Research Article Open Access

Fast Approach to Factorize Odd Integers with Special Divisors

Xingbo Wang1 and Junjian Zhong1
  • 1 Foshan University, China
Journal of Mathematics and Statistics
Volume 16 No. 1, 2020, 24-34

DOI: https://doi.org/10.3844/jmssp.2020.24.34

Submitted On: 4 December 2019
Published On: 27 January 2020

How to Cite: Wang, X. & Zhong, J. (2020). Fast Approach to Factorize Odd Integers with Special Divisors. Journal of Mathematics and Statistics, 16(1), 24-34. https://doi.org/10.3844/jmssp.2020.24.34

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 2αu +1 or 2αu-1 with u being an odd integer and α being a positive integer and the other divisor p satisfies 1 < p ≤ 2α+1 or 2α +1 < p ≤ 2α+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.

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Keywords

  • Cryptography
  • Integer Factorization
  • Binary Tree
  • Algorithm