Improving High-Meets-Low technique to generate odd-variable resilient Boolean functions with currently best nonlinearity

Abstract

This paper studies the construction of resilient Boolean functions in odd variables with strictly almost optimal (SAO) nonlinearity. Based on High-Meets-Low technique, we present a new construction to obtain Boolean functions with higher resiliency order and currently best known nonlinearity. It is shown that this method can achieve a better tradeoff between resiliency and nonlinearity than those of the previous research results.

Introduction

Cryptographic Boolean functions play an important role in the design of hash functions, stream and block ciphers. High nonlinearity and resiliency are two of the most important prerequisites of Boolean functions when used in certain families of stream ciphers. The nonlinearity, the distance from a Boolean function to the set of all affine functions, prevents from linear attacks in block ciphers [7]. Resiliency ensures the cipher is not susceptible to a divide-and-conquer attack [16]. Since the early 1990s, the construction of resilient functions with high nonlinearity has been an important challenge in cryptography and it was extensively studied, see [6], [10], [13], [25] and the references therein.

It is known that the nonlinearity of an $n$-variable Boolean function is bounded above by $2^{n-1}-2^{\frac{n}{2}-1}$, and this bound is tight when $n$ is an even positive integer. A Boolean function in even number of variables whose nonlinearity reaches this upper bound is called bent. Bent functions were first constructed by Rothaus [12] and Dillon [2]. For odd $n$, the nonlinearity value $2^{n-1}-2^{\frac{n-1}{2}}$ is known as the bent concatenation bound. Here , the functions with nonlinearity strictly greater than the bent concatenation bound are called strictly almost optimal (SAO) functions. For odd $n\le 7$, the maximum nonlinearity is equal to the bent concatenation bound [1], [9]. In 1983, Patterson and Wiedemann [11] constructed functions in $15$ variables with nonlinearity 16 276, so-called PW functions, exceeding the bent concatenation bound by 20. Later in [15], this technique was modified for the purpose of generating balanced Boolean functions with nonlinearity 16 272, which implies that for odd $n\ge 15$ there exist balanced $n$-variable Boolean functions with nonlinearity $2^{n-1}-2^{(n-1)∕2}+2^{(n-7)∕2}$ (obtained using the bent concatenation). And using those $15$ variables functions it is possible to obtain functions with nonlinearity $2^{n-1}-2^{(n-1)∕2}+20\cdot 2^{(n-15)∕2}$ for odd $n\ge 15$. However, for $n=9,11,13$, the maximum nonlinearity known until 2006 was still equal to the bent concatenation bound. Kavut, Maitra and Yücel [4] constructed Boolean functions in $n=9$ variables with nonlinearity $241$ which is SAO. Shortly afterwards, Kavut and Yücel [5] discovered $9$ variables Boolean functions with nonlinearity 242, so-called KY functions. This along with the secondary construction proposed by Patterson and Wiedemann demonstrates that there exist SAO functions in $n=9,11,13$ variables. In 2016, Kavut and Maitra [3] obtained Patterson–Wiedemann type functions on 21 variables with nonlinearity 1 047 613 which were SAO.

For even $n$, there are lots of results on constructing SAO resilient functions, see [18], [22], [23] and the references therein. However, for odd number of variables, constructing Boolean functions with maximum possible nonlinearity is an open problem in the area of cryptography and combinatorics. Especially, there are very few results on constructing SAO resilient functions for odd number of variables. The earliest SAO 1-resilient Boolean functions for odd $n\ge 41$ were obtained by Sarkar and Maitra [13], [14]. In 2008, Sarkar and Maitra [15] proposed 15-variable 1-resilient functions with SAO nonlinearity 16 264 by modifying a PW function. This result implies that an $n$-variable 1-resilient function with SAO nonlinearity $2^{n-1}-2^{(n-1)∕2}+2^{(n-9)∕2}$ can be obtained by using direct sum method for odd $n\ge 15$. A secondary construction of highly nonlinear resilient functions in [21] showed it is possible to get $(n+15)$-variable resilient functions with nonlinearity $2^{n+15-1}-2^{(n+15-1)∕2}+20\cdot 2^{n∕2}$. Using the generalized Maiorana–McFarland construction technique, Zhang and Pasalic [22] constructed SAO $t$-resilient Boolean functions for odd $n$. Zhang et al. [24] obtained resilient functions with very high nonlinearity. Recently, Zhang [20] proposed the “High-Meets-Low” construction technique via fragmentary Walsh transforms and Sun et al. [17] also proposed a method to construct SAO resilient functions. Both these methods generate resilient functions whose nonlinearity (in certain cases) achieves the currently best known nonlinearity value, i.e. $2^{n-1}-216\cdot 2^{(n-17)∕2}$ when using PW functions or $2^{n-1}-28\cdot 2^{(n-11)∕2}$ when using KY functions.

In this paper, we propose an improved version of the “High-Meets-Low” construction technique. It is shown that we can construct resilient Boolean functions with better resiliency order than those in [17], [20] and make the Boolean functions have the currently best known nonlinearity at the same time.

The remainder of this paper is organized as follows. Section 2 introduces basic definitions and preliminaries on Boolean functions. In Section 3, two constructions (PW case and KY case) of resilient functions with higher resiliency order are presented. In addition, we compare our results with those in [17], [20] for PW case and KY case respectively. Finally, Section 4 concludes this paper.