Efficient Computation of (3 n, 3 n) -Isogenies

Abstract

The parametrization of (3, 3)-isogenies by Bruin, Flynn and Testa requires over 37.500 multiplications if one wants to evaluate a single isogeny in a point. We simplify their formulae and reduce the amount of required multiplications by 94 %. Further we deduce explicit formulae for evaluating (3, 3)-splitting and gluing maps in the framework of the parametrization by Bröker, Howe, Lauter and Stevenhagen. We provide implementations to compute (3 n, 3 n) -isogenies between principally polarized abelian surfaces with a focus on cryptographic application. Our implementation can retrieve Alice’s secret isogeny in 11 s for the SIKEp751 parameters, which were aimed at NIST level 5 security.