Generalized Implicit Factorization Problem

TL;DR

This paper introduces a lattice-based algorithm for a generalized implicit factorization problem involving RSA moduli with shared bits at unknown positions, expanding previous work on shared bits in prime factors.

Contribution

It proposes a novel lattice-based approach for the generalized problem where shared bits are at different, unknown positions, and analyzes its efficiency with experimental validation.

Findings

Algorithm effectively handles unknown shared bit positions

Efficiency depends on specific conditions of shared bits

Experimental results support theoretical analysis

Abstract

The Implicit Factorization Problem was first introduced by May and Ritzenhofen at PKC'09. This problem aims to factorize two RSA moduli $N_1=p_1q_1$ and $N_2=p_2q_2$ when their prime factors share a certain number of least significant bits (LSBs). They proposed a lattice-based algorithm to tackle this problem and extended it to cover $k>2$ RSA moduli. Since then, several variations of the Implicit Factorization Problem have been studied, including the cases where $p_1$ and $p_2$ share some most significant bits (MSBs), middle bits, or both MSBs and LSBs at the same position. In this paper, we explore a more general case of the Implicit Factorization Problem, where the shared bits are located at different and unknown positions for different primes. We propose a lattice-based algorithm and analyze its efficiency under certain conditions. We also present experimental results to support our analysis.