Additive Autocorrelation of Resilient Boolean Functions

Abstract. In this paper, we introduce a new notion called the dual func-tion for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocor-relation. Second, we look at preferred functions which are Boolean func-tions with the lowest 3-valued spectrum. We prove that if a balanced preferred function has a dual function which is also preferred, then it is resilient, has high nonlinearity and optimal additive autocorrelation. We demonstrate four such constructions of optimal Boolean functions using the Kasami, Dillon-Dobbertin, Segre hyperoval and Welch-Gong Transformation functions. Third, we compute the additive autocorrela-tion of some known resilient preferred functions in the literature by using the dual function. We conclude that our construction yields highly non-linear resilient functions with better additive autocorrelation than the Maiorana-McFarland functions. We also analysed the saturated func-tions, which are resilient functions with optimized algebraic degree and nonlinearity. We show that their additive autocorrelation have high peak values, and they become linear when we fix very few bits. These potential weaknesses have to be considered before we deploy them in applications.

Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.Comment: 14 pages, 2 table

Artificial life techniques for cryptology

In this thesis, we investigate the applications of two swarm-inspired artificial life optimization techniques in cryptology. In particular, we investigate the use of both Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) for automated cryptanalysis of simple classical substitution ciphers. We also use PSO to construct Boolean functions with some desirable cryptographic properties. Both ACO and PSO based attacks proved to be effective for the cryptanalysis of simple substitution ciphers encoded with various sets of encoding keys. Purely uni-gram and bi-gram statistics are used for solving this problem. Boolean functions are vital components of symmetric-key ciphers such as block ciphers, stream ciphers and hash functions. When used in cipher systems, Boolean functions should satisfy several cryptographic properties such as balance, high nonlinearity, resiliency and high algebraic degree. Using PSO, with an unorthodox approach of spectral inversion, we are able to construct Boolean functions that achieve the maximum possible nonlinearity (Bent function) and several other important resilient functions. In fact, we were able to construct, for the first time, a 9-variable Boolean function with nonlinearity 240, algebraic degree 5, and resiliency degree 3. This construction affirmatively answers the open problem about the existence of such function