On Algebraic Immunity of Weight Symmetric H Boolean Functions

Using the derivative of Boolean functions and the e-derivative defined by ourselves as research tools, we discuss the relationship among a variety of cryptographic properties of the weight symmetric H Boolean functions in the range of the weight with the existence of H Boolean functions. We also study algebraic immunity and correlation immunity of the weight symmetric H Boolean functions and the balanced H Boolean functions. We obtain that the weight symmetric H Boolean function should have the same algebraic immunity, correlation immunity, propagation degree and nonlinearity. Besides, we determine that there exist several kinds of H Boolean functions with resilient, algebraic immunity and optimal algebraic immunity. The above results not only provide a theoretical basis for reducing nearly half of workload when studying the cryptographic properties of H Boolean function, but also provide a new research method for the study of secure cryptographic property of Boolean functions. Such researches are important in cryptographic primitive designs.

Construction of Odd-Variable Boolean Functions with Optimum Algebraic Immunity

Recently, algebraic attacks becomes a major attack method to threat to cryptography security. In order to resist algebraic attacks, algebraic immunity as a Boolean function cryptographic property has been put out. This makes that Boolean functions should have high algebraic immunity to resist algebraic attacks. In this paper, a specific decomposition method of the space is proposed. By the method, we construct a class of odd number of variables Boolean functions with optimal algebraic immunity.