A complete characterization of plateaued Boolean functions in terms of their Cayley graphs
AA complete characterization of plateaued Booleanfunctions in terms of their Cayley graphs
Constanza Riera , Patrick Sol´e , Pantelimon St˘anic˘a Department of Computing, Mathematics, and Physics,Western Norway University of Applied Sciences5020 Bergen, Norway; [email protected] CNRS/LAGA, University of Paris 8, 2 rue de la Libert´e,93 526 Saint-Denis, France; [email protected] , Department of Applied Mathematics, Naval Postgraduate School,Monterey, CA 93943, USA; [email protected]
Abstract.
In this paper we find a complete characterization of plateaued Boolean func-tions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function f is s -plateaued (of weight = 2 ( n + s − / ) if and only if the associated Cayley graph is acomplete bipartite graph between the support of f and its complement (hence the graphis strongly regular of parameters e = 0 , d = 2 ( n + s − / ). Moreover, a Boolean function f is s -plateaued (of weight (cid:54) = 2 ( n + s − / ) if and only if the associated Cayley graph isstrongly 3-walk-regular (and also strongly (cid:96) -walk-regular, for all odd (cid:96) ≥
3) with someexplicitly given parameters.
Keywords:
Plateaued Boolean functions, Cayley graphs, strongly regular, walk regular.
Boolean functions are very important objects in cryptography, coding theory, and communica-tions, and have connections with many areas of discrete mathematics [4,5]. In particular bentfunctions, which offer optimal resistance to linear cryptanalysis, when used in symmetric cryp-tosystems, have been extensively studied [13,15]. They were shown in [1,2] to be connectedto strongly regular graphs. This connection occurs through the Cayley graph with generatorset the support of the Boolean function (denoted by Ω f below). Namely, having two nonzerocomponents in the Walsh-Hadamard spectrum translates at the Cayley graph level as havingthree eigenvalues. This link is often referred to as the Bernasconi-Codenotti correspondence .In this paper, we extend this connection by relating semibent and, in general, plateauedfunctions with a special class of walk-regular graphs. Plateaued Boolean functions are char-acterized as having three values in their Walsh-Hadamard spectrum [12]. Their correspondingCayley graphs belong to a special class of regular graphs with either three or four eigenvaluesin their spectrum. The three eigenvalue case is dealt with by the strong regularity and thefour eigenvalues case corresponds to the strongly t -walk-regular graphs introduced by Fiol andGarriga [9]. The special case of four eigenvalues of these graphs was studied in particular in [8].The material is organized as follows. The next section compiles the necessary notions anddefinitions on Boolean functions and graph spectra. Section 3 derives the main characterizationresult of the paper. Let F be the finite field with two elements and Z be the ring of integers. For any n ∈ Z + ,the set of positive integers, let [ n ] = { , . . . , n } . The Cartesian product of n copies of F is a r X i v : . [ m a t h . C O ] J u l n = { x = ( x , . . . , x n ) : x i ∈ F , i ∈ [ n ] } which is an n -dimensional vector space over F , whichwe will denote by V n . We will denote by ⊕ , respectively, +, the operations on F n , respectively, Z . For any n ∈ Z + , a function F : V n → F is said to be a Boolean function in n variables. Theset of all Boolean functions will be denoted by B n . A Boolean function can be regarded as amultivariate polynomial over F , called the algebraic normal form (ANF) f ( x , . . . , x n ) = a ⊕ (cid:88) ≤ i ≤ n a i x i ⊕ (cid:88) ≤ i 1) = d ( v − r − . The distance in the graph Γ = ( V, E ) between two vertices x, y ∈ V , denoted by d ( x, y ),is given by the length of the shortest path between x and y . The diameter of a graph is D = max x,y ∈ V d ( x, y ). A connected graph is called distance-regular of parameters ( c i , a i , b i )(called intersection numbers), if, for all 0 ≤ i ≤ D , and for all vertices x, y with d ( x, y ) = i ,among the neighbors of y , there are c i that are at distance i − x , a i at distance i , and b i at distance i + 1 (thus Γ is regular of degree r = b ).Fiol and Garriga [9] introduced t -walk-regular graphs as a generalization of both distance-regular and walk-regular graphs. We call a graph Γ = ( V, E ) a t -walk-regular (assuming Γ has itsdiameter at least t ) if the number of walks of every given length (cid:96) between two vertices x, y ∈ V depends only on the distance between x, y , provided it is ≤ t . In [8], van Dam and Omidigeneralized this concept and called Γ a strongly (cid:96) -walk-regular with parameters ( σ (cid:96) , µ (cid:96) , ν (cid:96) ) ifthere are σ (cid:96) , µ (cid:96) , ν (cid:96) walks of length (cid:96) between every two adjacent, every two non-adjacent, andevery two identical vertices, respectively. Certainly, every strongly regular graph of parameters( v, r, e, d ) is a strongly 2-walk-regular graph with parameters ( e, d, r ).Similarly to Proposition 1, the adjacency matrix A of a strongly (cid:96) -walk-regular graph willsatisfy the following property. Proposition 2 ([8]). Let (cid:96) > , and A be the adjacency matrix of a graph Γ . Then Γ is astrongly (cid:96) -walk-regular with parameters ( σ (cid:96) , µ (cid:96) , ν (cid:96) ) if and only if A (cid:96) + ( µ (cid:96) − σ (cid:96) ) A + ( µ (cid:96) − ν (cid:96) ) I = µ (cid:96) J. In general, the spectrum of the Cayley graph of an s -plateaued Boolean function f : F n → F will be 4-valued, and therefore the graph will not be strongly regular (see [5, Theorem 9.7]).This can be easily deduced from the fact that, if the Walsh-Hadamard transform of a Booleanfunction takes values in { , ± k } (for s -plateaued functions, k = 2 ( n + s ) / ), then the Fouriertransform of f takes values in { wt ( f ) , , ± k } (recall that the Fourier transform of f gives thegraph spectrum of the corresponding Cayley graph), as the following argument shows.By [5, Eq. (2.15)], W f ( w ) = 2 n − δ ( w ) − W ˆ f ( w ) , where δ is the Kronecker delta. Note that, for w = , W f ( ) = wt ( f ). By Parseval’s identity(see [5]), 2 n = (cid:88) w ∈ F n |W ˆ f ( w ) | , the multiplicity of ± k is n k . Hence, the multiplicity of theseeigenvalues will be (assuming wt ( f ) (cid:54) = k ; the other case follows easily): i ) If f is balanced, then W ˆ f ( ) = 0, while W f ( ) = wt ( f ). Then, the multiplicity of λ = wt ( f )is 1, the multiplicity of λ = 0 is 2 n − n k − 1, while the multiplicities of λ , λ = ± k willsum to n k .( ii ) If f is not balanced, then W ˆ f ( ) = ± k , while W f ( ) = wt ( f ). Then, the multiplicity of λ = wt ( f ) is 1, the multiplicity of 0 is 2 n − n k , while the multiplicities of ± k will sum to n k − Example : n = 3, f = x x ⊕ x x ⊕ x x , which is semibent, since W ˆ f ( w ) = (0 4 4 0 4 0 0 − T . We compute that W f ( w ) = (4 − − − T , which is 4-valued.Certainly, if f is semibent, the multiplicities are more precisely known (see [12], for example).For instance, if n is odd (without loss of generality, we assume that f ( ) = 0), the multiplicitiesof the spectra coefficients of ˆ f are value multiplicity0 2 n − ( n +1) / n − + 2 ( n − / − ( n +1) / n − − ( n − / . We show in Figure 1 the Cayley graph of a semibent function. Fig. 1. Cayley graph associated to the semibent f ( x ) = x x ⊕ x x ⊕ x x x ⊕ x x x ⊕ x x x s -Plateaued Boolean functions f with wt ( f ) = 2 ( n + s − / Theorem 1 If f : F n → F is s -plateaued and wt ( f ) = 2 ( n + s − / , then G f (if connected) isthe complete bipartite graph between the vectors in Ω f and vectors in F n \ Ω f (if disconnected, itis a union of complete bipartite graphs). Moreover, G f is a strongly regular graph with ( e, d ) = (cid:0) , ( n + s − / (cid:1) .roof. We know that the Walsh-Hadamard spectra of ˆ f in this case is { , ± ( n + s ) / } andtherefore, the spectra of f is also 3-valued, that is, { wt ( f ) , , ± ( n + s − / } = { , ± ( n + s − / } ,and thus, the Cayley graph of f in this case is strongly regular. Now, from [6], we know that if G f has three distinct eigenvalues λ = wt ( f ) > λ = 0 > λ = − λ , then G f is the completebipartite graph between the nodes in Ω f and nodes in F n \ Ω f .Since the eigenvalues of the strongly regular graph G f of f can be expressed in terms of theparameters e, d , namely λ = wt ( f ) , λ , = 12 (cid:16) e − d ± (cid:112) ( e − d ) − d − wt ( f )) (cid:17) , or equivalently, e = r + λ λ + λ + λ , d = r + λ λ , and given the Walsh-Hadamard spectraof f , the last claim follows. (cid:117)(cid:116) s -plateaued Boolean functions We now assume that f is s -plateaued and wt ( f ) (cid:54) = 2 ( n + s − / , and, therefore, the spectrumof G f is 4-valued. It is known (see [11]) that if G is connected and regular with four distincteigenvalues, then G is walk-regular. In fact, in our case a result much stronger is true (see ourtheorem below). We will need the following two propositions (we slightly change notations, tobe consistent). Proposition 3 (van Dam and Omidi [8, Proposition 4.1]). Let Γ be a connected regulargraph with four distinct eigenvalues r > λ > λ > λ . Then Γ is strongly -walk-regular if andonly if λ + λ + λ = 0 . Proposition 4 (van Dam and Omidi [8, Proposition 3.1]). A connected r -regular graph Γ on v vertices is strongly (cid:96) -walk-regular with parameters ( σ (cid:96) , µ (cid:96) , ν (cid:96) ) if and only if all eigenvaluesexcept r are roots of the equation x (cid:96) + ( µ (cid:96) − σ (cid:96) ) x + µ (cid:96) − ν (cid:96) = 0 , and r satisfies r (cid:96) + ( µ (cid:96) − σ (cid:96) ) r + µ (cid:96) − ν (cid:96) = µ (cid:96) v. In our main theorem of this section we show the counterpart for the Bernasconi-Codenottiequivalence in the case of plateaued functions. Theorem 2 Let f : F n → F be a Boolean function, and assume that G f is connected, andthat r := wt ( f ) (cid:54) = 2 ( n + s − / . Then, f is s -plateaued (with -valued spectra for f ) if and onlyif G f is strongly -walk-regular of parameters ( σ, µ, ν ) = (2 − n r + 2 n + s − − s − r, − n r − s − r, − n r − s − r ) (hence µ = ν ).Proof. We first assume that f is s -plateaued and so, its spectra is { , ± ( n + s ) / } . Consequently,the spectra of G f is 4-valued (since r := wt ( f ) (cid:54) = 2 ( n + s − / ), namely { r = wt ( f ) , λ :=2 ( n + s − / , λ := 0 , λ := − ( n + s − / } . The fact that G f is strongly 3-walk-regular followsfrom Proposition 3, since λ + λ + λ = 0, which certainly happens for our graphs. Moreover,the parameters ( σ, µ, ν ) (we removed, for convenience, the subscripts (cid:96) = 3) can be foundusing Proposition 4 as solutions to the diophantine system (recall that in our case v = 2 n and r = wt ( f )) 0 = 2 n + s − / + ( µ − σ )2 ( n + s − / + µ − ν, − n + s − / − ( µ − σ )2 ( n + s − / + µ − ν,µ n = r + ( µ − σ ) r + µ − ν, amely, ( σ, µ, ν ) = (2 − n r + 2 n + s − − s − r, − n r − s − r, − n r − s − r ).Conversely, assuming G f is a 3-walk-regular graph with the above parameters, then theeigenvalues λ > λ > λ will satisfy the equation x + ( µ − σ ) x + µ − ν = 0 , which will render the roots, λ = 2 ( n + s − / , λ = 0 , λ = − ( n + s − / . The claim is shown. (cid:117)(cid:116) Remark 1. Using a result of Godsil [10] one can easily show (under mild conditions – thus re-moving strongly regular ones, for example) that the graphs corresponding to plateaued functionsare not distance-regular.In fact, from [8] we know that the graph with four distinct eigenvalues is (cid:96) -walk-regular forany odd (cid:96) ≥ 3, but in our case we can show a lot more, by finding the involved parametersprecisely. Theorem 3 If A is the adjacency matrix of the Cayley graph corresponding to an s -plateauedwith -valued spectra (of f ), then G f is strongly (cid:96) -walk-regular for any odd (cid:96) of parameters ( σ (cid:96) , µ (cid:96) , ν (cid:96) ) , where (cid:96) = 2 t + 1 , σ (cid:96) = µ ( n + s − t − r t n + s − − r + 2 ( n + s − t , µ (cid:96) = ν (cid:96) = µ ( n + s − t − r t n + s − − r . Further,the following identity holds, for all t ≥ , A t +1 = 2 ( n + s − t A + µ ( n + s − t − r t n + s − − r J , where ( σ, µ, ν ) = (2 − n r + 2 n + s − − s − r, − n r − s − r, − n r − s − r ) .Proof. From our Theorem 2, we know that A = ( σ − µ ) A + µJ, since we know that µ = ν . We will show our result by induction, and so, for simplicity we label x := σ − µ = 2 n + s − , y := µ = 2 − n r − s − r . Assume now that A t +1 = x t A + y t J. (1)First, observe that, since our graph is regular of degree r , then AJ = rJ , and more general, A k J = r k J . Multiplying (1) by A , we get A t +3 = x t A + y t A J = x t ( x A + y J ) + y t r J = x t x A + ( x t y + y t r ) J, and consequently, we get the recurrences x t +1 = x t x y t +1 = x t y + y t r . Solving the system, we get x t +1 = x t +11 = ( σ − µ ) t +1 = 2 ( n + s − t +1) and y t +1 = y x t +11 − r t +1) x − r = µ ( n + s − t +1) − r t +1) n + s − − r , and our claim is shown. (cid:117)(cid:116) eferences 1. 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