Rational Base Descent: A Deterministic Algorithm for Factoring Structured Semiprimes

TL;DR

This paper introduces a deterministic algorithm tailored for factoring certain structured semiprimes where one prime factor follows a specific exponential form, achieving efficient factorization under these conditions.

Contribution

The paper presents a novel, special-purpose algorithm for factoring structured semiprimes with a new search strategy and complexity analysis, including a practical Python implementation.

Findings

Algorithm isolates factors in O(log^3 N) time when parameters are correct.

The method is efficient for semiprimes with prime factors following a specific exponential pattern.

The approach poses no threat to balanced RSA semiprimes.

Abstract

We present a special-purpose algorithm for factoring semiprimes $N = pq$ in which one prime factor satisfies $p \approx c\,(a/b)^n$ for positive integers $a, b, c, n$ with $a > b$ and $\gcd(a,b) = 1$. Given the correct parameters $(a, b)$, the algorithm isolates a factor in ${O}(\log^3 N)$ time when $a/b$ is bounded away from $1$, and the cofactor $q$ is unconstrained beyond a mild size bound. We describe a search strategy over $(a, b)$ using primitivity filters, give a complexity analysis showing that the method poses no threat to balanced RSA semiprimes, and provide a gmpy2-based Python implementation.