Paper 2025/372

KLPT²: Algebraic Pathfinding in Dimension Two and Applications

Abstract

Following Ibukiyama, Katsura and Oort, all principally polarized superspecial abelian surfaces over $$ can be represented by a certain type of $$ matrix $$, having entries in the quaternion algebra $$. We present a heuristic polynomial-time algorithm which, upon input of two such matrices $$, finds a "connecting matrix" representing a polarized isogeny of smooth degree between the corresponding surfaces. Our algorithm should be thought of as a two-dimensional analog of the KLPT algorithm from 2014 due to Kohel, Lauter, Petit and Tignol for finding a connecting ideal of smooth norm between two given maximal orders in $$. The KLPT algorithm has proven to be a versatile tool in isogeny-based cryptography, and our analog has similar applications; we discuss two of them in detail. First, we show that it yields a polynomial-time solution to a two-dimensional analog of the so-called constructive Deuring correspondence: given a matrix $$ representing a superspecial principally polarized abelian surface, realize the latter as the Jacobian of a genus-$$ curve (or, exceptionally, as the product of two elliptic curves if it concerns a product polarization). Second, we show that, modulo a plausible assumption, Charles-Goren-Lauter style hash functions from superspecial principally polarized abelian surfaces require a trusted set-up. Concretely, if the matrix $$ associated with the starting surface is known then collisions can be produced in polynomial time. We deem it plausible that all currently known methods for generating a starting surface indeed reveal the corresponding matrix. As an auxiliary tool, we present an explicit table for converting $$-isogenies into the corresponding connecting matrix, a step for which a previous method by Chu required super-polynomial (but sub-exponential) time.