Fast Approach to Factorize Odd Integers with Special Divisors

Abstract

The paper proves that an odd composite integer N can be factorized in O((log₂N)⁴) bit operations if N = pq, the divisor q is of the form 2^αu +1 or 2^αu-1 with u being an odd integer and α being a positive integer and the other divisor p satisfies 1 < p ≤ 2^α+1 or 2^α +1 < p ≤ 2^α+1-1. Theorems and corollaries are proved with detail mathematical reasoning. Algorithm to factorize the odd composite integers is designed and tested in Maple. The results in the paper demonstrate that fast factorization of odd integers is possible with the help of valuated binary tree.