Hidden Number Problem Given Bound of Secret

Abstract

In this paper, we study the HNP problem only with constant Oracle queries, and transform the problem into solving the singular variable modular inequality (tx-u) modN>, basing on lattice techniques given the bound X of the hidden number x. Using these Oracle queries, we can successfully recover the secret with a sufficient large probability. Furthermore, as an application, we analyze the security of Schnorr signature scheme in the Random Oracle model, and give the effective reduction from security of the scheme to bits predictability of its nonce.