We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations.

翻译：我们研究一种整数因数分解的混合计算模型，其中唯一的非经典资源是访问有限图上的一个迭代扩散过程。具体而言，一个扩散步骤定义为将一个对称随机矩阵（半惰性游走算子）应用于一个ℓ¹归一化状态向量，随后可选择性地读取选定坐标的值。设N≥3为一个既非素数也非素数幂的奇数，并设b∈(ℤ/Nℤ)*具有奇数乘法阶r=ord_N(b)。我们在不预先知道r的情况下构造一个加权凯莱图，其顶点集为循环子群⟨b⟩，其边对应于t≤⌊log₂N⌋+1的幂b^(±2^t)。通过使用显式谱分解结合一个基本的倍增引理，我们证明r可以在最多O((log₂N)²)次扩散步骤后从单个热核值中恢复，并给出有效界。随后，我们将此阶寻找模型与从因数分解到阶寻找的标准归约（遵循Shor算法的框架）相结合，得到一个随机化因数分解过程，其成功概率仅取决于N的不同素因子个数m。我们与Shor算法的比较是概念性和基于模型的。我们用马尔可夫ℓ¹演化取代了酉ℓ²演化，并以两种成本度量报告复杂度：数字步骤和扩散步骤。最后，我们提供了示例说明并讨论了实际实现问题。