A Composition Construction of Bent-Like Boolean Functions from Quadratic Polynomials

Boolean functions used in cryptosystems are required to have good cryptographic properties, such as balancedness, high nonlinearity and high algebraic degree, to ensure the systems are resistant against linear cryptanalysis ([2]). Besides, it is a desirable property that a Boolean function has 3-valued spectra. This property provides a protection against the soft output joint attack ([3]). A framework unifies such cryptographic properties is to introduce the concept of near-bent function. Except balancedness, such a function behaves like a bent function, that is, it has 3-valued spectra and high nonlinearity close to the upper bound on nonlinearity, and may have high algebraic degree. It is important to study near-bent functions since bent functions are not balanced and can’t be directly used in a cryptosystem. There is an intrinsic relation between near-bent functions and bent functions. Bent functions can be constructed from near-bent functions ([4]), and vice versa. Another important aspect to study bent and near-bent functions is that the theory is closely related to combinatorics ([5-7]) and communication ([8-15]) in some useful fields such as difference set, partial spread, and sequences with low cross correlation.