Cryptographic Properties Of Boolean Functions And Its Applications In The Analysis Of Aes Algorithm

The security of cryptosystem depends on the cryptographic performance of the Boolean functions which is used in the cryptosystem. AES S box can be described by Boolean functions. The point in this thesis is that how to find out a good S box. This S box has good cryptographic performance and can be implemented more simply in hardware.This thesis performs analysis on the cryptographic properties of the AES S box, and finds out the AES S box's best approximate functions. According to the basic theory of Boolean functions, Boolean functions'representations of AES S box are calculated from S box's truth table. Based on the Boolean functions'representations and their Walsh circle spectrum, several cryptographic properties such as balanceness, correlation immunity, strict avalanche, propagation characteristic, stability, symmetric, orthotropic, linearity, non-linearity, differential uniformance, robust, and so on are calculated. Based on the Walsh circle spectrum, its best approximate functions are found out. The cryptographic properties for best approximate functions are calculated. At last, the new S box is generated by the best approximate functions.Eight Boolean function's representations of AES S box are complex. The number of polynomial in eight representations is 110, 112, 114, 131, 136, 145, 133 and 132. Based on the analysis of the Boolean functions'representations and their Walsh circle spectrum, several conclusions can be described as follows. Every Boolean function is balanceness. It is not correlation immunity. It doesn't satisfied strict avalanche. It is not propagation characteristic. It is not stability. It is not symmetric function. AES S box's Boolean function is orthogonal. Every function's algebraic degree is 7. Eight Boolean function's algebraic degree is 7. The degree of nonlinearity is 112. The degree of differential uniformance is 4. The degree of robust is 0.984375.Based on AES S box's Boolean function, its best approximate functions are found out. Its representations are simply. The number of polynomial in eight representations is 4, 2, 2, 2, 4, 4, 5 and 5. So its best approximate functions can be implemented more simply in hardware. Based on the analysis of the Boolean functions'representations and their Walsh circle spectrum, several conclusions can be described as follows. Every Boolean function is not balanceness. It is not correlation immunity. It doesn't satisfied strict avalanche. It is not propagation characteristic. It is not stability. It is not symmetric function. The Boolean function of the best approximate functions is not orthogonal. Every function's algebraic degree is 1. Eight Boolean function's algebraic degree is 0. The degree of nonlinearity in eight Boolean functions is 16, 1, 1, 1, 16, 16, 15 and 15. The degree of differential uniformance is 254. The degree of robust is 0.006866.