Intractability of Learning the Discrete Logarithm with Gradient-Based Methods

TL;DR

This paper demonstrates that gradient-based methods are fundamentally limited in learning the parity bit of discrete logarithms in finite cyclic groups, due to the concentration of gradients around fixed points, regardless of network complexity.

Contribution

It provides a theoretical and empirical analysis showing the intractability of using gradient-based learning for the discrete logarithm parity problem, highlighting fundamental limitations.

Findings

Gradient concentration around fixed points impedes learning.

Success rate decreases as group order increases.

Limitations hold regardless of neural network complexity.

Abstract

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.