Manuscript submitted to doi:10.3934/amc.2019038 AIMS' Journals Volume 13, Number 4, November 2019 pp. X{XX GOWERS U2 NORM AS A MEASURE OF NONLINEARITY FOR BOOLEAN FUNCTIONS AND THEIR GENERALIZATIONS Sugata Gangopadhyay Department of Computer Science and Engineering Indian Institute of Technology Roorkee Roorkee 247667, INDIA Constanza Riera Department of Computing, Mathematics, and Physics Western Norway University of Applied Sciences 5020 Bergen, Norway Pantelimon Stanic˘ a˘∗ Department of Applied Mathematics Naval Postgraduate School Monterey, CA 93943, USA Abstract. In this paper, we investigate the Gowers U2 norm for generalized Boolean func- tions, and Z-bent functions. The Gowers U2 norm of a function is a measure of its resistance to affine approximation. Although nonlinearity serves the same purpose for the classical Boolean functions, it does not extend easily to generalized Boolean functions. We first pro- vide a framework for employing the Gowers U2 norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers U2 norms, and an evaluation of the Gowers U2 norm of functions that are affine over spreads. We also give an introduction to Z-bent functions, as proposed by Dobbertin and Leander [8], to provide a recursive framework to study bent functions. In the second part of the paper, we concentrate on Z-bent functions and their U2 norms. As a consequence of one of our results, we give an alternate proof to a known theorem of Dobbertin and Leander, and also find necessary and sufficient conditions for a function obtained by gluing Z-bent functions to be bent, in terms of the Gowers U2 norms of its components. 1. Introduction. Boolean functions are functions mapping binary strings to 0 or 1. Over the years, several generalizations of Boolean functions have been proposed. In this paper, we consider such a generalization for which the domain set remains the same as for classical Boolean functions but the range is the set of integers modulo a positive integer q ≥ 2. These generalized Boolean functions have evolved to an active area of research [11, 12, 14, 16, 17, 18, 20, 23, 24, 25, 26, 27, 29] due to several possible applications in communications and cryptography. Boolean functions which are maximally resistant to affine approximation have special sig- nificance. The idea of nonlinearity is developed and extensively studied for classical Boolean functions. In the case of classical Boolean functions on an even number of variables, the func- tions with the highest possible nonlinearity are called bent functions [22] (see [3] and references therein, for more recent work on the subject). The concept of nonlinearity does not extend easily to the generalized setup. In the first part of the paper we investigate the Gowers U2 2010 Mathematics Subject Classification. Primary: 94C10: Secondary: 06E30. Key words and phrases. Gowers norms; Boolean functions; generalized Boolean functions; bent functions; Z-bent functions. ∗ Corresponding author: Pantelimon St˘anic˘a. 1 2 SUGATA GANGOPADHYAY, CONSTANZA RIERA, PANTELIMON STANIC˘ A˘ norm as a possible alternative to nonlinearity for measuring the resistance to affine approxima- tion. As examples, we provide the expressions of the Gowers U2 norms for the generalized bent functions, plateaued functions, functions that are affine over spreads, and a recurrence rule for the Gowers U2 norms. Characterization of bent Boolean functions is a longstanding open problem. One of the roadblocks faced by the researchers has been the absence of recurrence rules within the set of bent Boolean functions. Dobbertin and Leander [8] introduced the notion of Z-bent functions in order to put bent functions in a recursive framework at the cost of leaving the space of Boolean functions, and replacing it with the one of Z-bent functions of different levels. Here, we further obtain some recurrences of Gowers U2 norms of Z-bent functions, and a necessary and sufficient condition involving Gowers U2 norms of four Z-bent functions of level 1 so that bent functions are always obtained by the \gluing" process proposed by Dobbertin and Leander [8]. 2. Preliminaries. 2.1. Generalized Boolean functions. Let F2 be the finite field containing two elements; C, R, Z be the fields of complex numbers, real numbers, and the ring of integers respectively. The n cardinality of a set S is denoted by #S. For any positive integer n, let F2 = f(x1; : : : ; xn): xi 2 F2; 1 ≤ i ≤ ng be a vector space over F2. Let Zq be the ring of integers modulo q, where q is a positive integer. By `+' and `−' we respectively denote addition and subtraction n n modulo q, whereas `⊕' denotes the addition over F2 . Any function from F2 to F2, respectively, Zq; q > 2, is a Boolean, respectively, generalized Boolean function, in n variables, and the set q of all such functions is denoted by Bn, respectively, GBn. The character form of a generalized q n f(x) n Boolean function f 2 GBn, χf : F2 ! C, is defined by χf (x) = ζq , for all x 2 F2 , where 2π i ζq = e q . The algebraic normal form (ANF) of f 2 Bn is the polynomial representation M a1 an k f(x) = µax1 : : : xn ; where x = (x1; : : : ; xn), a = (a1; : : : ; an), and µa 2 F2. If q = 2 for n a2F2 q some k ≥ 1 we can associate to any f 2 GBn a unique sequence of Boolean functions ai 2 Bn, 0 ≤ i < k, such that k−1 n f(x) = a0(x) + 2a1(x) + ··· + 2 ak−1(x); for all x 2 F2 : n The (Hamming) weight of x 2 F2 , denoted by wt(x), is the number of nonzero coordinates n in x, and the (Hamming) weight of a Boolean function f is wt(f) = #fx 2 F2 : f(x) 6= 0g. The (Hamming) distance d(f; g) between two functions f; g is the weight of their sum. The n algebraic degree of f is deg(f) = maxfwt(a): a 2 F2 ; µa 6= 0g. The Boolean functions having algebraic degree at most one are affine functions. n For a (generalized) Boolean function f : F2 ! Zq we define the (generalized) Walsh- Hadamard transform to be the complex valued function (q) X f(x) u·x Hf (u) = ζq (−1) ; n x2F2 L where u·x = 1≤i≤n uixi For q = 2, we obtain the usual Walsh-Hadamard transform Wf (u) = X f(x) u·x q (q) X f(x)−f(x⊕u) (−1) (−1) : The autocorrelation of f 2 GBn is defined by Cf (u) = ζq : n n x2F2 x2F2 We shall use the identity [26] (q) −n X (q) 2 u·x Cf (u) = 2 jHf (x)j (−1) : (1) n x2F2 n (q) n=2 n A function f : F2 ! Zq is called generalized bent (gbent) if jHf (u)j = 2 for all u 2 F2 . q (q) (n+s)=2 n Further, we say that f 2 GBn is called s-plateaued if jHf (u)j 2 f0; 2 g for all u 2 F2 , for a fixed integer s depending on f. For simplicity of notation, when q is fixed, we sometimes use GOWERS U2 NORM FOR BOOLEAN FUNCTIONS AND THEIR GENERALIZATIONS 3 (q) (q) ζ, Hf , Cf instead of ζq, Hf , and Cf , respectively. We refer the reader to [11, 16, 15, 20] and references therein for more on generalized bent functions and their characterizations in terms of their components. 2.2. Gowers U2 norm. Let g : V ! C be any function on a finite set V and B ⊆ V . Then 1 P Ex2B[g(x)] := #B x2B g(x) is the average of f over B. Let f : V ! R be any function on 1 P a finite set V and B ⊆ V . Then Ex2B[f(x)] := #B x2B f(x) is the average of f over B. n + If f : F2 ! R, as in [5, Definition 2.2.1], for every ` 2 Z , we define the `th-order Gowers uniformity norm (the Gowers U` norm, or simply the U` norm) of f to be 1 0 2 !31 2` Y M kfk = n f x ⊕ h : (2) U` @Ex;h1;:::;h`2F2 4 i 5A S⊆[`] i2S n If f : F2 ! C is a complex-valued function, we define the Gowers norm by 1 0 2 !31 2` Y #S M kfk = n C f x ⊕ h U` @Ex;h1;h2;:::;hm2F2 4 i 5A S⊆[`] i2S where C is the complex conjugation operator, and [`] = f1; : : : ; `g. Since in this paper we shall n be using the Gowers U2 norm for f : F2 ! C, we give it explicitly below: 1=4 kfk = n [f(x)f(x ⊕ h ) f(x ⊕ h )f(x ⊕ h ⊕ h )] U2 Ex;h1;h22F2 1 2 1 2 1=4 2 = n j n [f(x)f(x ⊕ h )]j : Eh12F2 Ex2F2 1 We want to point out that the sum-of-squares indicator [30] is yet another measure of \non- linearity" defined for Boolean functions. There is a large body of literature dedicated to this concept (and many others), and one can find such references in [1,2,6, 19]. We want to point out [4], where a nice asymptotic bound has been obtained for the sum-of-squares indicator. Surely, using the Cauchy inequality, one can find an upper bound of the Gowers U2 norm in terms of the sum-of-squares indicator, but it will not achieve the purpose of this paper. n n It is known (cf. [5, pp. 22{24]) that for f : F2 ! R, if there is a polynomial P : F2 ! f0; 1g P (x) of degree d such that j n f(x)(−1) j ≥ , then kfk ≥ , for any > 0.