A result on the distribution of quadratic residues with applications to elliptic curve cryptography

Abstract

In this paper, we prove that for any polynomial function / of fixed degree without multiple roots, the probability that all the (f(x + l), f(x + 2),..., f(x + k)) are quadratic non-residue is ≈ 1/2k. In particular for f(x) = x3 + ax + b corresponding to the elliptic curve y2 = x3 +ax + b, it implies that the quadratic residues (f(x + 1), f(x + 2),... in a finite field are sufficiently randomly distributed. Using this result we describe an efficient implementation of El-Gamal Cryptosystem. that requires efficient computation of a mapping between plain-texts and the points on the elliptic curve. © Springer-Verlag Berlin Heidelberg 2007.