On Codes and Learning With Errors over Function Fields

TL;DR

This paper introduces a function field analogue of the LWE problem, establishing the first search-to-decision reduction for structured codes, and connects lattice-based and code-based cryptography through new theoretical insights.

Contribution

It proposes a novel function field version of LWE, enabling search-to-decision reductions for structured codes, and links lattice and code cryptography using Carlitz extensions.

Findings

First search-to-decision reduction for structured codes

Application to Ring-LPN variants in cryptography

Connections between lattice and code-based cryptography

Abstract

It is a long standing open problem to find search to decision reductions for structured versions of the decoding problem of linear codes. Such results in the lattice-based setting have been carried out using number fields: Polynomial-LWE, Ring-LWE, Module-LWE and so on. We propose a function field version of the LWE problem. This new framework leads to another point of view on structured codes, e.g. quasi-cyclic codes, strengthening the connection between lattice-based and code-based cryptography. In particular, we obtain the first search to decision reduction for structured codes. Following the historical constructions in lattice-based cryptography, we instantiate our construction with function fields analogues of cyclotomic fields, namely Carlitz extensions, leading to search to decision reductions on various versions of Ring-LPN, which have applications to secure multi party computation and to an authentication protocol.