On the zeta functions of two towers of function fields

Abstract

The discrete logarithm problem (DLP) on elliptic curves over finite field has been extensively studied as a cryptographic building block. The DLP recently was considered over other algebraic structures such as Jacobian of hyperelliptic curves, superelliptic curves, and Abelian varieties in general. The main objective is to determine a large subgroup of prime order for which no index calculus attack is known. We investigate the Jacobian of two towers of function fields that have good asymptotic property as another potential source of Abelian groups for the DLP. This paper is the first step in this direction and compute the size of the Jacobian via the zeta function.