The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences

TL;DR

This paper introduces three computationally hard problems related to elliptic divisibility sequences, establishing their equivalence to the elliptic curve discrete logarithm problem and connecting them to known cryptographic attacks.

Contribution

It defines new problems in elliptic divisibility sequences and proves their computational hardness is equivalent to the elliptic curve discrete logarithm problem, linking them to existing cryptographic attacks.

Findings

Three new hard problems in elliptic divisibility sequences are introduced.

The hardness of these problems is equivalent to the elliptic curve discrete logarithm problem.

Connections are established between these problems and known cryptographic attacks.

Abstract

We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in sub-exponential time if and only if the elliptic curve discrete logarithm problem is solvable in sub-exponential time. We also relate the problem of EDS Association to the Tate pairing and the MOV, Frey-R\"{u}ck and Shipsey EDS attacks on the elliptic curve discrete logarithm problem in the cases where these apply.