Nonlinearity and propagation characteristics of balanced boolean functions

Three of the most important criteria for cryptographically strong Boolean functions are the balancedness, the nonlinearity and the propagation criterion. The main contribution of this paper is to reveal a number of interesting properties of balancedness and nonlinearity, and to study systematic methods for constructing Boolean functions satisfying some or all of the three criteria. We show that concatenating, splitting, modifying and multiplying (in the sense of Kronecker) sequences can yield balanced Boolean functions with a very high nonlinearity. In particular, we show that balanced Boolean functions obtained by modifying and multiplying sequences achieve a nonlinearity higher than that attainable by any previously known construction method. We also present methods for constructing balanced Boolean functions that are highly nonlinear and satisfy the strict avalanche criterion (SAC). Furthermore we present methods for constructing highly nonlinear balanced Boolean functions satisfying the propagation criterion with respect to all but one or three vectors. A technique is developed to transform the vectors where the propagation criterion is not satisfied in such a way that the functions constructed satisfy the propagation criterion of high degree while preserving the balancedness and nonlinearity of the functions. The algebraic degrees of functions constructed are also discussed, together with examples illustrating the various constructions

Spectral and Nonlinear Properties of the Sum of Boolean Functions

Boolean functions are the mathematical basis of modern cryptographic algorithms. However, in practice, a set of interrelated Boolean functions is often used to construct a cryptographic algorithm. This circumstance makes the task of research of cryptographic quality, in particular, the distance of the nonlinearity of the sum of few Boolean functions important. The nonlinearity distance of a Boolean function is determined by the maximum value of its Walsh-Hadamard transform coefficients. In this paper, we proposed a formula that is the equivalent of the summation of Boolean functions in the Walsh-Hadamard transform domain. The application of this formula, as well as the Walsh-Hadamard spectral classification made it possible to determine the structure of WalshHadamard transform coefficients, and the distance of the nonlinearity when summing the Boolean functions lengths N 8 and N 16 , indicating valuable practical application for information protection