Two Classes Of Bent Functions Over The Finite Field Of Odd Characteristic

Nonlinear cryptographic functions are a core component of a cryptosystem since they are of great significance for design and analysis of cryptosystems. They include per-fect nonlinear functions, almost perfect nonlinear functions, bent functions, almost bent functions, algebraic immunity optimal functions etc.. Bent functions have widely ap-plications in cryptography, coding theory and sequence design since they have optimal nonlinearity. This thesis discusses two classes of p-ary bent functions, and its main contributions include:(1) Based on the properties of block circulant matrices, we give a concise charac-terization on a class of quadratic p-ary bent function with coefficients belong to some extended field of Fp by polynomial greatest common divisor computa-tions. Moreover, the enumeration formulas of the constructed bent polynomials are presented for some special cases of n(2) We give a necessary and sufficient condition such that the class of p-ary binomial functions proposed by Jia et al. in [17] are regular bent functions, and thus settle the open problem raised at the end of that paper. Moreover, we investigate the bentness of the proposed binomial functions under the case gcd(t/2,pn/2+1)=1for some even integers t and n. Computer experiments show that the new class contains bent functions that are affinely inequivalent to known ones.