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

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.

CONSTRUCTION OF SUBSETS OF BENT FUNCTIONS SATISFYING RESTRICTIONS IN THE REED-MULLER DOMAIN

Bent functionsare Booleanfunctionswith highestnonlinearitywhich makes them interesting for cryptography. Determination of bent functions is an importantbut hard problem, since the general structure of bent functions is still unknown. Various constructions methods for bent functions are based on certain deterministic procedures, which might result in some regularitythat is a feature undesired for applications in cryptography. Random generation of bent functions is an alternative, however, the search space is very large and the related procedures are time consuming. A solution is to restrict the search space by imposing some conditions that should be satisfied by the produced bent functions. In this paper, we propose three ways of imposing such restrictions to construct subsets of Boolean functions within which the bent functions are searched. We estimate experimentally the number of bent functions in the corresponding subsets of Boolean functions

ON DILLON\u27S CLASS H OF BENT FUNCTIONS, NIHO BENT FUNCTIONS AND O-POLYNOMIALS

One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well- known Maiorana-McFarland class. We first notice that H can be extended to a slightly larger class that we denote by H. We observe that the bent functions constructed via Niho power functions, which four examples are known, due to Dobbertin et al. and to Leander-Kholosha, are the univariate form of the functions of class H. Their restrictions to the vector spaces uF2n=2 , u 2 F? 2n, are linear. We also characterize the bent functions whose restrictions to the uF2n=2 \u27s are affine. We answer to the open question raised by Dobbertin et al. in JCT A 2006 on whether the duals of the Niho bent functions introduced in the paper are Niho bent as well, by explicitely calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing whether H (or H) is a subclass of the Maiorana-McFarland completed class. We then show that the condition for a function in bivariate form to belong to class H is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial and we deduce eight new cases of bent functions in H which are potentially new bent functions and most probably not affine equivalent to Maiorana-McFarland functions