检测同源图上的分裂曲面、RM曲面与最小路径
Detecting split surfaces, RM surfaces, and minimum walks on isogeny graphs
浏览论文内容
中文总结 AI 辅助
我们利用精细Humbert不变量将同源计算转化为本原表示问题，提出检测分裂曲面与RM嵌入的算法，并在小素数及千比特素数上验证，分裂度满足对数关系。
中文摘要 AI 辅助
我们研究了主极化超特殊阿贝尔曲面的同源图中分裂曲面的可检测性，这一问题与基于二维同源的密码学的安全性分析相关。我们的方法使用精细的Humbert不变量，将显式的同源计算替换为五元正定二次型的本原表示问题。我们开发了检测$(N,N)$-分裂的算法，并计算超特殊雅可比簇的最小$(N,N)$-分裂级别，而无需构造相应的同源路径。同一框架通过实二次序判别式的本原表示来检测实乘法（RM）序的嵌入。对于RM，我们使用详尽的小素数数据以及对于$227\leq p\leq1619$的每个素数$100{,}000$个随机极化，测试无平方因子判别式$D\leq100$的本原表示；第一个非平凡的本原表示判别式通常很小。我们在两种情况下实验性地应用这些方法。对于$11\leq p\leq251$，其中不可约主极化已被详尽地知道，从精细Humbert不变量恢复的自同构数据重现了Ibukiyama--Katsura--Oort对不可约极化的计数。对于大参数，我们达到1000比特的素数，其中产生的分裂度满足$\log_2N\approx\log_2p$，其分布形状在整个范围内稳定，因此检测到的$(N,N)$-同源度约为$p^{2}$；观察到的$N$的最小最大素数幂因子在250比特以内保持较小，并从300比特起急剧增加。
英文摘要
We study the detectability of split surfaces in isogeny graphs of principally polarized superspecial abelian surfaces, a question relevant to the security analysis of dimension-$2$ isogeny-based cryptography. Our approach uses refined Humbert invariants to replace explicit isogeny computations with primitive representation problems for positive definite quadratic forms in five variables. We develop algorithms to detect $(N,N)$-splittings and to compute the minimum $(N,N)$-splitting level of a superspecial Jacobian without constructing the corresponding isogeny path. The same framework detects embeddings of real multiplication (RM) orders through primitive representations of discriminants of real quadratic orders. For RM, we use exhaustive small-prime data together with $100{,}000$ random polarizations per prime for $227\leq p\leq1619$, testing primitive representations of square-free discriminants $D\leq100$; the first nontrivial primitively represented discriminant is typically small. We apply these methods experimentally in two regimes. For $11\leq p\leq251$, where the irreducible principal polarizations are known exhaustively, the automorphism data recovered from the refined Humbert invariant reproduces the counts of Ibukiyama--Katsura--Oort for the irreducible polarizations. For large parameters, we reach primes of $1000$ bits, where the splitting degrees produced satisfy $\log_2N\approx\log_2p$ with a distribution whose shape is stable across the whole range, so the detected $(N,N)$-isogeny has degree about $p^{2}$; the smallest observed largest prime-power divisor of $N$ remains small up to $250$ bits and increases sharply from $300$ bits onward.
发表机构
- University of Birmingham（伯明翰大学）
- University of Auckland（奥克兰大学）
机构由 AI 辅助整理，请以论文原文为准。
