Constructions Of Balanced Vectorial Boolean Functions And Permutation Functions

Cryptographic functions contain Boolean functions and vectorial Boolean functions.They are generally key parts in stream ciphers and block ciphers.The security of the whole encryption algorithm depends on the cryptographic property of the functions.Cryptographic functions which satisfy a variety of cryptographic properties can be used to resist various attacks.Two cryptographic properties of vectorial Boolean functions,namely the balanceness and nonlinearity,are mainly studied in this thesis.The following is the main work of this thesis:1.In this thesis,a design method of vectorial Boolean functions through the M-M class of Boolean functions is presented.A class of balanced vectorial Boolean functions with output dimension greater than half of input dimension is given by controlling the mapping ? of each coordinate functions.Their nonlinearity reach bent concatenation bound.The schemes will be different slightly by the inequable parity of input dimension.A considerable number of semi-bent vectorial Boolean functions can be provided with this method.Instances constructed by this means is given.Finally,the output dimension of the functions is increased by appropriately extending the structure.2.A design method of permutation functions with good nonlinearity over vector space through the M-M class of Boolean functions is presented.Firstly,a class of Plateaued permutation functions with special Walsh spectra are given.They are frameworks that can be flexibly extended to obtain significant amount of permutation functions.Secondly,a class of permutation functions over vector space with nonzero nonlinearity are constructed by concatenating Boolean functions with good properties,which make up the defect of the permutation functions above.The concatenated Boolean functions are chosen as the coordinate functions of permutations with high nonlinearity.The schemes will be different when the parity of the dimension of the permutations and concatenated Boolean functions changes,so they must be explain in the explicit case.Finally,a class of permutation functions with better nonlinearity is given.By changing the mapping ? of each coordinate functions,the nonlinearity of the permutation functions obtained are improved obviously compared with the construction above.