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.

翻译：我们首次研究了单层二值神经网络中寻找碰撞问题的算法复杂度。给定随机矩阵 $\mathbf{A} \in \mathbb{R}^{m\times n}$，输入 $\mathbf{x} \in \{-1,1\}^n$ 被映射为二值输出向量 $\varphi(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$，其中 $\varphi$ 为激活函数，其在阈值 $κ\geq 0$ 上具有区间 $[κ, \infty)$ 内的常值行为。我们确定阈值尺度 $κ=Θ(1/\sqrtα)$（其中 $α=m/n$）可作为区分两种互补现象的分界点。当 $κ\ll 1/\sqrtα$ 时，我们提出一种简单的在线算法，可高效生成大规模碰撞。当 $κ\gg 1/\sqrtα$ 时，对于自然的\emph{随机化}非周期激活函数和适当的振荡复杂度，我们证明该大规模碰撞空间具有重叠间隙性质（OGP），从而得到针对在线算法的指数级下界。本文首次将重叠间隙性质作为抗碰撞性的严格判据。碰撞寻找与平均情形搜索的关键区别在于，碰撞寻找具有新的“最坏情形”特征：碰撞寻找者可完全控制碰撞对的选择。我们的下界在在线模型下得到证明；将该保证推广至更广泛算法类别（包括谱方法、代数方法、格基方法或量子方法）仍是待探索的研究方向。