Reversed genetic algorithms for generation of bijective s-boxes with good cryptographic properties

Often the S-boxes are the only nonlinear components in a block cipher and as such play an important role in ensuring its resistance to cryptanalysis. Cryptographic properties and constructions of S-boxes have been studied for many years. The most common techniques for constructing S-boxes are: algebraic constructions, pseudo-random generation and a variety of heuristic approaches. Among the latter are the genetic algorithms. In this paper, a genetic algorithm working in a reversed way is proposed. Using the algorithm we can rapidly and repeatedly generate a large number of strong bijective S-boxes of each dimension from (8 × 8) to (16 × 16), which have sub-optimal properties close to the ones of S-boxes based on finite field inversion, but have more complex algebraic structure and possess no linear redundancy.

[1]  Hideki Imai,et al.  Relating Differential Distribution Tables to Other Properties of of Substitution Boxes , 2000, Des. Codes Cryptogr..

[2]  Eli Biham,et al.  On Matsui's Linear Cryptanalysis , 1994, EUROCRYPT.

[3]  Amr M. Youssef,et al.  On Some Algebraic Structures in the AES Round Function , 2002, IACR Cryptol. ePrint Arch..

[4]  Pulak Mishra,et al.  Mergers, Acquisitions and Export Competitive- ness: Experience of Indian Manufacturing Sector , 2012 .

[5]  Yongqiang Li,et al.  Constructing differential 4-uniform permutations from know ones , 2011, IACR Cryptol. ePrint Arch..

[6]  Vincent Rijmen,et al.  The Design of Rijndael: AES - The Advanced Encryption Standard , 2002 .

[7]  Yin Tan,et al.  More constructions of differentially 4-uniform permutations on F22k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\m , 2013, Designs, Codes and Cryptography.

[8]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[9]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[10]  Eli Biham,et al.  Differential cryptanalysis of DES-like cryptosystems , 1990, Journal of Cryptology.

[11]  Neil J. A. Sloane,et al.  The theory of error-correcting codes (north-holland , 1977 .

[12]  Jovan Dj. Golic,et al.  Fast Low Order Approximation of Cryptographic Functions , 1996, EUROCRYPT.

[13]  Jakub Töpfer.  Links Between Differential and Linear Cryptanalysis , 2015 .

[14]  Claude Carlet,et al.  Boolean Functions for Cryptography and Error-Correcting Codes , 2010, Boolean Models and Methods.

[15]  Willi Meier,et al.  Nonlinearity Criteria for Cryptographic Functions , 1990, EUROCRYPT.

[16]  Dominique De Werra.  Boolean Models and Methods in Mathematics, Computer Science, and Engineering , 2010, Boolean Models and Methods.

[17]  Roman Oliynykov,et al.  A Method For Generation Of High-Nonlinear S-Boxes Based On Gradient Descent , 2013, IACR Cryptol. ePrint Arch..

[18]  P. Tesar.  A New Method for Generating High Non-linearity S-Boxes , 2010 .

[19]  Mitsuru Matsui,et al.  Linear Cryptanalysis Method for DES Cipher , 1994, EUROCRYPT.

[20]  Bart Preneel,et al.  Cryptographic hash functions , 2010, Eur. Trans. Telecommun..

[21]  William Millan.  Low Order Approximation of Cipher Functions , 1995, Cryptography: Policy and Algorithms.

[22]  Kaisa Nyberg,et al.  Perfect Nonlinear S-Boxes , 1991, EUROCRYPT.

[23]  Alexandr Kuznetsov,et al.  The Design of Boolean Functions by Modified Hill Climbing Method , 2009, 2009 Sixth International Conference on Information Technology: New Generations.

[24]  Louis Goubin,et al.  Impact of Sboxes Size upon Side Channel Resistance and Block Cipher Design , 2013, AFRICACRYPT.

[25]  Claude Carlet,et al.  Vectorial Boolean Functions for Cryptography , 2006 .

[26]  Susan Stepney,et al.  The design of S-boxes by simulated annealing , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[27]  Joos Vandewalle,et al.  Correlation Matrices , 1994, FSE.

[28]  J. Seberry,et al.  Relationships among nonlinearity criteria , 1994 .

[29]  Yin Tan,et al.  More Constructions of Differentially 4-uniform Permutations on $\gf_{2^{2k}}$ , 2013, ArXiv.

[30]  Kaisa Nyberg,et al.  Differentially Uniform Mappings for Cryptography , 1994, EUROCRYPT.

[31]  Josef Pieprzyk,et al.  Cryptanalysis of Block Ciphers with Overdefined Systems of Equations , 2002, ASIACRYPT.

[32]  Yin Tan,et al.  Constructing Differentially 4-Uniform Permutations Over ${\BBF}_{2^{2k}}$ via the Switching Method , 2013, IEEE Transactions on Information Theory.

[33]  María Naya-Plasencia,et al.  Block Ciphers That Are Easier to Mask: How Far Can We Go? , 2013, CHES.

[34]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

[35]  Vincent Rijmen,et al.  The Design of Rijndael , 2002, Information Security and Cryptography.

[36]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[37]  Jennifer Seberry,et al.  Systematic generation of cryptographically robust S-boxes , 1993, CCS '93.

[38]  William Millan,et al.  How to Improve the Nonlinearity of Bijective S-Boxes , 1998, ACISP.

[39]  William Millan,et al.  Evolutionary Heuristics for Finding Cryptographically Strong S-Boxes , 1999, ICICS.

[40]  William Millan,et al.  Linear Redundancy in S-Boxes , 2003, FSE.