Analysis Of Several Classes Of Cryptographic Properties Of Boolean Functions

Boolean functions are widely used in communications and cryptography. Especially, they hold an extremely important position in the design and analysis of symmetric cryp-tographic algorithms. The analysis and study on the properties of Boolean functions are always one of the most active research areas. This paper investigates several cryptographic properties of Boolean functions, and obtains the following main results:(1) A new kind of algebraic attack on two classes of filter generators is proposed, in which the Boolean functions have high algebraic immunity. Firstly, the ability of symmetric Boolean functions with high algebraic immunity to resist the algebraic attack is analyzed. By using the method of splitting Boolean functions, it is proved theoretically that even symmetric Boolean functions with high algebraic immunity remain vulnerable to such algebraic attack if they are used inappropriately. Sec-ondly, a class of more extensive Boolean functions with high algebraic immunity are analyzed, which are the sum of rotation symmetric Boolean functions and the Boolean functions with low degree, it is proved theoretically that the construction methods of these functions has a possible weakness and these Boolean functions with high algebraic immunity also present a risk of suffering from algebraic attacks.(2) The value distributions of Walsh spectrum of the quadratic Plateaued functions with n variables are studied. All the possible value distributions of Walsh spectrum of the functions are given. Furthermore, the precise value of Walsh spectrum of several subclasses of these functions are determined, the corresponding distributions and the conditions to achieve these distributions are also given. Our results can be used to estimate the nonlinearities of these functions, their resiliency orders, their numbers of non-zero Walsh spectrum values and the lower bounds on the second order nonlinearity of cubic Boolean functions. And then it has a guiding role to construct functions with good cryptographic properties.(3) To improve and generalize the existing results, the lower bounds on the second or-der nonlinearity of general cubic Boolean functions are deduced. And, a subclass of functions are found, whose lower bounds on the second order nonlinearity are superior to those of the general Boolean functions under certain conditions. Fur-thermore, the exact values of Walsh spectrums of all the derivatives of three classes of functions are derived, and then the tight lower bounds on the second order non-linearity of these functions are given. Our results show that our bounds are better than the previously obtained general bounds. Especially, the lower bounds of the second order nonlinearity of the quartic bent and semi-bent Boolean functions con-structed by Charpin et al. are derived. Our results show that all the above Boolean functions have large second order nonlinearity when n is not too small, and can resist the affine and quadratic function approximation attacks.(4) The additive autocorrelations of two classes of semi-bent functions constructed by Charpin et al. are studied. The additive autocorrelations at all the nonzero points of the two classess of functions are deduced theoretically. We determine that they have a non-zero linear structure. Our results show that the two classes of semi-bent functions have the worst additive autocorrelations. Therefore, these functions can not resist differential-like attacks. On the other hand, the orders of their correlation immune are determined, we prove that these functions can not resist correlation attacks.