Constructions Of Generalized Nonlinear Functions

Cryptographic functions play an important role in designing cryptosystems, such as stream ciphers and block ciphers. Such functions are widely used and caught quite a lot of attentions. Cryptographic functions are referred to Boolean functions, vectorial Boolean functions and their generalized forms. To ensure the safety of cryptosystems, resist damage from linear attack, differential attack etc., the cryptographic functions used in the systems should gain high algebraic degree, high nonlinearity and low dif-ferential uniformity, and so on.Bent functions on finite fields with even characteristic are the ones of optimal nonlinearity. Extending the conception of Bent functions to finite fields with odd char-acteristic, generalized Bent functions are put forward. These yeas, study on these kinds of cryptographic functions have attracted a lot of attentions. In2006, Helleseth and Kholosha extended the Dillon index to finite fields with odd characteristic, and proved that a monomial function with generalized Dillon index is a generalized Bent function if and only if the value of the Kloosterman sums on its coefficient is zero. Based on the previous works, a class of binomial functions over finite fields with odd characteristic is investigated in this paper, and the bentness of the functions in the class is character-ized in terms of the Kloosterman sums. Moreover, it is also proven that the proposed class contains bent functions that are affinely inequivalent to all known monomial and binomial ones by some numerical results. All the bent functions in the proposed class are both regular and normal bent functions with highest algebraic degree. In particular, regular bent functions on finite fields with characteristic of5are found in the proposed class, which are not seen in previous works.Constructions of permutations with low differential uniformity (especially, perfect nonlinear permutations or almost perfect nonlinear permutations) have caught a lot of attentions. In2012, Guobiao Weng and Xiangyong Zeng found some important proper-ties of perfect nonlinear DO functions. Inspired by their discoveries, we define a kind of functions called DO-like. By investigating the relationship between DO-like functions and perfect nonlinear functions, a method using perfect nonlinear DO-like functions to induce permutation polynomials with low differential uniformity is proposed. The range of the differential uniformity of the constructed polynomials is discussed, and a necessary and sufficient condition for these polynomials being permutations is given. The proposed method is quite different from previous constructions, in which the con-structed permutations have explicit representations. It provides a new way in generating permutations with low differential uniformity.