The supersingular isogeny path and endomorphism ring problems are equivalent

TL;DR

This paper proves the polynomial-time equivalence of the supersingular isogeny path problem and the endomorphism ring problem, establishing a foundational link crucial for isogeny-based cryptography, under the assumption of the generalized Riemann hypothesis.

Contribution

It establishes the polynomial-time equivalence between two key problems in supersingular elliptic curve cryptography and develops a rigorous algorithm for the quaternion analog of the path-finding problem.

Findings

Proved the equivalence of the two problems under certain assumptions.

Developed a rigorous algorithm for the quaternion analog of the path-finding problem.

Provided insights into cryptanalytic tools and cryptosystem building blocks.

Abstract

We prove that the path-finding problem in $\ell$-isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis. The presumed hardness of these problems is foundational for isogeny-based cryptography. As an essential tool, we develop a rigorous algorithm for the quaternion analog of the path-finding problem, building upon the heuristic method of Kohel, Lauter, Petit and Tignol. This problem, and its (previously heuristic) resolution, are both a powerful cryptanalytic tool and a building-block for cryptosystems.