BENT AND VECTORIAL BENT FUNCTIONS, PARTIAL DIFFERENCE SETS, AND STRONGLY REGULAR GRAPHS

. Bent and vectorial bent functions have applications in cryptography and coding and are closely related to objects in combinatorics and ﬁnite geometry, like diﬀerence sets, relative diﬀerence sets, designs and divisible designs. Bent functions with certain additional properties yield partial diﬀerence sets of which the Cayley graphs are always strongly regular. In this article we continue research on connections between bent functions and partial diﬀerence sets respectively strongly regular graphs. For the ﬁrst time we investigate rela-tions between vectorial bent functions and partial diﬀerence sets. Remarkably, properties of the set of the duals of the components play here an important role. Seeing conventional bent functions as 1-dimensional vectorial bent func- tions, some earlier results on strongly regular graphs from bent functions follow from our more general results. Finally we describe a recursive construction of inﬁnitely many partial diﬀerence sets with a secondary construction of p -ary bent functions.


Introduction
Let p be a prime, n be a positive integer and let V n be an n-dimensional vector space over F p . For a function f from V n to F p , the Walsh transform of f is defined to be the complex valued function W f on V n where p = e 2πi/p and u · x denotes a (nondegenerate) inner product of V n . If V n = F n p , then we may choose the conventional dot product for u · x, the standard inner product when V n = F p n is u · x = Tr n (ux), where Tr n (z) denotes the absolute trace of z ∈ F p n .
The function f : V n → F p is called bent if the absolute value of its Walsh transform W f (u) is p n/2 for all u ∈ V n .
If p = 2, then p = −1 and W f (u) is an integer. Hence a Boolean bent function f can only exist when n is even, and then W f (u) = 2 n/2 (−1) f * (u) for some Boolean function f * , called the dual of f . As is well known, f * is bent as well. (Strictly speaking, f * is the dual of f with respect to the considered inner product since differently to the Walsh spectrum, the Walsh coefficient W f (u) at u ∈ V n is not independent from the inner product in (1).) If p is odd, then the Walsh coefficient W f (u) at u ∈ V n of a bent function f always satisfies (cf. [12,13]) (2) W f (u) = ± f * (u) p p n/2 : p n ≡ 1 mod 4; ±i f * (u) p p n/2 : p n ≡ 3 mod 4, where f * is a function from V n to F p , again called the dual of f . A bent function (which when p is odd, we call a p-ary bent function) is said to be weakly regular if W f (u) = ζ f * (u) p p n/2 , where ζ ∈ {±1, ±i} does not depend on u. If ζ = 1, we call f regular. A Boolean bent function is hence always regular. A bent function for which ζ changes with u is called non-weakly regular. Weakly regular bent functions f always appear in pairs since the dual function f * in (2) is also a (weakly regular) bent function, [12]. This is in general not true for non-weakly regular bent functions, see [5,6].
Motivated by connections to cryptography and coding, bent functions, and generalizations, have been investigated in many aspects, see for instance [16]. One approach which provides to the understanding of bent functions is to relate them to combinatorial objects, first of all they can be identified with relative difference sets in the elementary abelian group. In this paper we are interested in partial difference sets and strongly regular graphs obtained from bent functions.
Let G be a group of order v. A subset D of G with k elements is called a (v, k, λ, µ) partial difference set (PDS) of G, if every nonzero element of D can be represented as d 1 − d 2 , d 1 , d 2 ∈ D, in exactly λ ways, and every nonzero element of G \ D can be represented as d 1 − d 2 , d 1 , d 2 ∈ D, in exactly µ ways. Combinatorial objects related with partial difference sets are strongly regular graphs. A k-regular graph with v vertices is called a (v, k, λ, µ) strongly regular graph, if every two adjacent vertices have exactly λ common neighbours, and every two non-adjacent vertices have exactly µ common neighbours.
For a group G and a subset D of G such that D = −D and 0 ∈ D, we can define a graph, called Cayley graph generated by D, as follows: The vertices of the graph are the elements of G, and two vertices g, h are adjacent if g − h ∈ D. Let now G be a group of order v and D be a subset of G for which D = −D and 0 ∈ D. The Cayley graph generated by D is a (v, k, λ, µ) strongly regular graph if and only if D is a (v, k, λ, µ) partial difference set of G, see [15].
Connections between bent functions and strongly regular graphs were already observed in [1,2], where it is shown that the Cayley graph of D f 0 = {x : f (x) = 0} for a Boolean bent function f is strongly regular. Indeed, Boolean bent functions have been precisely characterized by this property. (In [1,2], 0 is included in the PDS). In the significant paper [19], constructions of strongly regular graphs from certain classes of ternary bent functions are proposed, and families of strongly regular graphs are obtained, some of which were previously unknown. In [8,11] the construction is generalized for p-ary bent functions.
In this article we for the first time investigate connections between vectorial bent functions and partial difference sets respectively strongly regular graphs. Notably, a property of the duals of vectorial bent functions which was investigated only recently in [7] plays here a crucial role. In this light, some results on connections between strongly regular graphs and conventional bent functions can be seen as a special case of our results on vectorial bent functions. Finally, in the last section we employ a secondary construction of bent functions to recursively generate infinite sequences of strongly regular graphs.

Preliminaries
In this section we recall vectorial bent functions and discuss some properties of their duals, which recently have been investigated in [7]. As it turns out, it exactly depends on these properties, whether a vectorial bent function induces partial difference sets.
. . , f m (x)) from V n to F m p is called a vectorial bent function if every nontrivial linear combination c 1 f 1 + c 2 f 2 + · · · + c m f m is bent, or equivalently, for every nonzero c ∈ F m p , the p-ary function f c (x) = c·F (x) is bent. We call f c a component function of F . In the framework of finite fields, a function F : F p n → F p m for a divisor m of n, is a vectorial bent function, if for all nonzero α ∈ F p m , the component function For a vectorial bent function we always must have m ≤ n. If all component functions are regular, then m ≤ n/2, see [17]. In particular when p = 2, then m ≤ n/2.
Let f 1 and f 2 be (weakly regular) bent functions, let f * 1 and f * 2 be their duals and suppose that f 1 + f 2 is also bent. In [7] it is pointed out that then f * 1 + f * 2 is in general not a bent function. In general, the collection of the duals of the component functions of a vectorial bent function, does not even form a vector space. Accordingly, in [7] the following definition was given. For Boolean bent functions, the concept of a vectorial dual is a generalization of a Boolean bent function and its dual seen as a 1-dimensional vector space over F 2 . This is different for a p-ary (weakly regular) bent function f , which seen as a 1-dimensional vector space over F p consists of all constant multiples of f . The set of the duals is in general not closed under addition. Hence, seen as a 1-dimensional vector space, a (weakly regular) p-ary bent function does in general not have a vectorial dual.
Though several classical constructions of vectorial bent functions are vectorial dual-bent, this property seems to be quite rare. Some examples are given in the following. For more details we refer to [7].

Strongly regular graphs from vectorial bent functions
In this section we investigate for the first time partial difference sets respectively strongly regular graphs obtained from vectorial bent functions F (x) = (f 1 (x), f 2 (x), . . . , f m ( x)) from V n to F m p for some odd prime p. Vectorial bent functions which we consider here, will always have the following properties: n is even, and all component functions f c of F are regular or all are weakly regular with = −1. Equivalently, for all component functions we have W fc (u) = p n/2 f * c (u) p where = 1 or = −1 is independent from c. Furthermore, we always suppose that F (0) = 0 = (0, . . . , 0), and we suppose that F satisfies the condition that F (x) = F (−x). The last condition seems quite restrictive, but every vectorial dual-bent function we know so far does satisfy this condition.
The main tool is the following characterization of partial difference sets via characters, see [15,19].
Then D is a (ν, k, λ, µ) partial difference set if and only if for each non-principal character χ, We will further use the following relation between the parameters of a partial difference set, see for instance [15,Proposition 1.4(b)]: . . , f m ( x)) be a vectorial bent function from V n to F m p (such that all component functions are regular or all are weakly regular with = −1), for each i = 1, 2, . . . , m, let D fi 0 be the set D fi and show that under some condition, D is a partial difference set, hence gives rise to a strongly regular graph. Most notably the property of F being vectorial dual-bent will play a central role. We start with a lemma, which relates χ(D) with the Walsh transforms of the component functions of F . For u ∈ V n , we denote by and for u = 0, Proof. Denote by D j1,...,jm the set Then we have Observing that c∈F m p c·j p = 0, j = 0, we get the assertion. For u = 0 this summation yields and the result on the cardinality of D 0 follows. From Lemma 2, for the cardinality of D F 0 for a vectorial bent function, we obtain the following corollary. Corollary 1. Let n be even and F (x) = (f 1 (x), f 2 (x), . . . , f m (x)) be a vectorial bent function from V n to F m p . Then the cardinality of Proof. By our assumption, As obvious, the smallest respectively the largest possible value for The latter condition equivalently means that f c is weakly regular but not regular respectively regular for all c ∈ F m p \ {0}. With Lemma 2 we also obtain the following proposition for a special class of vectorial bent functions, namely for vectorial dual-bent functions. We remark that then the condition f * c (0) = 0 for all c ∈ F m p \ {0} simply means that F * (0) = 0.
. . , f m (x)) be a vectorial dual-bent function, and let {g * 1 , g * 2 , . . . , g * m } be a basis for the space of the dual F * of F . For each non-principal character χ u of V n , Proof. By our assumption, for every ( Since {g * 1 , g * 2 , . . . , g * m } is a basis for the vectorial dual F * of F , the above sum can be written as As the (nonzero) u ∈ F n p is fixed by the non-principal character χ u , also g * 1 (u), g * 2 (u), . . . , g * m (u) are fixed. We write the above sum as As a consequence, otherwise.
Since χ u (D), u = 0, takes on two different values, the set D is a partial difference set in V n .
Note that |D F 0 | is maximal respectively minimal by the bound in Corollary 1. Remark 1. A strongly regular graph with parameters (n 2 , r(n − ), n + r 2 − 3 r, r 2 − r) is called Latin Square or Negative Latin Square type when = 1 or = −1, respectively. Comparing with the parameters of the partial difference sets in Theorem 1, we observe that the Cayley graph of D F 0 in Theorem 1 is of Latin Square type if = 1, hence all components f c are regular, and of Negative Latin Square type if = −1, i.e. all components f c are weakly regular but not regular. There are many constructions of graphs of Latin Square type, however new graphs of Negative Latin Square type obtained from bent functions, which are not equivalent to earlier known graphs with the same parameters, have been presented e.g. in [19], see also our discussion in the next section and [19] for details.
Since vectorial dual-bent functions always appear in pairs, assuming that F and F * both satisfy our conditions, also partial difference sets as obtained in Theorem 1 come in pairs. Recall that all vectorial dual-bent functions we know so far satisfy F (x) = F (−x). Recall from [8,11,19] that (weakly) regular bent functions f : V n → F p , n even, for which f (αx) = α f (x) for all α ∈ V n and some fixed positive integer with gcd( − 1, p − 1) = 1, give rise to strongly regular graphs. The next proposition connects our results on vectorial bent functions and strongly regular graphs with results in [8,11,19].
Proposition 2. Let f : V n → F p be a weakly regular bent function for which there exists an integer with gcd( − 1, p − 1) = 1 such that f (αx) = α f (x) for all α ∈ F p . Then f (seen as a 1-dimensional vector space of bent functions) is a vectorial dual-bent function.
Since gcd( −1, p−1) = 1, the mapping α → α 1− is a permutation of F * p . Therefore {(αf (x)) * : α ∈ F * p } = {αf * (x) : α ∈ F * p }, i.e. f is vectorial dual-bent. By Proposition 2 and Theorem 1 we can see the result of [8,11,19] as summarized below, as special case of the general result for vectorial dual-bent functions. Bent functions f with the above described properties in fact induce strongly regular graphs as these properties guarantee that f (seen as 1-dimensional vectorial bent function) is vectorial dual-bent.
We finish this section pointing to some interesting properties of partial difference sets obtained from vectorial bent functions.
Supposing that at least some of the subspaces of a vectorial dual-bent function F (satisfying our conditions) are also vectorial dual-bent, then Theorem 1 reveals several partial difference sets on the same group.
, the strongly regular graph G F one obtains by taking the edges which are in both strongly regular graphs, G f1 and G f2 .
A similar observation we will now deduce about the union of two related partial difference sets: If F = f 1 , f 2 , . . . , f m is an m-dimensional vectorial bent function (with F (x) = F (−x) and satisfying our regularity condition), and F 1 = f 1 , f 2 , . . . , f k , k < m, is a k-dimensional subspace of F , and both, F and F 1 are vectorial dual-bent, then D F1 0 \ {0} is a partial difference set containing the partial difference set D F 0 \ {0} as a subset. We have moreover the following result on the complement of D F 0 as a subset of D F1 0 .
Therefore χ u (D) takes on the two different values otherwise.

Again, with (3) we get
By Corollary 4 the partial difference set D F1 0 \ {0} is the disjoint union of two partial difference sets, andḠ F are the strongly regular graphs corresponding to the partial difference sets 0 \D F 0 , then the graph G F is a strongly regular subgraph of G F1 , and two vertices in the strongly regular graphḠ F are adjacent if and only if they are adjacent in G F1 , but not G F . Equivalently, G F andḠ F are two disjoint strongly regular graphs (no common edges), for which the union is again strongly regular. We remark, that for two partial difference sets D 1 , D 2 with D 2 ⊂ D 1 it does in general not hold that D 1 \ D 2 is again a partial difference set, but it is also not an unusual property, see e.g. [10]. Remark 3. In Corollary 4 we require that besides from F , also the k-dimensional subspace F 1 is vectorial dual-bent. All our examples of vectorial dual-bent bent functions have this property for any of their subspaces. However, duality of bent functions is a quite complex topic. By now we cannot exclude the existence of a vectorial dual-bent function F with a subspace F 1 for which the set of the duals of its components only forms a subset in the vector space of the duals of the components of F .
As for also many other objects, examples for vectorial-dual bent functions are obtained with spreads of V n . For these functions all components have algebraic degree n/2. The unions of certain numbers of spread elements are then the partial difference sets described in this section. We here described connections between partial difference sets and vectorial bent functions in terms of properties of duals. This may also raise interest in finding more classes of vectorial dual-bent functions. Some examples (also of algebraic degree smaller than n/2) are 1-dimensional vectorial dual-bent functions investigated in connection with strongly regular graphs in [8,19], and functions in the list in the preliminaries. Some of the 1-dimensional examples yield new graphs, see [19] or the construction in the next section. We checked some graphs for the functions in the list in the introduction for some small parameters, all of which are of Latin Square type and solely yield new interpretations of known graphs. To further investigate vectorial dual-bent functions and to test the graphs for (non-quadratic) examples in also higher dimension should be a worthwhile topic.

A secondary (recursive) construction of strongly regular graphs
In [8,11,19] several p-ary bent functions satisfying the properties in Corollary 3, i.e. bent functions which -seen as vectorial bent functions in dimension 1are dual-bent, have been considered. By this, even some new strongly regular graphs have been found, [19]. However, in general p-ary bent functions do not have these properties. In this section we construct from bent functions that satisfy the conditions in Corollary 3 new such bent functions in a higher dimension. By this we obtain a recursive procedure to generate (infinitely many) bent functions for strongly regular graphs. Finally we analyse the resulting graphs and give some examples.
For the convenience of the reader and for the sake of completeness, we first summarize some facts on p-ary bent functions and strongly regular graphs. For a function f from V n to F p , let D f i = {x ∈ V n : f (x) = i}, i = 0, 1, . . . , p − 1, and let S := {x ∈ F * p : x is a square in F * p } and N := {x ∈ F * p : x is a nonsquare in F * p }. We then define the sets D S and D N as The following summarizes Theorem 1, Theorem 2 and Remark 3(i) in [8]. Let p be an odd prime, n be an even integer, and let f : V n → F p be a (weakly) regular bent function, hence W f (u) = p n/2 f * (u) p for all u ∈ V n , and fixed = ±1. Suppose that for some positive integer with gcd( − 1, p − 1) = 1 we have f (αx) = α f (x) for any α ∈ F p .
Recall that a strongly regular graph with parameters (n 2 , r(n − ), n + r 2 − 3 r, r 2 − r) is called Latin Square or Negative Latin Square type when = 1 or = −1, respectively. The Cayley graphs of D f S , D f N and the Cayley graph of D f 0 \{0} (see also Remark 1) are of Latin Square type if = 1, hence f is regular, and of Negative Latin Square type if = −1, i.e. f is weakly regular but not regular.
There are many constructions of strongly regular graphs of Latin Square type, quadratic bent functions yield affine polar graphs, which can also be constructed using projective two-weight codes, see Result 7 in [19] and the discussion thereafter. Hence graphs from weakly regular, but not regular, non-quadratic bent functions seem most interesting. Several examples of partial difference sets respectively strongly regular graphs can be found in [8,19].
The following theorem enables a recursive construction of bent functions which induce strongly regular graphs.
Theorem 2. Let g 0 , g 1 : V n → F p be two (distinct) bent functions for which there exists an integer with gcd( − 1, p − 1) = 1, such that g i (αx) = α g i (x), i = 0, 1, for all α ∈ F p . Then the function F : V n × F 2 p → F p given by for a non-zero element u ∈ F p , is a bent function satisfying F (α(x, y, z)) = α F (x, y, z) for all α ∈ F p .
Proof. For a ∈ V n and b, c ∈ F p we have Consequently F is bent. The property F (α(x, y, z)) = α F (x, y, z) for all α ∈ F p follows with the assumption that g i (αx) = α g i (x) holds for all α ∈ F p . .

Remark 4.
If g 1 − g 0 has degree d, then F has degree d + p − 1. The obtained function can be used in a further step to generate bent functions for strongly regular graphs in dimension n + 4.
Before we give some examples, we interpret the construction in Theorem 2 as a construction of a strongly regular graph from the graphs obtained with g 0 and g 1 .
By definition of the graph obtained from F , the vertices (x, y, z), (x 1 , y 1 , z 1 ), x, x 1 ∈ V n , y, y 1 , z, z 1 ∈ F p , are adjacent if and only if F (x − x 1 , y − y 1 , z − z 1 ) is a nonzero square (nonsquare, equal zero). With the observation that we obtain that (x, y, z), (x 1 , y 1 , z) are adjacent if and only if g 0 (x − x 1 ) is a nonzero square (nonsquare, equal zero), i.e.
x and x 1 are adjacent in the strongly regular graph of g 0 . We remark that for every (y, z) ∈ F p × F p the set of vertices {(x, y, z) | x ∈ F n p } forms a subgraph which is a copy of the strongly regular graph of g 0 .
Again with (7), we obtain that (x, y, z), (x 1 , y 1 , z 1 ), z 1 = z, are adjacent if and only if g 1 (x − x 1 ) + u(y − y 1 )(z − z 1 ) −1 is a nonzero square (nonsquare, equal zero), or equivalently We finish with some examples for the construction. We will employ the very well understood class of quadratic bent functions. Note that a quadratic bent function f always satisfies the condition f (αx) = α f (x) with = 2. Recall that a quadratic bent function is always weakly regular, and a quadratic function from F n p to F p , n even, p odd, of the form c 1 x 2 1 +c 2 x 2 2 +· · ·+c n x 2 n , is a weakly regular, but not a regular bent function, if and only if n ≡ 2 mod 4 and ∆ := n i=1 c i is a nonzero square in F p , or n ≡ 0 mod 4 and ∆ is a non-square in F p , see [12]. In our examples we will only consider the (more interesting) case of weakly regular but not regular bent functions, hence they give rise to strongly regular graphs of Negative Latin Square type. The articles [8,19] contain tables of strongly regular graphs with relevant parameters, which are based on the comprehensive lists [3,9]. Using MAGMA we determine the size of their automorphism group and the p-rank of their adjacency matrix, to examine equivalence to the graphs listed in [8,19], respectively to graphs in the comprehensive list in [3]. Example 1. g 0 (x 1 , x 2 ) = x 2 1 + x 2 2 , g 1 (x 1 , x 2 ) = 2x 2 1 + 2x 2 2 ∈ F 3 [x 1 , x 2 ], u = 1. Then F (x 1 , x 2 , x 3 , x 4 ) = 2x 2 4 (2x 2 1 + 2x 2 2 ) + x 3 x 4 + x 2 1 + x 2 2 .
The examples are of affine polar type, the latter equivalent to an example in [14], see also [19,Section 4.2].