发表机构
Universitat de Lleida（莱里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对素域F_p设计模因算法，在40至51位素数规模下搜索超奇异椭圆曲线，成功发现精确或近超奇异曲线，验证了NMD驱动搜索的有效性。
AI 中文摘要
超奇异椭圆曲线的搜索是基于同源的密码学中的基本计算问题。最近在有限域F_{p^2}上的一种元启发式方法引入了非多重距离（NMD）目标函数，用于测量弗罗贝尼乌斯迹与p的倍数的偏差，结果显示当候选数量超过约10^13时，无信息的随机搜索会失效。本研究针对素域F_p上的元启发式搜索展开研究，该场景适用于CSIDH、OSIDH和SQISign等定向同源协议。尽管候选空间从p^2缩小到p，但超奇异轨迹渐近稀疏（含O(√p log p)条曲线），使得搜索仍具有指数级难度。我们针对F_p设计了一种模因算法，采用一维j不变量染色体、比特级重组、自适应变异，以及在NMD目标下的周期性局部搜索。在40位、46位和51位素数规模（p≈1.13×10^15）下，对30个独立种子的基准测试显示，该算法在46位时可发现精确的超奇异曲线，并始终收敛到“近超奇异”的普通曲线，其弗罗贝尼乌斯迹非常接近零：40位时最佳NMD值为19，51位时为3，对应在哈塞区间上的相对迹偏差分别为1.3×10^-5和4.5×10^-8。这些结果表明，NMD驱动的模因搜索可有效探索稀疏的F_p空间，并系统定位近超奇异结构。
英文摘要
The search for supersingular elliptic curves is a fundamental computational problem in isogeny-based cryptography. A recent metaheuristic formulation over $\mathbb{F}_{p^2}$ introduced the NonMultiplicity Distance (NMD) objective, measuring the deviation of the Frobenius trace from a multiple of $p$, and showed that uninformed random search fails beyond $\approx 10^{13}$ candidates. This work investigates metaheuristic search over the prime field $\mathbb{F}_p$, the setting for oriented isogeny protocols such as CSIDH, OSIDH, and SQISign. Although the candidate space decreases from $p^2$ to $p$, the supersingular locus is asymptotically sparse ($O(\sqrt{p}\log p)$ curves), keeping the search exponentially difficult. We formulate a memetic algorithm tailored to $\mathbb{F}_p$ using a one-dimensional $j$-invariant chromosome, bit-level recombination, adaptive mutation, and periodic local search under the NMD objective. Benchmarks across 30 independent seeds at 40-bit, 46-bit, and 51-bit prime sizes ($p \approx 1.13\times 10^{15}$) show that the algorithm discovers an exact supersingular curve at 46 bits and consistently converges to ``near-supersingular'' ordinary curves with Frobenius traces remarkably close to zero: best NMD values of 19 at 40 bits and 3 at 51 bits, corresponding to relative trace deviations of $1.3\times 10^{-5}$ and $4.5\times 10^{-8}$ across the Hasse interval. These results demonstrate that NMD-driven memetic search effectively navigates the sparse $\mathbb{F}_p$ landscape and systematically locates near-supersingular structures.
