Private key sieves for improved attacks on RSA

Abstract

The RSA cryptosystem has been studied extensively and its insecurity in specific scenarios is well known. For instance, a sufficiently small private key allows to factorize the modulus $N$ and break the system. Attacks like this are often based on lattice reduction for finding small roots of polynomials, but high lattice dimensions can make it difficult to achieve the theoretically proved bounds in practice. In this paper, we adapt a sieve technique originally developed for deterministic integer factorization and apply it to RSA parameters. The sieve allows for an asymptotically significant reduction of the private key search space, and can be used to decrease the bounds for the polynomial roots. We also provide a square-root speed-up of the best currently known runtime bound for factoring semiprime $N=pq$ with small prime difference $q-p$.