Exp function for Edwards curves over local fields

Abstract

The difficulty of solving the ECDLP is the foundation of the security of elliptic curve cryptography (ECC). A connection between the lifting problem on a curve in Weierstrass form and the ECDLP has been stressed by Silverman. Based on this, in order to study the ECDLP, we compute the exponential map for Edwards curves, which are more efficient for cryptographic use, as proved by Bernstein and Lange. In this paper, using the birational equivalence between Edwards curves and elliptic curves in Weierstrass form, we extend the map Exp to Edwards curves over local fields. Furthermore, we compute the map Exp for the specific case of p-adic numbers. More recently, curves over local fields have been investigated by Tang, Xu, and Qi, who introduced a cryptosystem based on their quotient groups.