Nonlinear Balanced Boolean Functions Research Articles

Balanced crossover operators in Genetic Algorithms

Abstract In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction. In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators (each with two variants: “left-to-right” and “shuffled”), two of which have never been published before, and compare their performances with classic one-point crossover. We are able to confirm that the balanced crossover operators perform better than one-point crossover. Furthermore, in two out of three crossovers, the “left-to-right” version performs better than the “shuffled” version.

A combined DCA: GA for constructing highly nonlinear balanced boolean functions in cryptography

Substitution boxes, aka S-boxes, are a key component of modern crypto-systems. Several studies and developments were carried out on the problem of building high-quality S-boxes in the last few years. Qualities of such boxes, such as nonlinearity and balance, steer the robustness of modern block ciphers. This work is concerned with the construction of highly nonlinear balanced Boolean functions. A deterministic optimization model which is the minimization of a polyhedral convex function on a convex polytope with 0---1 variables is introduced. A local deterministic optimization approach called DCA (Difference of Convex functions Algorithm) is investigated. For finding a good starting point of DCA we propose two versions of a combined DCA---GA (Genetic Algorithm) method. Numerical simulations prove that DCA is a promising approach for this problem. Moreover the combination of DCA---GA improves the efficiency of DCA and outperforms other standard approaches.

On maximally nonlinear and extremal balanced Boolean functions

We prove a new sufficient condition for a Boolean function to be extremal balanced or maximally nonlinear, in odd or even dimension. Under this condition, we deduce the balanced covering radius ρB(n) and the covering radius ρ(n). We prove some general properties about maximally nonlinear or extremal balanced functions. Finally, an application to even weights Boolean functions is given.

Highly nonlinear balanced Boolean functions with good local and global avalanche characteristics

Here we deal with an interesting subset of n-variable balanced Boolean functions which satisfy strict avalanche criteria. These functions achieve the sum-of-square indicator value (a measure for global avalanche criteria) strictly less than 2 2 n+1 and nonlinearity strictly greater than 2 n−1 −2 ⌊ n/2⌋ . These parameters are currently best known. Moreover, these functions do not possess any nonzero linear structure. The technique involves a well-known simple construction coupled with very good initial functions obtained by computer search, which were not known earlier.

Highly Nonlinear Balanced Boolean Functions with Very Good Autocorrelation Property

Abstract Constructing highly nonlinear balanced Boolean functions with very good autocorrelation property is an interesting open question. In this direction we use the measure Δ f , the highest magnitude of all autocorrelation coefficients for a function f . We provide balanced functions f with currently best known nonlinearity and Δ f values together. Our results for 15-variable functions disprove the conjecture proposed by Zhang and Zheng (1995) for different ranges of nonlinearity, where our constructions are based on modifications of Patterson-Wiedemann (1983) functions. Also we propose a simple bent based construction technique to get functions with very good Δ f values for odd number of variables. This construction has a root in Kerdock Codes. Moreover, our construction on even number of variables is a recursive one and we conjecture (similar to Dobbertin's conjecture (1994) with respect to nonlinearity) that this provides the minimum possible value of Δ f for a balanced function f on even number of variables.

Nonlinearity and Propagation Characteristics of Balanced Boolean Functions

Three important criteria for cryptographically strong Boolean functions are balance, nonlinearity, and the propagation criterion. The main contributions of this paper are to reveal a number of interesting properties of balance and nonlinearity, and to study systematic methods for constructing Boolean functions that satisfy 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 balance and nonlinearity of the functions. The algebraic degrees of functions constructed are also discussed.