Integer factorization via spatial photonic Ising machine using a lattice sieve method.

Integer factorization, a cryptographic cornerstone, remains exponentially hard for classical computers, while quantum approaches, though theoretically polynomial-time, still face hardware limitations. We propose an Ising-accelerated lattice sieve method using a photonic Ising machine to solve the NP-hard closest vector problem efficiently. Our approach requires only sub-linear growth in spin number and stable dynamic range of the interaction matrix as integer bit-length increases, minimizing hardware demands. We construct a programmable spatial photonic Ising machine based on a Dammann grating. Through multi-path simulated annealing experiments on the constructed SPIM, we experimentally achieved the complete factorization of a 32-bit integer and demonstrated the identification of multiple distinct smooth relation pairs required for factorizing a 100-bit integer. To further validate the generality and scalability of the proposed method, we performed classical computer simulations of the multi-path annealing process, achieving factorization of integers up to 120 bits and establishing a benchmark for lattice sieve-based factorization.