On randomness extraction in elliptic curves

Abstract

A deterministic extractor for an elliptic curve, that converts a uniformly random point on the curve to a random k-bit-string with a distribution close to uniform, is an important tool in cryptography. Such extractors can be used for example in key derivation functions, in key exchange protocols and to design cryptographically secure pseudorandom number generator. In this paper, we present a simple and efficient deterministic extractor for an elliptic curve E defined over double-struck Fqn, where q is prime and n is a positive integer. Our extractor, denoted by Dk, for a given random point P on E, outputs the k-first double-struck Fq-coordinates of the abscissa of the point P. This extractor confirms the two conjectures stated by R. R. Farashahi and R. Pellikaan in [6] and by R. R. Farashahi, A. Sidorenko and R. Pellikaan in [7], related to the extraction of bits from coordinates of a point of an elliptic curve. © 2011 Springer-Verlag.