Common use of Elliptic Curve Cryptography Clause in Contracts

Elliptic Curve Cryptography. Elliptic curve cryptography makes use of elliptic curves in which the variables and coefficients are all restricted to elements of a finite field. Typically, elliptic curves are defined over either the integers modulo a prime number (GF(p)) or over binary polynomials (GF(2m)). An elliptic curve is a cubic equation of the form: y2 +axy + by = x3+ cx2 + dx + e. (1) where a, b, c, d, and e are real numbers. In an elliptic curve cryptosystem (ECC), the elliptic curve equation is defined as the form of Ep(a, b): y2 = x3+ax+b( mod p) (2) over a prime finite field Fp, where a, b ε Fp, p > 3, and 4a3 + 27b2 (mod p) ≠ 0. Generally, the security of ECC relies on the difficulties of the following problems [10]. Definition 1 Given two points P and Q over Ep(a, b), the elliptic curve discrete logarithm problem (ECDLP) is to find an integer s ε Fp* such that Q = s.P. Definition 2 Given three points P, s.P, and t.P over Ep(a, b) for s; t ε Fp*, the computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ problem (CDHP) is to find the point (s.t).P over Ep(a, b). Definition 3 Given two points P and Q = s.P + t.P over Ep(a, b) or s; t ε Fp*, the elliptic curve factorization problem (ECFP) is to find two points s. P and t.P over Ep(a, b). Up to now, there is no algorithm to be able to solve any of the above problems [10]