Factorization using binary decision diagrams

Abstract

We address the factorization problem in this paper: Given an integer $N=pq$, find two factors $p$ and $q$ of $N$ such that $p$ and $q$ are of same bit-size. When we say integer multiplication of $N$, we mean expressing $N$ as a product of two factors $p$ and $q$ such that $p$ and $q$ are of same bit-size. We work on this problem in the light of Binary Decision Diagrams (BDD). A Binary Decision Diagram is an acyclic graph which can be used to represent Boolean functions. We represent integer multiplication of $N$ as product of factors $p$ and $q$ using a BDD. Using various operations on the BDD we present an algorithm for factoring $N$. All calculations are done over $GF(2)$. We show that the number of nodes in the constructed BDD is $\mathcal {O}(n^{3})$ where $n$ is the number of bits in $p$ or $q$. We do factoring experiments for the case when $p$ and $q$ are primes as in the case of RSA modulus $N$, and report on the observed complexity. The multiplication of large RSA numbers (that cannot be factored fast in practice) can still be easily represented as a BDD.