[Resource Topic] 2021/1133: Multiradical isogenies

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Multiradical isogenies

Authors: Wouter Castryck, Thomas Decru


We argue that for all integers N \geq 2 and g \geq 1 there exist “multiradical” isogeny formulae, that can be iteratively applied to compute (N^k, \ldots, N^k)-isogenies between principally polarized g-dimensional abelian varieties, for any value of k \geq 2. The formulae are complete: each iteration involves the extraction of g(g+1)/2 different $N$th roots, whence the epithet multiradical, and by varying which roots are chosen one computes all N^{g(g+1)/2} extensions to an (N^k, \ldots, N^k)-isogeny of the incoming (N^{k-1}, \ldots, N^{k-1})-isogeny. Our group-theoretic argumentation is heuristic, but it is supported by concrete formulae for several prominent families. As our main application, we illustrate the use of multiradical isogenies by implementing a hash function from (3,3)-isogenies between Jacobians of superspecial genus-2 curves, showing that it outperforms its (2,2)-counterpart by an asymptotic factor \approx 9 in terms of speed.

ePrint: https://eprint.iacr.org/2021/1133

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