Bent functions and strongly regular graphs

The family of bent functions is a known class of Boolean functions, which have a great importance in cryptography. The Cayley graph defined on $\mathbb{Z}_{2}^{n}$ by the support of a bent function is a strongly regular graph $srg(v,k\lambda,\mu)$, with $\lambda=\mu$. In this note we list the parameters of such Cayley graphs. Moreover, it is given a condition on $(n,m)$-bent functions $F=(f_1,\ldots,f_m)$, involving the support of their components $f_i$, and their $n$-ary symmetric differences.


Introduction
A cryptosystem is an encryption and decryption algorithm for a message.If Alice wants to send a message p to Bob, the encryption algorithm E computes the cyphertext z starting from a key K A , i.e. z = E(p, K A ). Bob uses the decryption algorithm D to recover p from a key K B , i.e. p = D(z, K B ). Necessairily, for all p, K A , K B , D(E(p, K A ), K B ) = p.Cryptosystems are called private key, if the parties know each other and have shared informations about their private keys, or public key if it is not necessary that the two parties know each other, and they have two public keys.The best known private key algorithms are DES (Data Encryption System) and its successor AES (Advanced Encryption System).The reader can find more information on cryptography in [1] .One of the most importand features for cryptographic algorithms is the confusion, i.e. the relation between any bit and all the plaintext appearing random.After the linear cryptanalysis techniques of A.
Matsui [2] , one of the research item in cryptograph was to find functions as far as possible from the linear functions, i.e. maximizing the Hamming distance, in order to resist to linear attacks, see [3] .Among the family of Boolean functions, such functions are called bent functions.In [4] [5] it is given a characterization of bent functions in terms of strongly regular graphs.Here, we give considerations on parameters of such strongly regular graphs, and a first characterization of (n, m)-

Preliminaries
Let Z 2 be the binary field.A Boolean function is a function f:Z n 2 ⟶ Z 2 and to denote f we will use two different notations: the classical notation, where the input string is given by n binary variables, and the 2 n -tuple vector representation f = (f 0 f 1 …f 2 n − 1 ) where f i = f(b(i)) and b(i) is the binary expansion of the integer i.We will denote by Ω f the support of f, i.e.
Definition 2.1.Let l be a Boolean function.
We say that l is a linear function if ∀x, We say that l is an affine function if it is a linear function plus a constant in Z 2 .

We denote with A the set of all affine functions
The nonlinearity of a Boolean function f is the minimum Hamming distance between f and an affine function, i.e.
Here we define the Abstract Fourier Transform of a Boolean function f as the rational valued function f * which defines the coefficients of f with respect to the orthonormal basis of the group characters Q w (x) = ( − 1) (w⋅x ) , when " ⋅ " is the standard inner product and w . The Walsh spectrum is the set of values of f * (w).Here we investigate the spectrum in terms of a graph eigenvalue problem.
Since eigenvectors of the Cayley graph are exactly the group characters Q w (x) = ( − 1) Tr n m (wx ) , see [6] , the following two results give a characterization of the spectrum of G f from the Walsh spectrum of f.Result 3.2. [[4]Theorem 1] The i-th eigenvalue λ i of the Cayley graph, which corresponds to the eigenvector Q b(i) , is given by 1.The largest spectral coefficients is 2. The number of non zero spectral coefficients is the rank of the adjacency matrix of G f .
3. If G f is connected, f has a spectral coefficient equal to −λ 0 if and only if its Walsh spectrum is symmetric with respect to 0.

Strongly regular graphs
A strongly regular graph with parameters (v, k, λ, μ), denoted by srg(v, k, λ, μ), is a graph with v vertices, each vertex lies on k edges, any two adjacent vertices have λ common neighbours and any two non-adjacent vertices have μ common neighbours.We give now some folklore results on strongly regular graphs, see [7] for more details.
The spectrum of the adjacency matrix of an srg(v, k, λ, μ) is fully determined by its parameters.
We write the spectrum as k, θ 2 .On the other hand, we can express the parameters of a strongly regular graph starting from its spectrum Result 4.4.The parameters λ and μ of a srg(v, k, λ, μ) may be derived from its spectrum, since: In [4][5] is given a characterization of bent functions in a graph theoretical point of view. ).
Note that in each case graphs have the parameters of the complements of the affine polar graphs VO ∓ (2n, 2), which is the graph arising from a quadric Q in the vector space V = V(2n, 2) and two points u, v ∈ V represent adjacent vertices if and only if Q(u − v) = 0. Note that the quadric is elliptic or hyperbolic while we consider the first or the second example, respectively.See the table of strongly regular graphs in [8] for more details.

Result 4 . 5 .Result 4 . 6 .Proposition 4 . 7 .
[[4],Lemma 12]  If f is a bent function, the graph G f is a strongly regular graph with λ = μ.[[5],Theorem 3] Bent functions are the only functions whose associated Cayley graph G f is a strongly regular graph with λ = μ.The Cayley graph G f of a bent function is exactly one of the following:

Consider now functions F:Z n 2 ⟶ Z m 2 , 2 ⟶ Z 2 .Definition 5 . 1 .2 n − 2 n 2 2Definition 5 . 2 .Proposition 5 . 3 .
F(x 1 , …, x n ) = (f 1 , …, f m ), where for each i, f i :Z n The set of affine vectorial functions A n, m is defined as in the case m = 1.We can introduce two different way to express the nonlinearity of a vectorial Boolean function:nl(F) = min v ∈Z n 2 ∖ {0} Nl(F ⋅ v) Nl(F) = min ϕ∈A n , m | {x ∈ Z n 2 | F(x) ≠ ϕ(x)} | A (n, m)-bent function, or vectorial bent function, is a function F = (f 1 , …, f m ) such that nl(F) = , orequivalently each linear combination of f 1 , …, f m is a bent function.In order to give graph based properties of (n, m)-bent functions we need now to define the set operation symmetric difference, which is the equivalent of the logical operation XOR.The symmetric difference between two sets A and B isA△B = (A ∖ B) ∪ (B ∖ A) = (A ∪ B) ∖ (A ∩ B).The power set of any set X is an elementary abelian 2-group under the operation of symmetric