# Efficient computation of \$(3^n,3^n)\$-isogenies

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Mar 17, 2023
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 seconds for the SIKEp751 parameters, which were aimed at NIST level 5 security.

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Author Of this post: Thomas Decru