椭圆曲线离散对数的不动点

Fixed Points for Elliptic Curve Discrete Logarithms

椭圆曲线离散对数问题在密码领域有着重要的应用.本文中我们将模素数p的离散对数的不动点问题推广到有限域上椭圆曲线离散对数的不动点问题.对于有限域p?上的任意椭圆曲线()pE?,证明了当p足够大时,以大概率存在一点(,)()pQ?x y?E?使得logPQ?x,即Q?x?P,其中logP被看成是以点()pP?E?为底的离散对数,且点P满足ord(P)?n,而x被看成是在区间[0,n?1][0,p?1]?上的整数.

Elliptic Curve Discrete Logarithm Problem(ECDLP) has very important applications in cryptography. By extending the fixed point problem for the discrete logarithm modulo prime p, this paper studies the fixed point problem for discrete logarithms on elliptic curves over finite fields. For any elliptic curve E(Fp), we show that when p is sufficiently large, there exists, with a high probability, a point pQ=(x,y)∈E(Fp) such that logPQ=x, i.e., Q =x ·P, where logP is considered as the discrete logarithm function to the base P∈E(Fp) with ord(P) =n and x is also regarded as an integer in the interval [0,n-1]∩[0, p-1].