Refined Humbert Invariants in Supersingular Isogeny Degree Analysis

2026/07/28 by Eda Kırımlı, Gaurish Korpal
Mathematics · #math.NT

paper · pdf

Abstract

We focus on refined Humbert invariants of principally polarized superspecial abelian surfaces, introduced by Kani in 1994. The main contributions are to enumerate principal polarizations on a superspecial surface, and for each polarization, to compute the refined Humbert invariant of a principally polarized superspecial abelian surface. Then, we present several applications of computing this invariant for isogeny-based cryptography. First, we provide a decision algorithm to check if two given polarizations are isomorphic. Second, we present an efficient algorithm to determine the geometric type of a principally polarized superspecial surface. Third, we prove an upper bound on the largest minimal isogeny degree among pairs of supersingular elliptic curves, independent of their endomorphism-ring structures, and our experimental evidence verifies this claim up to p=659, p≡ 11\pmod12. Fourth, we present experimental evidence for a minimum isogeny frequency within the proven upper bounds. Lastly, we provide a different perspective on the fixed isogeny degree problem using refined Humbert invariants and analyze it without explicit endomorphism rings.

Citations

• The Refined Humbert Invariant for an Automorphism Group of a Genus 2 Curve
• The supersingular Endomorphism Ring and One Endomorphism problems are equivalent
• Kneser's method of neighbors
• The Refined Humbert Invariant for Imprimitive Ternary Forms
• The supersingular isogeny path and endomorphism ring problems are\n equivalent
• Lecture Notes on Quadratic Forms and their Arithmetic
• Supersingular Curves With Small Non-integer Endomorphisms
• Reduced norms and the Riemann-Roch theorem for Abelian varieties
• On the quaternion ℓ-isogeny path problem
• Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms
• Lattice methods for algebraic modular forms on classical groups

Related

• Unconditional foundations for supersingular isogeny-based cryptography
• Counting Richelot isogenies between superspecial abelian surfaces
• Failing to hash into supersingular isogeny graphs
• Computing isogeny classes of typical principally polarized abelian surfaces over the rationals
• Higher-degree supersingular group actions