Additive Autocorrelation of Resilient Boolean Functions

Abstract

Abstract. In this paper, we introduce a new notion called the dual function for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocorrelation. Second, we look at preferred functions which are Boolean functions 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 autocorrelation of some known resilient preferred functions in the literature by using the dual function. We conclude that our construction yields highly nonlinear resilient functions with better additive autocorrelation than the Maiorana-McFarland functions. We also analysed the saturated functions, 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.

Time-Memory-Data Trade-Off Attack on Stream Ciphers Based on Maiorana-McFarland Functions

“…In this case, the equivalent keylength is only half of what is claimed. As shown by Gong and Khoo [9], this case correspond to the:…”

“…Such functions were constructed by Sarkar and Maitra in [19]. It was shown by Gong and Khoo in [9] that the saturated functions correspond to n-bit Maiorana-McFarland functions as follows.…”

“…and let the tap points to (x 0 , x 1 , x 2 , x 3 ) be LF SR[0, 3,5,9]. f (x) is linear when the first two tap points 0 and 3 are known, so k = 4.…”

Generalized Semi-bent and Partially Bent Boolean Functions

“…In Boolean case, Gong and Khoo [19] have introduced the concept of dual of a Boolean function and provided a relationship between the autocorrelation of the s-plateaued functions and the Walsh-Hadamard Spectrum of the dual of the s-plateaued functions. Also, if the function f ∈ B n , for n odd, is a balanced semi-bent function such thatf ∈ B n also semi-bent, then f = 2 n+1 2…”

“…2]. Several classes Boolean functions such as Dillon-Dobbertin, Kasami, Segre hyperoval and Welch-Gong Transformation functions for which the bounds are optimal is discussed in [19]. Several research papers are available in literature on these indicators, for details we refer [17,19,22,23] and the references of these papers.…”