Extractors for Jacobian of hyperelliptic curves of genus 2 in odd characteristic

Abstract

We propose two simple and efficient deterministic extractors for J(double-struck F signq), the Jacobian of a genus 2 hyperelliptic curve H defined over double-struck F signq, for some odd q. Our first extractor, SEJ, called sum extractor, for a given point D on .J(double-struck F signq), outputs the sum of abscissas of rational points on H in the support of D, considering D as a reduced divisor. Similarly the second extractor, PEJ, called product extractor, for a given point D on the J(double-struck F signq), outputs the product of abscissas of rational points in the support of D. Provided that the point D is chosen uniformly at random in J(double-struck F signq), the element extracted from the point D is indistinguishable from a uniformly random variable in double-struck F signq. Thanks to the Kummer surface K, that is associated to the Jacobian of H over double-struck F signq, we propose the sum and product extractors, SEK and PEK, for K(double-struck F signq). These extractors are the modified versions of the extractors SEJ and PEJ. Provided a point K is chosen uniformly at random in K, the element extracted from the point K is statistically close to a uniformly random variable in double-struck F signq. © Springer-Verlag Berlin Heidelberg 2007.