Abstract
We develop a framework for solving polynomial equations with size constraints on solutions. We obtain our results by showing how to apply a technique of Coppersmith for finding small solutions of polynomial equations modulo integers to analogous problems over polynomial rings, number fields, and function fields. This gives us a unified view of several problems arising naturally in cryptography, coding theory, and the study of lattices. We give (1) a polynomial-time algorithm for finding small solutions of polynomial equations modulo ideals over algebraic number fields, (2) a faster variant of the Guruswami-Sudan algorithm for list decoding of Reed-Solomon codes, and (3) an algorithm for list decoding of algebraic-geometric codes that handles both single-point and multi-point codes. Coppersmith's algorithm uses lattice basis reduction to find a short vector in a carefully constructed lattice; powerful analogies from algebraic number theory allow us to identify the appropriate analogue of a lattice in each application and provide efficient algorithms to find a suitably short vector, thus allowing us to give completely parallel proofs of the above theorems.
Highlights
Many important problems in areas ranging from cryptanalysis to coding theory amount to solving polynomial equations with side constraints or partial information about the solutions
To extend the method beyond the integers, we examine the analogous structures for polynomial rings, number fields, and function fields
We prove a more general result about finding short vectors under arbitrary non-Archimedean norms, which may have further applications beyond list decoding of algebraic-geometric codes
Summary
Many important problems in areas ranging from cryptanalysis to coding theory amount to solving polynomial equations with side constraints or partial information about the solutions. We show how the ideas of Coppersmith’s theorem can be extended to a more general framework encompassing the original number-theoretic problem, list decoding of Reed-Solomon and algebraic-geometric codes, and the problem of finding solutions to polynomial equations modulo ideals in rings of algebraic integers. Many theorems in number theory and algebraic geometry have parallel versions for the integers and for polynomial rings, or more generally for number fields and function fields, and translating statements or techniques between these settings can lead to valuable insights. Multivariate versions of Coppersmith’s theorem correspond to list decoding of Parvaresh-Vardy and Guruswami-Rudra codes [14]
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