Using evolutionary computation to create vectorial Boolean functions with low differential uniformity and high nonlinearity

The two most important criteria for vectorial Boolean functions used as S-boxes in block ciphers are differential uniformity and nonlinearity. Previous work in this field has focused only on nonlinearity and a different criterion, autocorrelation. In this paper, we describe the results of experiments in using simulated annealing, memetic algorithms, and ant colony optimisation to create vectorial Boolean functions with low differential uniformity.

[1]  Eli Biham,et al.  Differential Cryptanalysis of the Full 16-Round DES , 1992, Annual International Cryptology Conference.

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

[3]  Eli Biham,et al.  Differential Cryptanalysis of the Full 16-Round DES , 1992, CRYPTO.

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

[5]  Joo Yeon Cho,et al.  Linear Cryptanalysis of Reduced-Round PRESENT , 2010, CT-RSA.

[6]  Alex Biryukov,et al.  On Multiple Linear Approximations , 2004, IACR Cryptol. ePrint Arch..

[7]  Susan Stepney,et al.  Searching for cost functions , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[8]  Ross Anderson,et al.  Serpent: A Proposal for the Advanced Encryption Standard , 1998 .

[9]  Darrell Whitley,et al.  A genetic algorithm tutorial , 1994, Statistics and Computing.

[10]  Luca Maria Gambardella,et al.  Ant colony system: a cooperative learning approach to the traveling salesman problem , 1997, IEEE Trans. Evol. Comput..

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

[12]  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).

[13]  Mitsuru Matsui,et al.  The First Experimental Cryptanalysis of the Data Encryption Standard , 1994, CRYPTO.

[14]  Eli Biham,et al.  Enhancing Differential-Linear Cryptanalysis , 2002, ASIACRYPT.

[15]  Susan Stepney,et al.  Evolving Boolean Functions Satisfying Multiple Criteria , 2002, INDOCRYPT.

[16]  John A. Clark,et al.  Almost Boolean functions: the design of Boolean functions by spectral inversion , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[17]  Claude Carlet,et al.  New classes of almost bent and almost perfect nonlinear polynomials , 2006, IEEE Transactions on Information Theory.

[18]  John A. Clark,et al.  Results on Rotation Symmetric Bent and Correlation Immune Boolean Functions , 2004, FSE.

[19]  Marco Dorigo,et al.  AntNet: Distributed Stigmergetic Control for Communications Networks , 1998, J. Artif. Intell. Res..

[20]  Carlos Cotta,et al.  A Primer on Memetic Algorithms , 2012, Handbook of Memetic Algorithms.

[21]  Kaisa Nyberg,et al.  Multidimensional Extension of Matsui's Algorithm 2 , 2009, FSE.

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

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

[24]  Dong Hoon Lee,et al.  Resistance of S-Boxes against Algebraic Attacks , 2004, FSE.

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

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

[27]  Luca Maria Gambardella,et al.  An Ant Colony Optimization Approach to the Probabilistic Traveling Salesman Problem , 2002, PPSN.

[28]  Thomas Stützle,et al.  The Ant Colony Optimization Metaheuristic: Algorithms, Applications, and Advances , 2003 .

[29]  Matthew J. B. Robshaw,et al.  PRINTcipher: A Block Cipher for IC-Printing , 2010, CHES.

[30]  John A. Clark,et al.  Two-Stage Optimisation in the Design of Boolean Functions , 2000, ACISP.

[31]  Alasdair McAndrew.  Data Encryption Standard (DES) for Sage , 2009 .

[32]  Andrey Bogdanov,et al.  PRESENT: An Ultra-Lightweight Block Cipher , 2007, CHES.

[33]  Susan K. Langford,et al.  Differential-Linear Cryptanalysis , 1994, CRYPTO.

[34]  Luca Maria Gambardella,et al.  Ant Algorithms for Discrete Optimization , 1999, Artificial Life.

[35]  L. Darrell Whitley,et al.  A Comparison of Genetic Sequencing Operators , 1991, ICGA.

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

[37]  Meiqin Wang,et al.  Algebraic Techniques in Differential Cryptanalysis Revisited , 2011, ACISP.

[38]  Pablo Moscato,et al.  On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts : Towards Memetic Algorithms , 1989 .

[39]  Gregor Leander,et al.  On the Classification of 4 Bit S-Boxes , 2007, WAIFI.

[40]  Nicolas Courtois,et al.  How Fast can be Algebraic Attacks on Block Ciphers ? , 2006, IACR Cryptol. ePrint Arch..

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

[42]  I ScottKirkpatrick.  Optimization by Simulated Annealing: Quantitative Studies , 1984 .

[43]  Marco Dorigo,et al.  Ant system: optimization by a colony of cooperating agents , 1996, IEEE Trans. Syst. Man Cybern. Part B.