On Codes and Learning with Errors over Function Fields

• Published in: Advances in Cryptology – CRYPTO 2022 pp. 513 - 540
• Main Authors: Bombar, Maxime, Couvreur, Alain, Debris-Alazard, Thomas
• Format: Book Chapter
• Language: English
• Published: Cham Springer Nature Switzerland
• Series: Lecture Notes in Computer Science
• Subjects: Carlitz modules; Code-based cryptography; Function fields; Search to decision reductions
• ISBN: 9783031159787; 3031159780
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-031-15979-4_18

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{LWE}$$\end{document}, Module–LWE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{LWE}$$\end{document} and so on. We propose a function field version of the LWE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{LWE}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{LPN}$$\end{document}, which have applications to secure multiparty computation and to an authentication protocol.