Maximizing Small Root Bounds by Linearization and Applications to Small Secret Exponent RSA

Abstract

We present an elementary method to construct optimized lattices that are used for finding small roots of polynomial equations. Former methods first construct some large lattice in a generic way from a polynomial f and then optimize via finding suitable smaller dimensional sublattices. In contrast, our method focuses on optimizing f first which then directly leads to an optimized small dimensional lattice.

Using our method, we construct the first elementary proof of the Boneh-Durfee attack for small RSA secret exponents with d ≤ N 0.292. Moreover, we identify a sublattice structure behind the Jochemsz-May attack for small CRT-RSA exponents \(d_p, d_q \leq N^{0.073}\) . Unfortunately, in contrast to the Boneh-Durfee attack, for the Jochemsz-May attack the sublattice does not help to improve the bound asymptotically. Instead, we are able to attack much larger values of d p ,d q in practice by LLL reducing smaller dimensional lattices.