a normal basis, said multiplier having a pair of m celled recirculating shift registers connected to a m celled recirculating accumulating register to generate in each of said accumulating register a respective grouped term of the normal basis representation of the product of a pair of elements located in respective ones of said recirculating shift registers, said method comprising the steps of a) representing the number x as a vector of binary digits xi where xi is the coefficient of A2i in the normal basis representation of x, b) loading in to each of said shift registers the vector of binary digits xi representing the normal basis representation of x2, c) cyclically shifting the binary digits of a first of said registers one cell to provide in said first register a vector representing X4, d) rotating said vectors in said shift registers and cojointly rotating said accumulating register with a m fold cyclic shift to generate in the cells of said accumulating register the m grouped terms representing the vector of the product of x2 and X4, e) loading the vector from the accumulating register to a second of said shift registers, f) repeating the steps of (c), (d), and (e) (g-2) times where g is a factor of m-1 to provide in said accumulating register a vector .gamma. which is the normal basis representation of the exponentiation of , g) loading the vector representing the normal basis representation of .gamma. in each of said shift registers, h) performing a g-fold cyclic shift the binary digits of the vector in one of said shift registers where g is a factor of m-1 and g.h = m-1 to provide a vector representing .gamma.2g in said one register, i) rotating said bit elements in said shift registers and said accumulating register to generate grouped terms of the vector representing the product of .gamma.
and .gamma.2g, j) loading the vector from the accumulating register to the other of said shift registers, k) repeating steps h), i), and j) a total of g(h-1)-1 times to provide in said accumulating cell a vector of binary digits of the coefficients of the normal basis representation of the inverse of x.

2. A method according to claim 1 including the step of loading the vector representing x into one of said registers, performing a 1 cell cyclic shift to provide x2 and copying the resultant vector in to the other of said registers.

3. In a data encryption system in which the data is considered with an encryption key to produce ciphertext, a method of generating a key comprising the steps of a) selecting an elliptic curve lying in the finite field GF2m and having elements A2i (oim) that constitute a normal basis, b) representing the coordinates of a point on said curve as a set of vectors, each vector representing a coordinate of said point and having m binary digits, each of which represents the coefficient of A2i in the normal basis representation of said vector, c) computing from addition of at least two sets of vectors an additional set of vectors to represent the coordinates of further point on said curve, and d) utilising said additional set of vectors to derive a key for encrypting data,