Efficiently representing the integer factorization problem using binary decision diagrams

Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log2(α)3 + 16.5log2(α)2 + 6log2(α) whereas previous work on the subject has only produced systems in which at least one of the polynomials has an ordered binary decision diagram representation with size exponential in log2(α). Using a different approach based on the Chinese remainder theorem we prove that for α ≥ 4 there is an alternative system of boolean equations whose solutions correspond to nontrivial factorizations of α such that there exists a C > 0, independent of α, such that for any order σ on the variables in the system every function in the system can be represented by a σ-OBDD with size less than C log2(log2(α))2log2(α)4 .