有限域上Markoff图中路径查找的启发式子指数算法
A Heuristic Subexponential Algorithm to Find Paths in Markoff Graphs Over Finite Fields
AI总结：
针对Markoff图在哈希函数构造中的路径查找难题，提出启发式子指数算法，证明在两个随机性假设下，该问题难度等价于分解q-1及求解三个有限域离散对数问题。
摘要：
Charles、Goren与Lauter[《密码学杂志》22(1)，2009]阐述了如何利用扩张图构造哈希函数，这类图中指定顶点间的路径难以查找。有限域$\u211d_q$上经典Markoff方程$X^2+Y^2+Z^2=XYZ$的解集具有自然的三分图结构，通过三个非交换多项式自同构连接各点。据推测这类图构成扩张图族，Fuchs、Lauter、Litman与Tran[《数学密码学》1(1)，2022]提出将该Markoff图族用于CGL构造。本注记表明，在两个随机性假设下，无论从理论还是实际角度，$\u211d_q$上Markoff图的路径问题都可在次指数时间内求解，其难度大致与分解$q-1$以及在$\u211d_q^*$中求解三个离散对数问题相当。
英文摘要：
Charles, Goren, and Lauter [J. Cryptology 22(1), 2009] explained how one can construct hash functions using expander graphs in which it is hard to find paths between specified vertices. The set of solutions to the classical Markoff equation $X^2+Y^2+Z^2=XYZ$ in a finite field $\mathbb{F}_q$ has a natural structure as a tri-partite graph using three non-commuting polynomial automorphisms to connect the points. These graphs conjecturally form an expander family, and Fuchs, Lauter, Litman, and Tran [Mathematical Cryptology 1(1), 2022] suggest using this family of Markoff graphs in the CGL construction. In this note we show that in both a theoretical and a practical sense, assuming two randomness hypotheses, the path problem in a Markoff graph over $\mathbb{F}_q$ can be solved in subexponential time, and is more-or-less equivalent in difficulty to factoring $q-1$ and solving three discrete logarithm problem in $\mathbb{F}_q^*$.
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