1970-01-01

(2026-02) On the conversion of module representations for higher dimensional supersingular isogenies

• [[Aurel Page]]
• [[Damien Robert]]
• [[Julien Soumier]]

2026-02-16

• #cryptography
• #paper

Abstract

We expand the well developed toolbox between quaternionic ideals and supersingular elliptic curves into its higher dimensional version, namely (Hermitian) modules and maximal supersingular principally polarised abelian varieties. One of our main result is an efficient algorithm to compute an unpolarised isomorphism $A \simeq E_0^g$ given the abstract module representation of $A$. This algorithm relies on a subroutine that solves the Principal Ideal Problem in matrix rings over quaternion orders, combined with a higher dimensional generalisation of the Clapotis algorithm. To illustrate the flexibility of our framework, we also use it to reduce the degree of the output of the KLPT$^2$ algorithm, from $O(p^{25})$ to $O(p^{15.5})$.