Paper 2015/642 Original: eprint.iacr.org/2015/642 ↗

A New Partial Key Exposure Attack on Multi-power RSA

Abstract

An important attack on multi-power RSA ($N=p^rq$) was introduced by Sarkar in 2014, by extending the small private exponent attack of Boneh and Durfee on classical RSA. In particular, he showed that $N$ can be factored efficiently for $r=2$ with private exponent $d$ satisfying $d<N^{0.395}$. In this paper, we generalize this work by introducing a new partial key exposure attack for finding small roots of polynomials using Coppersmith's algorithm and Gröbner basis computation. Our attack works for all multi-power RSA exponents $e$ (resp. $d$) when the exponent $d$ (resp. $e$) has full size bit length. The attack requires prior knowledge of least significant bits (LSBs), and has the property that the required known part of LSB becomes smaller in the size of $e$. For practical validation of our attack, we demonstrate several computer algebra experiments.