AI 中文总结
研究主极化超特殊阿贝尔曲面的精细洪堡不变量，通过计算该不变量，在基于同构的密码学中实现检查极化同构、确定曲面几何类型、证明同构度上界及提供同构频率证据等应用，并从新视角分析固定同构度问题。
AI 中文摘要
我们聚焦于1994年卡尼引入的主极化超特殊阿贝尔曲面的精细洪堡不变量。主要贡献包括枚举超特殊曲面上的主极化，并为每个极化计算主极化超特殊阿贝尔曲面的精细洪堡不变量。接着给出该不变量在基于同构的密码学中的几个应用。一是提供检查两个给定极化是否同构的判定算法；二是给出确定主极化超特殊曲面几何类型的高效算法；三是证明超奇异椭圆曲线对中最大最小同构度的上界，实验证据验证至\(p = 659\)，\(p\equiv 11\pmod{12}\)；四是给出已证上界内最小同构频率的实验证据；最后用精细洪堡不变量对固定同构度问题提供不同视角并在无明确自同态环情况下进行分析。
英文摘要
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 $p=659$, $p\equiv 11\pmod{12}$. 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.
Comments24 pages, 2 figures
