Continued fractions and RSA with small secret exponent

TL;DR

This paper generalizes classical results on continued fractions and applies them to improve Wiener's attack on RSA with small secret exponents, providing a more comprehensive understanding of the solutions and attack modifications.

Contribution

It extends Legendre's classical result to a broader class of inequalities and applies this to refine RSA attack strategies involving small secret exponents.

Findings

All solutions of |x - a/b| < c/b^2 are characterized via continued fraction convergents.

A modified attack on RSA with small secret exponents is proposed based on the new theoretical insights.

The results offer a deeper understanding of the structure of solutions relevant to cryptanalysis.

Abstract

Extending the classical Legendre's result, we describe all solutions of the inequality |x - a/b| < c/b^2 in terms of convergents of continued fraction expansion of x. Namely, we show that a/b = (rp_{m+1} +- sp_m) / (rq_{m+1} +- sq_m) for some nonnegative integers m,r,s such that rs < 2c. As an application of this result, we describe a modification of Verheul and van Tilborg variant of Wiener's attack on RSA cryptosystem with small secret exponent.