Recent results and problems on constructions of linear codes from cryptographic functions
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Recent results and problems on constructions of linear codes from cryptographic functions Nian Li1,2 · Sihem Mesnager3,4,5 Received: 20 September 2019 / Accepted: 8 April 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Linear codes have a wide range of applications in the data storage systems, communication systems, consumer electronics products since their algebraic structure can be analyzed and they are easy to implement in hardware. How to construct linear codes with excellent properties to meet the demands of practical systems becomes a research topic, and it is an efficient way to construct linear codes from cryptographic functions. In this paper, we will introduce some methods to construct linear codes by using cryptographic functions over finite fields and present some recent results and problems in this area. Keywords Bent function · Linear code · Perfect nonlinear function · Plateaued function Mathematics Subject Classification (2010) 12E20 · 94A60 · 94B05
1 Introduction Linear codes are an important class of codes in coding theory and have been extensively studied due to their significant applications in practical systems. How to construct linear
This article is part of the Topical Collection on Boolean Functions and Their Applications IV Guest Editors: Lilya Budaghyan and Tor Helleseth This paper is dedicated to celebrate Prof. Claude Carlet’s 70 birthday. Nian Li
[email protected] Sihem Mesnager [email protected] 1
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China
2
State Key Laboratory of Cryptology, P.O. Box 5159, Beijing 100878, China
3
Department of Mathematics, University of Paris VIII, 93526 Saint-Denis, France
4
University of Paris XIII, Sorbonne Paris Cit´e 93430, LAGA, UMR 7539, CNRS, France
5
France Telecom Paris, 91120 Palaiseau, France
Cryptography and Communications
codes with good parameters is an interesting research problem in this area. The dimension of a code determines the size of the number of codewords and the minimum distance of a code directly determines its error correction capability. In addition, the weight distribution of a code not only characterizes its error correction capability, but also can be used to calculate the error probabilities of the error detecting and error correcting [52]. Thus, it is desirable either to construct linear codes with good parameters or to determine the weight distribution of the codes. Throughout this paper, let Fpm denote the finite field with p m elements, where p is a prime and m is a positive integer. An [n, k, d] linear code C over Fp is a k-dimensional subspace of Fnp with minimum Hamming distance d. Let Ai denote the number of codewords with Hamming weight i in a code C of length n. The weight enumerator of C is defined by 1 + A1 x + · · · + An x n . The sequence (1, A1 , A2 , · · · , An ) is called the weight distribution of C and it gives the minimum distance and the error correcting capability of C . A
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