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Title:
Refined Humbert Invariants in Supersingular Isogeny Degree Analysis
Authors: Eda Kırımlı, Gaurish Korpal
Abstract:In this paper, we discuss refined Humbert invariants of principally polarized superspecial abelian surfaces. Kani introduced the refined Humbert invariant of a principally polarized abelian surface in 1994. The main contribution of this paper is to calculate the refined Humbert invariant of a principally polarized superspecial abelian surface. We present three applications of computing this invariant in the context of isogeny-based cryptography. First, we discuss the maximum of the minimum degrees of isogenies between two uniformly random supersingular elliptic curves independent of their endomorphism ring structures. Second, we provide a different perspective on the fixed isogeny degree problem using refined Humbert invariants, and analyze this problem on average without endomorphism rings. Third, we give experimental evidence for the proven upper bounds that the minimum distance is \approx \sqrt{p}; our work verifies this claim up to p=727.
ePrint: https://eprint.iacr.org/2025/1605
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