Cryptology ePrint Archive: Report 2016/869

Cryptographic applications of capacity theory: On the optimality of Coppersmith's method for univariate polynomials

Ted Chinburg and Brett Hemenway and Nadia Heninger and Zachary Scherr

Abstract: We draw a new connection between Coppersmith's method for finding small solutions to polynomial congruences modulo integers and the capacity theory of adelic subsets of algebraic curves. Coppersmith's method uses lattice basis reduction to construct an auxiliary polynomial that vanishes at the desired solutions. Capacity theory provides a toolkit for proving when polynomials with certain boundedness properties do or do not exist. Using capacity theory, we prove that Coppersmith's bound for univariate polynomials is optimal in the sense that there are no auxiliary polynomials of the type he used that would allow finding roots of size $N^{1/d+\epsilon}$ for any monic degree-$d$ polynomial modulo $N$. Our results rule out the existence of polynomials of any degree and do not rely on lattice algorithms, thus eliminating the possibility of improvements for special cases or even superpolynomial-time improvements to Coppersmith's bound. We extend this result to constructions of auxiliary polynomials using binomial polynomials, and rule out the existence of any auxiliary polynomial of this form that would find solutions of size $N^{1/d+\epsilon}$ unless $N$ has a very small prime factor.

Category / Keywords: Coppersmith's method, lattices, polynomial congruences, capacity theory, RSA

Original Publication (in the same form): IACR-ASIACRYPT-2016

Date: received 6 Sep 2016, last revised 10 Sep 2016

Contact author: ted at math upenn edu, fbrett at cis upenn edu, nadiah at cis upenn edu, zscherr at math upenn edu

Available format(s): PDF | BibTeX Citation

Version: 20160910:154759 (All versions of this report)

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