University of Wollongong Research Online University of Wollongong Thesis Collection University of Wollongong Thesis Collections 2000 Boolean functions in cryptography Cheng-Xin Qu University of Wollongong Recommended Citation Qu, Cheng-Xin, Boolean functions in cryptography, Doctor of Philosophy thesis, Department of Computer Science, University of Wollongong, 2000. http://ro.uow.edu.au/theses/1292 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] NIVERSITY u OF w OLLONGONG oolean Functions In Cryptography i A thesis submitted in fulfillment of the requirements for the award of the degree Doctor of Philosophy from UNIVERSITY OF WOLLONGONG by Cheng-Xin QU Computer Science Department November 2000 / © Copyright 2000 by Cheng-Xin QU All Rights Reserved n Dedicated to my mother, wife and son. iii Declaration This is to certify that the work reported in this thesis was done •f by the author, unless specified otherwise, and that no part of it has been submitted in a thesis to any other university or similar institution. Cheng-Xin QU November 29, 2000 iv Abstract This thesis is about Boolean functions and their cryptographic properties. Two kinds of Boolean functions are discussed - balanced functions and bent functions. In addition to surveying recent activities of research into Boolean functions, a new representation of bent functions - degree-3 homogeneous bent functions are discovered. The complete 7 set of degree-3 homogeneous bent functions on the lowest dimension Boolean spaces V6 is given. By using bent functions, some ways to construct highly nonlinear balanced Boolean functions are shown in this thesis, which yield a new property of bent func­ tions. The structure of degree-3 highly nonlinear homogeneous balanced functions is also discussed. These results are based on computer searching. The theory of sym­ metric groups is applied in the research. In this study symmetric groups are applied to Boolean functions. Any Boolean function on Vn has its own symmetric properties associated with the symmetric group Sn. The relations between Boolean functions and symmetric groups are highlighted. This may lead to a new way to design good S-boxes by using an additive group of Boolean functions which is a subset of the function group generated by the symmetric group. Because good symmetric properties have the poten­ tial to be faster for implementation, the applications of homogeneous Boolean functions taken as rotation functions are discussed. Bent-like balanced functions are very good candidates of Boolean functions for good S-box design. In a degree-3 homogeneous bent or balanced Boolean function, each term is considered as a three variety block. Then it is found that the homogeneous Boolean function is tightly related with block designs BIBD and PBIBD. So in this thesis, the method of combinatorial block designs to discuss Boolean functions is also used. The connection of symmetric group theory with Boolean functions is established. v Publications During the study, the author, in cooperation with supervisors and colleagues, has published and submitted some papers. There is the list of them showing how much work the author did in these papers. 1. J. Pieprzyk and C. Qu, Rotation-symmetric functions and fast hashing, Infor­ mation Security and Privacy - ACISP'98, Lecture Note in Computer Science, Springer-Verlag, Berlin Heidelberg New York Toyko, 1438:169-180, 1998. — This paper shows the symmetric properties of Boolean functions in fast im­ plementations. The author did about 50 percent of the work. 2. C. Qu and J. Seberry and J. Pieprzyk, On the symmetric properties of homoge­ neous Boolean functions, Information Security and Privacy - ACISP'99, Lecture Note in Computer Science, Springer-Verlag, Berlin Heidelberg New York Toyko, 1587:26-35, 1999. — In this paper, the symmetric properties of homogeneous Boolean functions are studied. The author did about 80 percent of the work. 3. J. Pieprzyk and C. Qu, Fast hashing and rotation-symmetric functions, Journal of Universal Computer Science, 5.1:20-31, 1999. — In this paper further study was taken on the symmetric properties of Boolean functions in fast implementations. The author did about 50 percent of the work. 4. C. Qu and J. Seberry and J. Pieprzyk, Homogeneous bent functions, Discrete Applied Mathematics, 102:133-139, 2000. — In this paper the bent functions that do not contain any quadratic terms were discovered. Block designs were involved to analyze homogeneous bent functions. The author did 80 percent of the work. (Note: The paper was finished in 1998 and was accepted in 1999.) vi 5. C. Qu, J. Seberry and J. Pieprzyk, Construction of highly nonlinear balanced Boolean, functions, (Submit to ASIACRYPT'2000). — This paper gave a new property of bent sequences and shown a few ways to construct highly nonlinear balanced Boolean functions. The author did 80 percent of the work. 6. C. Qu, J. Seberry and J. Pieprzyk, Relationships between Boolean functions and symmetric groups, (Accepted by ICS'2000), Taiwan, 2000. — In this paper symmetric group theory was applied for studying homogeneous Boolean functions and some relations between Boolean functions and symmetric groups were established. The author did about 80 percent of the work. 7. J. Seberry, T. Xia, C. Qu and J. Pieprzyk, Construction of highly non-linearity cubic homogeneous Boolean functions on GF(2)2n+l and their properties, (Sub­ mitted to Designs, Codes, and Cryptography), 1999. — In this paper, The author did about 25 percent of the work. vn Symbols Galois field with parameter 2 An n entry boolean space ) A vector in Vn Nonlinearity of the boolean function f(x) on Vn An affine boolean function on Vn A sequence of a linear boolean function Hamming weight, the number of Is in the sequence £ Hamming weight, the number of solutions of f(x) = l over Vn ) A variable vector in the boolean space Vn A boolean function on Vn A boolean function on the subspace Vs(c Vn) The Walsh-Hadamard transform of a boolean function f(x) on The 2n x 2" Sylvester-Hadamard matrix n Sequences (binary or ±1) with length 2 on Vn Boolean addition, 101=0, 100 = 1, 0©0 = 0 S-box, a mapping S(x) : Vn —> Vm The dual space of a subspace Vm C Vn Subset of Vn The dual set of the subset E The number -of all vectors in the space Vn Set The A;-th order propagation criteria Balanced incomplete block design Partial balanced incomplete block design vm Acknowledgements I wish to acknowledge the help of my supervisor Professor Jennifer Seberry whose knowledge, patience and enthusiasm have been a driving force behind this work. She devoted many hours to my studies not only on academic research but also with English language help. I would like to thank Associate,Professor Josef Pieprzyk, Co-supervisor, who gave me many ideas and suggestions which greatly benefited to my studies. I have also appreciated our talks with Dr M. Zhang, Dr C. Charnes and Mr T. Xia which gave me many hints that enriched the thesis. Here I express my deep appreciation to them. I wish to thank all the staff and students in the Center for Computer Security Research and School of IT & CS, University of Wollongong for the help, both academic and technical. IX Contents Abstract j v Publications vi / Acknowledgements ix 1 Introduction 1 1.1 Contributions to this thesis 1 1.2 Contents of the thesis 2 1.3 Further research problems .....' 4 2 Boolean spaces and Boolean functions 5 2.1 Review of Boolean functions in cryptography 5 2.2 Boolean space and Boolean functions 12 2.3 Cryptographic desirable characteristics of Boolean functions 16 2.4 Hadamard matrix and Walsh-Hadamard transformation 21 2.5 Construction of affine sequences 26 3 Bent Boolean functions and their constructions 29 3.1 Bent functions and their basic properties 29 3.2 Constructions for bent functions 34 3.3 Constructing bent sequences 40 3.4 Notes on the propagation criterion of degree / and order k 43 3.5 Partially-bent functions 45 3.6 Plateaued Boolean functions 49 4 The excess of Boolean functions and Hadamard transform 52 x 5 On the symmetric properties of Boolean functions 60 5.1 Symmetric group and Boolean functions 60 5.2 Homogeneous Boolean functions 67 5.3 Degree-3 homogeneous Bent Functions 73 5.4 Degree-3 homogeneous Balanced Functions 78 5.5 Degree-3 homogeneous Boolean functions 81 6 Balanced Boolean functions 84 6.1 Balanced Boolean functions with high nonlinearity and good propaga­ tion criteria , 84 6.2 Concatenating functions 88 6.3 Constructions for highly nonlinear balanced Boolean functions by bent functions 93 6.4 Constructions for highly nonlinear balanced Boolean functions by highly nonlinear non-balanced Boolean functions 96 7 Block designs and degree-3 homogeneous functions 98 7.1 Introduction of BIBD and PBIBD 98 7.2 Designs for highly nonlinear homogeneous Boolean functions 102 8 The applications of degree-3 homogeneous Boolean functions 108 8.1 Motivation 109 8.2 Definition of Rotation-Symmetric Boolean Functions 110 8.3 Properties of Rotation-Symmetric Functions Ill 8.4 Balanced Rotation-Symmetric Boolean Functions 113 8.5 Evaluation of Functions 115 8.6 Extensions and Further Research 116 A Homogeneous bent functions on 14 118 B Homogeneous balanced functions 121 C Homogeneous functions with the highest nonlinearity 137 Bibliography 139 xi Chapter 1 Introduction This chapter contains three parts. In the first part, the author's contributions to this thesis are declared. In the second part, the contents in the following chapters are summarized. In the last part, some open problems related to the thesis are listed. 1.1 Contributions to this thesis In chapter 2 and 3, Boolean functions and bent functions are characterized from the viewpoint of cryptography.