A study on the construction and analysis of substitution boxes for symmetric cryptosystems

S(ubstitution)-boxes are quite important components of modern symmetric cryptosystems (in particular, block ciphers) in the sense that S-boxes bring nonlinearity to block ciphers and strengthen their cryptographic security. An S-box is said to satisfy the strict avalanche criterion (SAC), if and only if for any single input bit of the S-box, the inversion of it changes each output bit with probability one half. In this thesis, with the concrete proof of cryptographical properties of S-boxes satisfying the SAC, we propose a variety of provable construction methods for S-boxes satisfying the SAC. For Boolean S-boxes satisfying the SAC, we can construct and enlarge them by using concatenation, Kronecker (or direct) product, and dyadic shift. For bijective S-boxes satisfying the SAC, when an n-bit input Boolean function and an n-bit input bijective function satisfying the SAC are given, the combined function is proved to become an (n+1)-bit bijective function satisfying the SAC as well. Also, we propose one simple construction method to construct bijective functions satisfying the maximum order SAC. Until now, bent functions have been given great attention in coding theory, logic synthesis and spread spectrum communications. We show that there exists an interesting relationship between bent functions and Boolean functions satisfying the (maximum order) SAC. All Boolean functions satisfying the maximum order SAC are always bent and all bent functions satisfy at least the 0-th order SAC. For practical applications, we apply these cryptographically useful functions to construct DES-like S-boxes according to our design criteria including the possibility of differential attack. Compared with DES S-boxes, we found that our designed DES-like S-boxes exhibit better cyptographic properties than those of DES S-boxes. Moreover, as an experimental work of symmetric cryptosystems, we examine the statistical properties of DES-like cryptosystems (FEAL-4, FEAL-8, Multi2, DES, and s2DES ). s2DES is the DES-like cryptosystem in which all the S-boxes of DES are replaced by our designed 8 DES-like S-boxes. These experiments again indicate to us that our designed DES-like S-boxes have good cryptographic performances. Finally this thesis will aid us in designing and analyzing block cipher algorithms directly and stream cipher algorithms indirectly. iii Acknowledgements I would like to heartily thank my chief advisor, Professor Hideki Imai, for his guidance, encouragement, and support during my 3-year research work. I am also grateful to Professor Yasunori Dohi, Professor Rokuya Ishii, and Associate Professor Ryuji Kohno for their advice. I would like to express my special thanks to Associate Professor Tsutomu Matsumoto for his useful suggestions and helpful comments. I would like to acknowledge many foreign students in Imai Research Laboratory and, in particular, Mr. Manuel Cerecedo (from Spain) and Mr. Young Yoon (from Canada) for their proof reading of my rst draft and correcting of its grammatical mistakes. Special appreciation and thanks go to my wife, Jongsun Kang, for her patience and understanding throughout the course of this endeavor. Moreover, I feel thankful to my two lovely sons, 6-year-old Sunghak Kim and 4-year-old Jaehak Kim for growing up favorably. Finally, nancial support from the Japanese Ministry of Education (Monbusho) and my workplace in Korea, Electronics and Telecommunications Research Institute (ETRI), is gratefully acknowledged. iv