The discrete logarithm problem is a fundamental challenge in number theory with significant implications for cryptographic protocols. In this paper, we investigate the limitations of gradient-based methods for learning the parity bit of the discrete logarithm in finite cyclic groups of prime order. Our main result, supported by theoretical analysis and empirical verification, reveals the concentration of the gradient of the loss function around a fixed point, independent of the logarithm's base used. This concentration property leads to a restricted ability to learn the parity bit efficiently using gradient-based methods, irrespective of the complexity of the network architecture being trained. Our proof relies on Boas-Bellman inequality in inner product spaces and it involves establishing approximate orthogonality of discrete logarithm's parity bit functions through the spectral norm of certain matrices. Empirical experiments using a neural network-based approach further verify the limitations of gradient-based learning, demonstrating the decreasing success rate in predicting the parity bit as the group order increases.

翻译：离散对数问题是数论中的基础难题，对密码协议具有重要影响。本文研究了在素数阶有限循环群中，使用梯度方法学习离散对数奇偶位的局限性。通过理论分析与实验验证，我们的主要结果揭示了损失函数梯度在固定点附近的集中现象——该现象与所使用的对数底数无关。这种集中特性导致无论训练网络架构的复杂度如何，梯度方法均难以高效学习奇偶位。证明过程运用了内积空间中的Boas-Bellman不等式，并通过特定矩阵的谱范数建立了离散对数奇偶位函数的近似正交性。基于神经网络的实证实验进一步验证了梯度学习的局限性，表明随着群阶增大，奇偶位预测的成功率显著降低。