Analysis of affine equivalent boolean functions for cryptography

Analysis of Affine Equivalent Boolean Functions for Cryptography by Joanne Elizabeth Fuller Bachelor of Applied Science (Mathematics), 1998 Bachelor of Information Technology (Honours), 1999 Thesis submitted in accordance with the regulation for Degree of Doctor of Philosophy Information Security Research Centre Faculty of Information Technology Queensland University of Technology December, 2003 i ii Keywords Boolean functions, affine transformation, equivalence class, local connectivity, nonlinearity, algebraic order, autocorrelation, S-boxes, Advanced Encryption Standard iii iv Abstract Boolean functions are an important area of study for cryptography These functions, consisting merely of one’s and zero’s, are the heart of numerous cryptographic systems and their ability to provide secure communication Boolean functions have application in a variety of such systems, including block ciphers, stream ciphers and hash functions The continued study of Boolean functions for cryptography is therefore fundamental to the provision of secure communication in the future This thesis presents an investigation into the analysis of Boolean functions and in particular, analysis of affine transformations with respect to both the design and application of Boolean functions for cryptography Past research has often been limited by the difficulties arising from the magnitude of the search space The research presented in this thesis will be shown to provide an important step towards overcoming such restrictions and hence forms the basis for a new analysis methodology The new perspective allows a reduced view of the Boolean space in which all Boolean functions are grouped into connected equivalence classes so that only one function from each class need be established This approach is a significant development in Boolean function research with many applications, including class distinguishing, class structures, self mapping analysis and finite field based s-box analysis The thesis will begin with a brief overview of Boolean function theory; including an introduction to the main theme of the research, namely the affine transformation This will be followed by the presentation of a fundamental new theorem describing the connectivity that exists between equivalence classes The theorem of connectivity will form the foundation for the remainder of the research presented in this thesis A discussion of efficient algorithms for the manipulation of Boolean functions will then be presented The ability of Boolean function research to achieve new levels of analysis and understanding is centered on the availability of computer based programs that can perform various manipulations The development and optimisation of efficient algorithms specifically for execution on a computer will be shown to have a considerable advantage compared to those constructed using a more traditional approach to algorithm optimisation The theorem of connectivity will be shown to be fundamental in the provision of v many avenues of new analysis and application These applications include the first non-exhaustive test for determining equivalent Boolean functions, a visual representation of the connected equivalence class structure to aid in the understanding of the Boolean space and a self mapping constant that enables enumeration of the functions in each equivalence class A detailed survey of the classes with six inputs is also presented, providing valuable insight into their range and structure This theme is then continued in the application Boolean function construction Two important new methodologies are presented; the first to yield bent functions and the second to yield the best currently known balanced functions of eight inputs with respect to nonlinearity The implementation of these constructions is extremely efficient The first construction yields bent functions of a variety of algebraic order and inputs sizes The second construction provides better results than previously proposed heuristic techniques Each construction is then analysed with respect to its ability to produce functions from a variety of equivalence classes Finally, in a further application of affine equivalence analysis, the impact to both s-box design and construction will be considered The effect of linear redundancy in finite field based s-boxes will be examined and in particular it will be shown that the AES s-box possesses complete linear redundancy The effect of such analysis will be discussed and an alternative construction to s-box design that ensures removal of all linear redundancy will be presented in addition to the best known example of such an s-box vi Contents Certificate Recommending Acceptance i Keywords iii Abstract v List of Figures xi List of Tables xiii Declaration xvii Published Papers xix Introduction 1.1 Aims and Outcomes of Thesis 1.2 Overview of Thesis Preliminaries 2.1 Representation 2.1.1 Truth Tables 2.1.2 Algebraic Normal Form The Walsh-Hadamard Transform 10 2.2.1 Nonlinearity 12 2.2.2 Correlation Immunity and Resilience 12 2.2.3 Subfunction Hamming Weight 14 Autocorrelation 16 2.3.1 The Propagation Criteria 18 2.4 Bent Functions 19 2.5 Affine Transformations 20 2.2 2.3 vii 2.6 2.5.1 Equivalence Classes 20 2.5.2 Invariance Analysis 22 2.5.3 Local Connectivity 22 Conclusion 25 Tools for Efficient Boolean Function Analysis 3.1 27 General Optimisation Issues 28 3.1.1 Algorithm Development 28 3.1.2 Operation Minimisation 29 3.1.3 ModularProgramming 30 Implementation Issues 31 3.2.1 Boolean Function Structures 31 3.2.2 The Algebraic Normal Form 32 3.2.3 The Walsh-Hadamard Transform 34 3.2.4 The Autocorrelation Function 36 3.3 A Survey of Boolean Functions of Five Inputs 37 3.4 Conclusion 37 3.2 Analysis of Affine Equivalent Boolean Functions 4.1 4.2 4.3 4.4 4.5 39 Distinguishing Affine Equivalence Classes 40 4.1.1 Basic Class Distinguishing Properties 40 4.1.2 m-step Analysis 42 4.1.3 Identifying the Affine Transform 43 4.1.4 Experimental Analysis 45 Equivalence Class Structures 45 4.2.1 Exploration of the Class Structure 46 4.2.2 A Visual Representation 47 4.2.3 Bent Function Analysis 52 Self Mappings 55 4.3.1 Self Mapping Analysis 58 4.3.2 Counting Boolean Functions 59 A Survey of Boolean Functions of Six Inputs 60 4.4.1 Local and Global Maxima 60 4.4.2 Highly Nonlinear and Balanced Boolean Functions 61 4.4.3 Correlation Immune Boolean Functions 62 Conclusion 63 viii Appendix J Power Mapping Classes Appendix J provides a listing of the specific polynomials and powers corresponding to each possible equivalence class for s-boxes generated using power mappings Table J.1: × Finite Field Power Mappings # (nl,ord,acmax) powers for each valid MOD 0,1,256 0x3,0x7,0xF,0x1F,0x3F,0x7F,0xFF 80,4,112 0x2A,0x55,0x58,0x64,0x94,0xAB,0xB1,0xC9 96,3,64 0x6,0xD,0x1B,0x24,0x28,0x37,0x48,0x49, 0x51,0x6F,0x82,0x91,0x93,0xA3,0xC0,0xDF 96,3,64 0xA,0xC,0x15,0x19,0x2B,0x33,0x42,0x57, 0x60,0x67,0x84,0x85,0xA0,0xAF,0xC1,0xCF 96,4,64 0x16,0x2D,0x5B,0x70,0x8A,0xB7,0xC4,0xE1 96,4,64 0x34,0x4C,0x52,0x69,0x99,0xA5,0xA8,0xD3 96,4,96 0x1C,0x39,0x46,0x73,0x8D,0xA2,0xD0,0xE7 96,5,64 0x3A,0x66,0x75,0x9C,0xB2,0xCD,0xD8,0xEB 96,5,96 0x3C,0x4E,0x79,0x9D,0xA6,0xD2,0xE8,0xF3 10 104,3,128 0x12,0x25,0x30,0x4B,0x61,0x88,0x97,0xC3 11 104,5,56 0x2E,0x5D,0x78,0x96,0xBB,0xCA,0xE4,0xF1 12 112,5,48 0x5A,0x6A,0x6C,0xAC,0xB4,0xB5,0xD5,0xD9 13 112,5,56 0x1E,0x3D,0x7B,0x8E,0xC6,0xE2,0xF0,0xF7 14 112,7,32 0x7E,0xBE,0xDE,0xEE,0xF6,0xFA,0xFC,0xFD 151 152 Appendix K Replacement S-boxes The following s-box was generated from the two-step tweak algorithm It has nonlinearity 106, algebraic order 7, contains no fixed points and no linear redundancy The sbox has a DDT of The distribution of properties over all 255 XOR combinations of the sbox output functions is as follows Table K.1: Frequency Distribution of Sbox Properties nonlinearity frequency maximum autocorrelation frequency 106 32 108 76 40 93 110 147 48 134 112 24 56 27 SBox[256]={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} 153 The following s-box was generated from the four-step tweak algorithm It has nonlinearity 108, algebraic order 7, contains no fixed points and no linear redundancy The s-box is also a self inverse The distribution of properties over all 255 bα are as follows Table K.2: Distribution of S-box Properties nonlinearity frequency maximum autocorrelation frequency 108 78 32 110 146 40 104 112 30 48 125 56 18 SBox[256]={ 01,00,BD,D6,E3,9B,6B,7E,CC,2A,F0,64,88,63,3F,A0, 66,60,15,79,78,12,32,21,44,2C,8C,CF,A2,F5,50,39, 33,17,30,68,B7,43,81,53,3C,7B,09,CD,19,45,AD,D1, 22,69,16,20,46,FE,84,57,51,1F,C7,BA,28,7A,A1,0E, A4,D3,B6,25,18,2D,34,FF,E6,FA,9C,EC,FD,BF,94,71, 1E,38,80,27,B9,C9,DB,37,B1,E4,9F,DA,DC,76,D5,83, 11,67,89,0D,0B,F1,10,61,23,31,7F,06,6D,6C,96,E8, 95,4F,B2,90,DE,EE,5D,EA,14,13,3D,29,ED,92,07,6A, 52,26,D4,5F,36,9E,AF,A7,0C,62,AB,D8,1A,CE,C2,99, 73,B3,7D,EB,4E,70,6E,E9,C3,8F,E2,05,4A,DD,85,5A, 0F,3E,1C,F4,40,D2,AE,87,E1,F6,D9,8A,D0,2E,A6,86, E5,58,72,91,F2,C0,42,24,C8,54,3B,C6,D7,02,FC,4D, B5,F3,8E,98,F9,CA,BB,3A,B8,55,C5,F8,08,2B,8D,1B, AC,2F,A5,41,82,5E,03,BC,8B,AA,5B,56,5C,9D,74,EF, F7,A8,9A,04,59,B0,48,FB,6F,97,77,93,4B,7C,75,DF, 0A,65,B4,C1,A3,1D,A9,E0,CB,C4,49,E7,BE,4C,35,47 } 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Outcomes of Thesis This thesis presents a study of Boolean functions and in particular, analysis of an affine transformation with respect to both the design and application of Boolean functions for cryptography. .. and analysis of Boolean functions for cryptography The level of security achieved in applications based on Boolean functions is measured by the quality of combinatorial properties within the functions. .. Boolean functions have application in a variety of such systems, including block ciphers, stream ciphers and hash functions The continued study of Boolean functions for cryptography is therefore