Collision Resistance of Single-Layer Neural Nets

2026-06-02

AI Summary

This work investigates the computational complexity of collision finding in single-layer binary neural networks and its implications for cryptographic collision resistance. By introducing the Overlap Gap Property (OGP) as a rigorous criterion for collision resistance and leveraging tools from random matrix theory together with an analysis of threshold activation functions, the study characterizes the worst-case behavior of the problem within an online algorithmic framework. The main contributions are twofold: when the threshold is significantly smaller than \(1/\sqrt{\alpha}\), an efficient online algorithm for generating collisions is presented; conversely, when the threshold is substantially larger than \(1/\sqrt{\alpha}\), an exponential query lower bound is established, proving that generating many collisions efficiently is infeasible. This delineates a sharp phase transition between regimes where collisions are easy to find and those where the network exhibits strong collision resistance.

Abstract

We initiate the study of the algorithmic complexity of finding collisions in single-layer binary neural networks. Given a random matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$, an input $\mathbf{x} \in \{-1,1\}^n$ is mapped to a binary output vector $\varphi(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$, where $\varphi$ is an activation function with constant behavior on $[κ, \infty)$ for some threshold $κ\geq 0$. We identify the threshold scale $κ=Θ(1/\sqrtα)$, where $α=m/n$, as separating two complementary phenomena. When $κ\ll 1/\sqrtα$, we give a simple online algorithm that efficiently produces extensive collisions. When $κ\gg 1/\sqrtα$, for a natural \emph{randomized} non-periodic activation and suitable oscillation complexity, we prove that the extensive-collision space exhibits an overlap gap property (OGP), yielding an exponential lower bound against online algorithms. Ours is the first work to use the overlap gap property as a rigorous criterion for collision resistance. The key difference between collision finding and average-case search is that collision finding has a new ``worst-case'' aspect: the collision finder has full control over the choice of colliding pairs. Our lower bound is proved in the online model; extending such guarantees to broader classes of algorithms, including spectral, algebraic, lattice-based, or quantum methods, remains an open direction.

Problem

• collision resistance
• single-layer neural networks
• binary neural networks
• overlap gap property
• algorithmic complexity

Innovation

• overlap gap property
• collision resistance
• single-layer neural networks
• binary activation
• online algorithms