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Boolean functions with optimal algebraic immunity (OAI functions) are important cryptographic primitives in the design of stream ciphers. During the past decade, a lot of work has been done on constructing such functions, among which mathematics, especially finite fields, play an important role. Notably, the approach based on decompositions of additive or multiplicative groups of finite fields turns out to be a very successful one in constructing OAI functions, where some original ideas are contributed by Tu and Deng (2012), Tang, et al. (2017), and Lou, et al. (2015). Motivated by their pioneering work, the authors and their collaborators have done a series of work, obtaining some more general constructions of OAI functions based on decompositions of finite fields. In this survey article, the authors review our work in this field in the past few years, illustrating the ideas for the step-by-step generalizations of previous constructions and recalling several new observations on a combinatorial conjecture on binary strings known as the Tu-Deng conjecture. In fact, the authors have obtained some variants or more general forms of Tu-Deng conjecture, and the optimal algebraic immunity of certain classes of functions we constructed is based on these conjectures. [ABSTRACT FROM AUTHOR]

This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2. The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form $${x^{1 + {2^\alpha }}}$$ is almost perfect nonlinear. It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions. [ABSTRACT FROM AUTHOR]

This paper first proposes an infinite class of 2 k-variable Boolean functions with high nonlinearity and high algebraic degree. Then an infinite class of balanced Boolean functions are proposed by modifying the above Boolean functions. This class of balanced Boolean functions have optimal algebraic degree and high nonlinearity. Both classes have optimal algebraic immunity based on a general combinatorial conjecture. [ABSTRACT FROM AUTHOR]

Tu and Deng proposed a class of bent functions which are of optimal algebraic immunity under the assumption of a combinatorial conjecture. In this paper, the authors compute the dual of the Tu-Deng functions and then show that they are still of optimal algebraic immunity under the assumption of the same conjecture. For another class of Boolean functions constructed by Tang, et al. which are of optimal algebraic immunity with similar forms to Tu-Deng functions, the authors show that they are not bent functions by using some basic properties of binary complete Kloosterman sums. [ABSTRACT FROM AUTHOR]

A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of participants can recover the secret. Secret sharing is useful in management of cryptographic keys. Based on identity, we analyze the secret sharing scheme among weighted participants. Then we present a dynamic scheme about secret sharing among weighted participants. At last, we analyze the secret sharing scheme among weighted participants, which can make all weighted participants verifiable and dynamic. [ABSTRACT FROM AUTHOR]