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.

On propagation characteristics of resilient functions

In this paper we derive several important results towards a better understanding of propagation characteristics of resilient Boolean functions. We first introduce a new upper bound on nonlinearity of a given resilient function depending on the propagation criterion. We later show that a large class of resilient functions admit a linear structure; more generally, we exhibit some divisibility properties concerning the Walsh-spectrum of the derivatives of any resilient function. We prove that, fixing the order of resiliency and the degree of propagation criterion, a high algebraic degree is a necessary condition for construction of functions with good autocorrelation properties. We conclude by a study of the main constructions of resilient functions. We notably show how to avoid linear structures when a linear concatenation is used and when the recursive construction introduced in [11] is chosen

On the Constructing of Highly Nonlinear Resilient Boolean Functions by Means of Special Matrices

. In this paper we consider matrices of special form introduced in [11] and used for the constructing of resilient functions with cryptographically optimal parameters. For such matrices we establish lower bound 1 log 2 ( p 5+1) = 0:5902::: for the important ratio t t+k of its parameters and point out that there exists a sequence of matrices for which the limit of ratio of these parameters is equal to lower bound. By means of these matrices we construct m-resilient n-variable functions with maximum possible nonlinearity 2 n 1 2 m+1 for m = 0:5902 : : : n+O (log 2 n). This result supersedes the previous record. Keywords: stream cipher, Boolean function, nonlinear combining function, correlation-immunity, resiliency, nonlinearity, special matrices.

Construction of cryptographically important Boolean functions

Boolean functions are used as nonlinear combining functions in certain stream ciphers. A Boolean function is said to be correlation immune if its output leaks no information about its input values. Balanced correlation immune functions are called resilient functions. Finding methods for easy construction of resilient functions with additional properties is an active research area. Maitra and Pasalic [3] have constructed 8-variable 1-resilient Boolean functions with nonlinearity 116. Their technique interlinks mathematical results with classical computer search. In this paper we describe a new technique to construct 8-variable 1-resilient Boolean functions with the same nonlinearity. Using a similar technique, we directly construct 10-variable (resp. 12-variable), 1-resilient functions with nonlinearity 488 (resp. 1996). Finally, we describe some results on the construction of n-variable t-resilient functions with maximum nonlinearity

Autocorrelation Coefficients and Correlation Immunity of Boolean Functions

Abstract. We apply autocorrelation and Walsh coefficients for the investigation ofcorrelation immune and resilient Boolean functions. We prove new lower bound for the absolute indicator of resilient functions that improves significantly (for m> (n − 3)/2) the bound ofZheng and Zhang [18] on this value. We prove new upper bound for the number ofnonlinear variables in high resilient Boolean function. This result supersedes the previous record. We characterize all possible values of resiliency orders for quadratic functions and give a complete description ofquadratic Boolean functions that achieve the upper bound on resiliency. We establish new necessary condition that connects the number ofvariables, the resiliency and the weight ofan unbalanced nonconstant correlation immune function and prove that such functions do not exist for m>0.75n − 1.25. For high orders of m this surprising fact supersedes the well-known Bierbrauer–Friedman bound [8], [1] and was not formulated before even as a conjecture. We improve the upper bound ofZheng and Zhang [18] for the nonlinearity ofhigh order correlation immune unbalanced Boolean functions and establish that for high orders of resiliency the maximum possible nonlinearity for unbalanced correlation immune functions is smaller than for balanced

Classification of Boolean functions of 6 variables or less with respect to cryptographic properties. Cryptology ePrint Archive, Report 2004/248

Abstract. This paper presents an efficient approach for classification of the affine equivalence classes of cosets of the first order Reed-Muller code with respect to cryptographic properties such as correlationimmunity, resiliency and propagation characteristics. First, we apply the method to completely classify all the 48 classes into which the general affine group AGL(2, 5) partitions the cosets of RM(1, 5). Second, we describe how to find the affine equivalence classes together with their sizes of Boolean functions in 6 variables. We perform the same classification for these classes. Moreover, we also determine the classification of RM(3, 7)/RM(1, 7). We also study the algebraic immunity of the corresponding affine equivalence classes. Moreover, several relations are derived between the algebraic immunity and other cryptographic properties. Finally, we introduce two new indicators which can be used to distinguish affine inequivalent Boolean functions when the known criteria are not sufficient. From these indicators a method can be derived for finding the affine relation between two functions (if such exists). The efficiency of the method depends on the structure of the Walsh or autocorrelation spectrum.

Classification of Boolean Functions of 6 Variables or Less with Respect to Cryptographic Properties

This paper presents an efficient approach for classification of the affine equivalence classes of cosets of the first order Reed-Muller code with respect to cryptographic properties such as correlation-immunity, resiliency and propagation characteristics. First, we apply the method to completely classify all the 48 classes into which the general affine group AGL(2, 5) partitions the cosets of RM(1, 5). Second, after distinguishing the 34 affine equivalence classes of cosets of RM(1, 6) in RM(3, 6) we perform the same classification for these classes. We also study the algebraic immunity of the corresponding affine equivalence classes. Moreover, several relations are derived between the algebraic immunity and other cryptographic properties. Finally, we introduce two new indicators which can be used to distinguish affine inequivalent Boolean functions when the known criteria are not sufficient. From these indicators a method can be derived for finding the affine relation between two functions (if such exists). The efficiency of the method depends on the structure of the Walsh or autocorrelation spectrum

Almost Boolean Functions: The Design of Boolean Functions by Spectral Inversion.

The design of Boolean functions with properties of cryptographic significance is a hard task. In this paper, we adopt an unorthodox approach to the design of such functions. Our search space is the set of functions that possess the required properties. It is "Boolean-ness" that is evolved