Can we factor an integer $$N$$ unconditionally, in deterministic polynomial time, given the value of its Euler totient $$\varphi (N)$$ ? We show that this can be done under certain size conditions on prime factors . The key technique is lattice basis reduction using LLL algorithm. Among our results, if has a $$p > \sqrt{N}$$ , then recover $$p$$ time also shed some light analogous factorization...

Prime factoring is important for multiple reasons. One is its central role in several cryptographic schemes. The central element of factoring is to find two prime numbers p, q where n = p ∗ q, given the value of n. A mathematical expression is developed that is amenable to iterative resolution of prime factors for a composite. The expression is then shown to admit a factoring process of P-compl...

We describe a modification to the well-known large prime variant of the multiple polynomial quadratic sieve factoring algorithm. In practice this leads to a speed-up factor of 2 to 2.5. We discuss several implementation-related aspects, and we include some examples. Our new variation is also of practical importance for the number field sieve factoring algorithm. 1. Factoring with two large prim...

We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of fac...

Factoring Boolean functions is one of the basic operations in algorithmic logic synthesis. Current algorithms for factoring Boolean functions are based on some kind of division (Boolean or algebraic). In this paper, we present an algorithm for factoring that uses graph partitioning rather than division. Our algorithm is recursive and operates on the function and on its dual, to obtain the bette...