Semiprime Factorization with Discrete Diffusion and Beam Search: Scaling Laws

Brendan O Donoghue, Nenad Tomasev, Volodymyr Mnih, Timothy Nguyen, Dalin Guo +10

We demonstrate that transformer architectures trained with discrete denoising diffusion and decoded via guided test-time beam search can learn to recover the prime factors of semiprime integers directly from data. Our approach establishes a new frontier for learning-based semiprime factorization, successfully factoring 68-bit integers-a substantial advance over the prior 56-bit state of the art. While computationally less efficient than classical factoring algorithms, our model requires no domain-specific number-theoretic priors and learns purely from training examples. In practice, the diffusion model acts as a generative prior, identifying promising factor candidates where exact verification or lattice-based post-processing can complete the factorization. Furthermore, we derive empirical scaling laws relating approximation quality to training data volume, model size, and test-time computation. In contrast to language, problem size is the key driver of scaling behavior. Achieving a fixed accuracy on a held-out test set requires compute that scales exponentially with bit length. Consequently, our findings pose no threat to security, as the projected compute cost required to…