On The Problems Of Some Cryptographic Functions With Good Properties

Cryptography is in the core content of information security technology. One of the keyproblems in Cryptography is the security on cryptographic functions. In this thesis we discusssome important problems on cryptographic functions. The main results are as follows:1. The balancedness of symmetric Boolean functions is investigated. By the complexrepresentation of the Walsh spectrum at zero, we prove Canteaut’s conjecture on symmetricBoolean functions, which says all balanced symmetric Boolean functions of fixed algebraicdegree are trivial when the number of variables grows large enough.2. We also present the nonexistence of trivial balanced elementary symmetric Booleanfunction except for n=2t+1l–1and d=2t, where t and l are any nonnegative integers, whichshows Cusick’s conjecture for balanced elementary symmetric Boolean function is exactly theconjecture that all balanced elementary symmetric Boolean functions are trivial. In additional,we obtain a lower bound ndby the proof of Canteaut’s conjecture, such that Cusick’s conjectureholds for any n> nd.3. We also discuss Dobbertin’s problem whether near bent functions could be bent whenrestricted to a hyperplane. A necessary and sufficient condition is given in Chapter4.Furthermore, most of Kasami functions are proved to be not bent when restricted to anyhyperplane.4. The support set of the Walsh spectra of Kasami-Welch functions is investigated. Inadditional, Boolean functions with five valued Walsh spectra are characterized by second-ordercovering sequences.