Critical Groups for Cayley Graphs of Bent Functions

Transcription

1 Critical Groups for Cayley Graphs of Bent Functions Thomas F. Gruebl Adviser: David Joyner December 9,

2 Introduction This paper will study the critical group of bent functions in the p-ary case. Let f be a p-ary function from GF (p) m GF (p). Suppose f(0) = 0 and suppose f is even. Any such f can be described using its corresponding edge-weighted additive Cayley graph. Throughout this paper, we will refer to the Cayley graph of a function as Γ, and we will assume that all Cayley graphs are connected. In general, if α is a function theoretic property of f, does there exists a graph theoretic property of Γ that corresponds to property α? Γ generally encodes everything we need to know about f such that each of its function theoretic properties is reflected by some graph theoretic property of Γ. But are there any known graph theoretic properties of Γ that tell us something about a property of f? For example, let f have the property that it is constant everywhere except at f(0) = 0. What property of Γ does that correspond to? Suppose it corresponds to property β. Do we know anything about β from graph theory that tells us something about f? This idea that the natural function theoretic properties of f are connected to the natural graph theoretic propoerties of Γ is what we will explore in this paper. The ability to include examples for higher dimensions is limited due to the sheer size of these groups, forcing the paper to be of a more theoretical nature. However, low dimensional examples will be included. These examples will be Boolean functions from GF (2) m GF (2), and their additive Cayley graphs will be utilized to compute the critical groups of these bent functions. In the case of Boolean functions, Bernasconi shows that the Cayley graph of f is a strongly regular graph with parameters (2 m, k = supp(f), λ, µ) when f is bent. In fact, this is an if and only if correspondence. However, this correspondence fails when p > 2. We will look for higher dimension analogs to this correspondence: when p > 2, to what kind of graphs do bent functions correspond? Another question is whether there is a property of the critical groups that arises from bent functions. In particular, does the critical group of the Cayley graph of a bent function depend solely upon the size of the support of f? It will be necessary to give the reader a foundational education on the topics of Boolean functions, additive Cayley graphs, strongly regular graphs, p-ary functions, edge-weighted additive Cayley graphs, bent functions, and the idea of a critical group. These topics will be discussed at length to educate 2

3 the reader in the specific theory needed to pursue the intent of the paper. All of the aforementioned topics will use techniques garnered from Dino Lorenzini s Smith normal form and Laplacians, Graph eigenvalues and Walsh spectrum of Boolean functions by Pantelimon Stanica, and Edge-weighted Cayley graphs and p-ary bent functions by Celerier et al. 3

4 Definitions and Background For now we take these definitions verbatim from [CJMPW]. We make a small change in notation and let V m = GF (p) m. GF (p) is the finite field with p elements, which may be regarded as the set of integers 0, 1,..., p 1, where addition and multiplication are performed mod p. Definition 1. A p-ary function is a function f : V m GF (p). When p = 2, we call this a Boolean function. Note that a p-ary function may be regarded as a vector of length p m with entries in GF (p). Denote supp(f) = {v V m f(v) 0}. Note the number of non-zero coordinates of the vector representation of f is equal to supp(f). Definition 2. The Walsh(-Hadamard) transform of a function f : V m GF (p) is a complex-valued function on V m that can be defined as W f (u) = x V m ζ f(x) u,x, where ζ = e 2πi/p and, is the usual inner product. Definition 3. When f is complex-valued, we define the analogous Fourier transform of the function f as F T f (y) = x V where f C : V m C is obtained by identifying GF (p) = {0, 1,..., p 1} C. f C (x)ζ x,y, (1) In the Boolean case, the Fourier Transform is related to the Walsh Transform as follows: F T f (x) = f(y)( 1) x y = 1 (1 ( 1) f(y) )( 1) x y 2 y GF (2) m y GF (2) m (because f(y) = 1 ( 1)f(y) 2 for any Boolean function) 4

5 = 1 ( 1) x y y GF (2) m y GF (2) m ( 1) f(y)+x y = 1 2 W f(x) + 2 m 1 δ 0 (x). Definition 4. We call f bent if for all u V m. W f (u) = p m/2, Recall that for a graph, Γ = (V, E), two vertices in V are neighbors if they are connected by an edge in E. Unless otherwise stated, all graphs in this paper will be undirected, simple graphs. Definition 5. A connected (undirected) graph Γ = (V, E) with vertex set V and edge set E is a (ν, k, λ, µ)-strongly regular graph provided it has the following properties: (a) Γ has v vertices and each vertex g V has k neighbors, i.e., its degree is k, deg(g) = k. (b) Distinct vertices g 1 and g 2 have λ common neighbors if g 1 and g 2 are neighbors, and µ common neighbors if g 1 and g 2 are not neighbors. Definition 6. Let f be an even GF (p)-valued function on V m. The Cayley graph of f is defined to be the edge-weighted graph Γ f = (GF (p) m, E f ), whose vertex set is V = GF (p) m and whose set of edges E = E f is defined by E f = {(u, v) V V f(u v) 0}, where the edge (u, v) E f has weight f(u v). We routinely identify GF (p) with {0, 1,..., p 1} Z when referring to the edge-weights of Γ f. In other words, the edge-weighted Cayley graph Γ = Γ f associated to p-ary function f is the edge-weighted graph constructed as follows. Let the vertices of the graph be the elements of GF (p) m. Two vertices g 1 and g 2 5

6 are connected by an edge of weight i if f(g 2 g 1 ) = i. Let D i be the set of all v GF (p) m, such that f(v) = i. If D 1 i = D i for all i, the graph Γ is undirected. Recall that vertex v is incident to edge e if e = (v, w) E for some w V. The incidence matrix of a graph Γ is a V E matrix where V and E are the number of vertices and edges respectively, such that b ij = 1 if the vertex v i and edge e j are incident and 0 otherwise. Definition 7. Suppose the graph Γ has n vertices. Let B denote the incidence matrix of the graph Γ. The Laplacian of Γ can be defined as the (n n) matrix L = BB T. Definition 8. The reduced Laplacian is the matrix obtained by removing the first row and first column of the Laplacian, L. We denote this by L. Definition 9. If Γ is a graph on m vertices with Laplacian L, then the critical group of the graph Γ, denoted K(Γ), is Z m 1 /Im(L ), where Im(L ) is the Z-span of the columns of L. We recall finite Abelian groups by referencing Theorem 13.5 (the fundamental theorem of finite Abelian groups) in [J]. Every finite Abelian group is isomorphic to a direct product of cyclic groups of prime power order. So for an Abelian group, G, there are integers a i 1 and primes p 1,..., p k (not necessarily distinct) such that G = k (Z/p a i i Z). i=1 This decomposition is unique up to the order of the factors. For example, Z/180Z = (Z/4Z) (Z/5Z) (Z/9Z). 6

7 Boolean bent functions and their respective Cayley graphs The purpose of this paper is to see if there are connections between two well-known fields: critical groups and strongly regular Cayley graphs. We want to know if the graph theory can be reconciled with what is known about critical groups. Is there a connection to be made, and if so what is the connection? In order to connect critical groups to graph theory, we must understand properties of the Cayley graph of bent functions. But why use bent functions? The study of bent functions is gaining popularity currently because of their usefulness in cryptography. Cryptographers want to introduce as much controlled randomness in their systems as possible in order to make their code very hard to break while keeping it relatively simple for those who have the key. Bent functions are very useful to this end because they are the furthest functions from being linear. Their non-linearity makes it hard for an adversary to detect any kind of pattern, but the fact that they are functions allows an ally to easily decode any hidden messages. The process for solving this problem starts with reviewing work that others have done in this area. Pantelimon Stanica [S] has published a paper relating graph theoretic properties to bent functions and Dino Lorenzini [L] has published a paper relating graph theoretic properties to critical groups. Stanica gives formulas for the eigenvalues, and their respective multiplicities, of the Cayley graph of bent functions in the Boolean case. We can later take this information and combine it with results from Lorenzini in order to make Lorenzini s results more concrete. Applying formulas in Stanica to Lorenzini will also help computationally, so we can provide more examples for understanding the theoretic properties being discussed. Before we can review Lorenzini s paper entitled Smith normal form and Laplacians, we must first introduce some theorems pulled from Stanica s Graph eigenvalues and Walsh spectrum of Boolean functions, specifically Theorem 2.1 (items 1 and 2) and Theorem 3.1 [S]. Let r i be the binary representation of i regarded as a vector in GF (2) m for each i between 0 and 2 m 1. 7

8 Theorem 10. (Theorem 2.1, [S]) 1 The following are equivalent: 1. Let f : GF (2) m GF (2), and let γ 0,..., γ N, where N = 2 m 1, be the eigenvalues of its associated Cayley graph Γ f. Then λ i = F T f (r i ), for any i. 2. The multiplicity of the largest spectral coefficient of f, F T f (r 0 ), is equal to 2 m dim supp(f). Theorem 11. (Theorem 3.1, [S]) 2 When f is bent, Γ f with parameters (ν, k, λ, µ) where is strongly regular ν = 2 m, k = supp(f), λ = µ, (2) and so the eigenvalues of the adjacency matrix are λ 3 = supp(f) µ, λ 2 = supp(f) µ, λ 1 = supp(f), (3) with multiplicities m 3 = λ 2(2 m + 1) + λ 1 2λ 2, m 2 = λ 2(2 m 1) λ 1 2λ 2, m 1 = 1. (4) In order to calculate µ, we first must calculate λ 2 by a different formula. Theorem 10 gives that λ 2 = F T f (r 2 ). Recall that F T f (x) = 1 2 W f(x) + 2 m 1 δ 0 (x). Here W f (x) = 2 m/2 if and only if f is bent. So for our case F T f (r 2 ) = 1 2 W f(r 2 ) λ 2 = 2 m/2 1. We can now use formula (3) and our new value for λ 2 in order to solve for µ: 2 m/2 1 = λ 2 = supp(f) µ µ = supp(f) 2 m 2. Because f is bent, supp(f) = 2 m 1 ± 2 m/2 1, so µ = 2 m 2 ± 2 m/ Stanica uses V m to denote GF (2) m, F 2 to denote GF (2), and W (f)(b(i)), where b(i) is the binary representation of i, as notation for what we call the Fourier Transform, F T f (r i ). It follows that his Walsh spectrum is our Fourier spectrum. Stanica s wt(f), the cardinality of Ω f, is what we call the size of the support of f, or supp(f) 2 We change Stanica s notation as follows: the parameters (v, r, e, d) become (ν, k, λ, µ); Ω f = supp(f); Stanica uses n for the dimension, while we use m for the dimension. 8

9 We would like to verify that formulas (3) - (4) are correct in the p = 2, m = 4 case for specific examples of bent functions. After using Sagemath to compute supp(f), we can compute the eigenvalues of the adjacency matrix using the formulas. We then compare these answers with what Sagemath gives us as the eigenvalues. Example 12. Let p = 2, let m = 4, and let f(x 0, x 1, x 2, x 3 ) = x 0 x 1 + x 2 x 3. Here, Sagemath gives that supp(f) = 6. eigenvalues of the adjacency matrix: Using this, we compute the λ 1 = supp(f) = 6, λ 2 = supp(f) µ = 6 2 = 2, λ 3 = λ 2 = 2. We now utilize Sagemath to check if our eigenvalues derived from the formulas match what Sagemath computes the eigenvalues to be. Sagemath gives: λ 1 (A) = 6, λ 2 (A) = 2, λ 3 (A) = 2. The computed eigenvalues agree with what Sagemath gives for the eigenvalues, so we compute the multiplicities m 1 = 1, m 2 = λ 2(2 m 1) λ 1 = 2(24 1) 6 2λ 2 2(2) m 3 = λ 2(2 m 1) + λ 1 = 2(24 1) + 6 2λ 2 2(2) = 6, = 9, and compare to the actual multiplicity of each eigenvalue given by Sagemath: m 1 = 1, m 2 = 6, m 3 = 9. The computed and given multiplicities also agree. This example gives us a better understanding of the results from Stanica and allows us to move into Lorenzini. 9

10 We can now apply (3) and (4) for the adjacency eigenvalues to the formulas given by Lorenzini for the Laplacian eigenvalues. This yields: when i = 1 λ 1 (L) = deg(γ f ) λ 1 (A) = supp(f) supp(f) = 0. (5) When i = 2 we have λ 2 (L) = supp(f) supp(f) µ = 2 m 1 ± 2 m/2 2 m/2 1. (6) When i = 3 we have λ 3 (L) = supp(f) + supp(f) µ = 2 m 1 ± 2 m/2 + 2 m/2 1. (7) Again we use the example where p = 2, m = 4, and f(x 0, x 1, x 2, x 3 ) = x 0 x 1 + x 2 x 3 to check that (5), (6), (7) are correct, as well as to give a more concrete understanding. Example 13. Using the results from Example 12 we compute the eigenvalues of the Laplacian: λ 1 (L) = 0, λ 2 (L) = 2 m 1 ± 2 m/2 2 m/2 1 = ± 2 4/2 2 4/2 1 = 4 or 8, λ 3 (L) = 2 m 1 ± 2 m/2 + 2 m/2 1 = ± 2 4/ /2 1 = 8 or 12. In comparison, the eigenvalues of the Laplacian given by Sagemath are λ 1 (L) = 0, λ 2 (L) = 4, λ 3 (L) = 8. Both sets of eigenvalues are the same, showing that (5), (6), (7) hold true in this example. Now that we have verified that (5), (6), (7) are true, we can move deeper into Lorenzini and start to apply our findings to critical groups. 10

11 Critical Groups Having extracted all we can from Lorenzini in relation to the eigenvalues for the Laplacian of the Cayley graph of a bent function, we can now inspect his work for relationships to critical groups. Lorenzini immediately presents a relevant result in Proposition 2.1 [L]: Proposition 14. (Proposition 2.1, [L]) 3.Let γ 1,..., γ m 1 be the complete list m 1 of all non-zero eigenvalues of the Laplacian of Γ f. Then γ i = K(Γ f ) p m. We continue on to apply formulas (5), (6), and (7) to Proposition 14, giving us the following formula K(Γ f ) = λ 2 (L) m 2(L) λ 3 (L) m 3(L) p m. (8) We test the result of (8) by continuing to use the function and results from Example 13. Example 15. We compute K(Γ f ) as follows: K(Γ f ) = λ 2 (L) m 2(L) λ 3 (L) m 3(L) 2 m = = = Since Sagemath gives we get K(Γ f ) = (Z/8Z) 5 (Z/32Z) 4, K(Γ f ) = (Z/8Z) 5 (Z/32Z) 4 = = = 2 35, and formula (8) holds for our example. Lorenzini s next relevant result is Proposition 2.3 [L]: 3 We replace Lorenzini s M with the special case of the Laplacian matrix of a bent function because it satisfies the assumptions on M and is the only case we are concerned about in the context of this paper. This allows us to say that R, the integer vector generating the kernel of M, is the all ones vector, R (the integer vector generating the kernel of M T ) = R, and R R = n = p m by reasons explained in [L]. Instead of defining the characteristic polynomial we simply list the eigenvalues of the Laplacian. His definition of Φ(M) is equivalent to our definition of the critical group, K(Γ f ) 11 i=1

12 Theorem 16. (Proposition 2.3, [L]) 4 Let λ i (L) ±1 be a non-zero integer eigenvalue of the Laplacian, L, with multiplicity m i (L). 1. Let w i be an integer eigenvector of L for λ i (L). Assume that the greatest common divisor of its coefficients is 1. Then the order of the class of w i in K(Γ f ) is divisible by λ i (L)/gcd(λ i (L), p m ) and divides λ i (L) (2.2 in [L]). Let w i be any integer eigenvector of L T for λ i (L), with w i w i 0. Then the order of the class of w i in K(Γ f ) is divisible by λ i (L)/gcd(λ i (L), w i w i). 2. If there exists a prime d such that d λ i (L) but d p m, then K(Γ f ) contains a subgroup isomorphic to (Z/d ord d(λ i (L)) Z) m i(l). 3. K(Γ f ) contains a subgroup isomorphic to (Z/λ i (L)Z) m i(l) 1. Proof: See Lorenzini [L]. We find nothing of a non-trivial nature from part one, nothing of interest from part 2, but something interesting in part 3 of the proposition. The result from part 3 gives that K(Γ f ) (Z/λ i (L)Z) m i(l) 1. (9) We next present Proposition 2.6 from Lorenzini [L] because it will be used in the proof of Theorem 18, a result closely related to Equation 9 Theorem 17. (Proposition 2.6, [L]) 5 Let λ 1,..., λ t denote the distinct nonzero Laplacian eigenvalues of Γ f. Then the product of λ 1,..., λ t is an integer, d, and d has the property that dg = 0 in K(Γ f ) g K(Γ f ). 4 We replace Lorenzini s M with the special case of the Laplacian matrix of a bent function because it satisfies the assumptions on M and is the only case we are concerned about in the context of this paper. We specify that Lorenzini s λ is the ith eigenvalue of the Laplacian, λ i (L), and that µ(λ) is the geometric multiplicity m i (L). His definition of Φ(M) is equivalent to our definition of the critical group, K(Γ f ). Lorenzini previously defines r = R R ; recall R = R = 1, the all ones vector, and R R = p m 5 We replace Lorenzini s M with the special case of the Laplacian matrix of a bent function because it satisfies the assumptions on M and is the only case we are concerned about in the context of this paper. His definition of Φ(M) is equivalent to our definition of the critical group, K(Γ f ). ord d (x) is the highest power of d that divides x 12

13 Proof: See Lorenzini [L]. For our purposes, Corollary 3.2 in Lorenzini [L] gives the same result as item 3 in Theorem 16. Theorem 18. (Corollary 3.2, [L]) 6 Let Γ f be a strongly regular Cayley graph for a bent function f with parameters (ν, k, λ, µ). Then K(Γ f ) is killed by νµ. Because both eigenvalues λ 1 (A) and λ 2 (A) are integers, K(Γ f ) contains subgroups isomorphic to (Z/(λ 1 (L))Z) m1(l) 1 and (Z/(λ 2 (L))Z) m2(l) 1. If there exists a prime d p with the property that d λ 1 (L), then K(Γ f ) contains a subgroup isomorphic to (Z/d ord d(λ 1 (L)) Z) m1(l). The results from each of these findings should be the same for us, since the Laplacian for the Cayley graph of a bent function is very specific and satisfies the assumptions for both conjectures. We seek to prove Theorem 18 using Theorem 16 and Theorem 17. Proof. Let λ 1 (A), λ 2 (A), and λ 3 (A) denote the distinct eigenvalues of the adjacency matrix A of Γ f. The eigenvalues λ 2 (A) and λ 3 (A) are the roots of x 2 (λ µ)x (k µ). Since Γ f is k-regular, the non-zero eigenvalues of the Laplacian matric of Γ f are λ 2 (L) = k λ 2 (A) and λ 3 (L) = k λ 3 (A). The eigenvalues λ 2 (L) and λ 3 (L) are the roots of x 2 (λ µ 2k)x + νµ. It follows from Theorem 17 that K(Γ f ) is killed by νµ. It follows from Theorem 16 that K(Γ f ) contains subgroups isomorphic to (Z/(λ 1 (L))Z) m 1(L) 1 and (Z/(λ 2 (L))Z) m 2(L) 1. not just studied in relation to graph theory and bent functions. It has been used by mathematicians to investigate the analogy between algebraic curves and graphs, computer scientists use it to investiagte network routing models, and physicists use it to investigate sandpile dynamics (how a sindpile settles). Because of its various other uses, the critical group is known by various other names, such as the sandpile group in physics and the Jacobian of the graph when used in relation to algebraic 6 We replace Lorenzini s parameters (n, k, a, c) with (ν, k, λ, µ) for consistency. His definition of Φ(G) is equivalent to our definition of the critical group, K(Γ f ). Lorenzini uses θ and τ as eigenvalues for the adjacency matrix, and k θ and k τ as eigenvalues for the corresponding Laplacian matrix; we will use λ 2 (A), λ 3 (A), λ 2 (L), and λ 3 (L), respectively. ord d (x) is the highest power of d that divides x 13

14 curves. A good reference for more information on uses of the critical group is Algebraic Graph Theory by Godsil and Royle (ISBN ). 14

15 Critical groups for Cayley graphs of p-ary bent functions The results found in the last section all applied specifically to the Boolean case. In this section, we seek to apply them to the p-ary case. In order to test (8) and (9) in the p-ary case, we look at the 18 even bent functions f : GF (3) 2 GF (3) such that f(0) = 0. The algebraic normal form of these are (see [CJMPW] for details): b 1 = x x 2 1, b 2 = x x 2 1, b 3 = x 2 0 x 2 1, b 4 = x 2 0 x 0 x 1, b 5 = x 0 x 1 x 2 1, b 6 = x x 0 x 1, b 7 = x 0 x 1 + x 2 1, b 8 = x x 0 x 1, b 9 = x 0 x 1 x 2 1, b 10 = x 2 0 x 2 1, b 11 = x 0 x 1, b 12 = x 2 0 x 0 x 1 + x 2 1, b 13 = x 2 0 x 0 x 1 x 2 1, b 14 = x 2 0 x 0 x 1, b 15 = x 0 x 1 + x 2 1, b 16 = x 0 x 1, b 17 = x x 0 x 1 + x 2 1, b 18 = x x 0 x 1 x 2 1. Using Sagemath, we prove by inspection that (8) and (9) are true in the case where p = 3, m = 2, f is bent, f(0) = 0, and f is even. This process of proof requires special care in order to correctly interpret the results since Lorenzini was looking at unweighted graphs but the p = 3 examples have weighted Cayley graphs. The correct interpretation is to weight the adjacency matrix, thereby creating different eigenvalues for both the adjacency and Laplacian matrices. Notation: In the following examples the table is included to show the value of the function b i (x) and the Walsh Transform W f (x) at each point x in GF (3) 2. The result of the Walsh Transform is always of the form W f (x) = aζ3 2 + bζ 3 + c. For brevity the table entries will be given in the form {a, b, c}. For a fixed u GF (p) m and for a fixed p-ary function f, let T j = {x GF (p) m f(x) u, x = j}. Note that the T j s partition GF (p) m, for j = 0,..., p 1. Therefore, a + b + c = 9. Example 19. The bent function b 1 satisfies the following: b 1 (x) W b1 (x) {4, 4, 1} {1, 4, 4} {1, 4, 4} {1, 4, 4} {4, 1, 4} {4, 1, 4} {1, 4, 4} {4, 1, 4} {4, 1, 4} 15

16 Notice that there are 4 occurences of (4, 1, 4), 4 occurrences of (1, 4, 4), and 1 occurrence of (4, 4, 1). The Cayley graph Γ 1 has 9 vertices and an edge weight b 1 (x 0 x 1 ) on the edge (x 0, x 1 ). (To illustrate the difference weights make on a graph, note that the unweighted graph is K 9, the complete graph on 9 vertices.) The edge-weighted adjacency matrix is A 1 = with corresponding Laplacian matrix L 1 = The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 12, λ 3 = 15, m 1 = 1, m 2 = 4, m 3 = 4. K(Γ 1 ) = (Z/60Z) 2 (Z/180Z) 2. 16

17 The verification of (8) in the case of b 1 is presented in the calculation below: K(Γ 1 ) = = = λ m 2 3 p m. The verification of (9) in the case of b 1 is shown below: and K(Γ 1 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/12Z) 3 K(Γ 1 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/15Z) 3. Example 20. The bent function b 2 (x, y) = x 2 + y 2 satisfies the following: b 2 (x) W b2 (x) {2, 2, 5} {2, 5, 2} {2, 5, 2} {5, 2, 2} {2, 2, 5} {2, 2, 5} {5, 2, 2} {2, 2, 5} {2, 2, 5} Notice that there are 5 occurences of (2, 2, 5), 2 occurrences of (2, 5, 2), and 2 occurrences of (5, 2, 2). The edge-weighted adjacency matrix is A 2 = with corresponding Laplacian matrix L 2 =

18 The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 2 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 2 is presented in the calculation below: K(Γ 2 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 2 is shown below: K(Γ 2 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 2 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 2 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 21. The bent function b 3 (x, y) = x 2 y 2 satisfies the following: b 3 (x) W b3 (x) {2, 2, 5} {5, 2, 2} {5, 2, 2} {2, 5, 2} {2, 2, 5} {2, 2, 5} {2, 5, 2} {2, 2, 5} {2, 2, 5} The edge-weighted adjacency matrix is given by A 3 = A 2, and therefore the corresponding Laplacian matrix is given by L 3 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 3 ) = (Z/9Z) 2 (Z/54Z) 2. 18

19 The verification of (8) in the case of b 3 is presented in the calculation below: K(Γ 3 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 3 is shown below: K(Γ 3 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 3 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 3 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 22. The bent function b 4 (x, y) = x 2 xy satisfies the following: b 4 (x) W b4 (x) {2, 2, 5} {2, 2, 5} {2, 2, 5} {5, 2, 2} {2, 2, 5} {2, 5, 2} {5, 2, 2} {2, 5, 2} {2, 2, 5} The edge-weighted adjacency matrix is given by A 4 = A 2, and therefore the corresponding Laplacian matrix is given by L 4 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 4 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 4 is presented in the calculation below: K(Γ 4 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 4 is shown below: K(Γ 4 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 4 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 4 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. 19

20 Example 23. The bent function b 5 (x, y) = xy y 2 satisfies the following: b 5 (x) W b5 (x) {2, 2, 5} {5, 2, 2} {5, 2, 2} {2, 2, 5} {2, 2, 5} {2, 5, 2} {2, 2, 5} {2, 5, 2} {2, 2, 5} The edge-weighted adjacency matrix is given by A 5 = A 2, and therefore the corresponding Laplacian matrix is given by L 5 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 5 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 5 is presented in the calculation below: K(Γ 5 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 5 is shown below: K(Γ 5 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 5 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 5 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 24. The bent function b 6 (x, y) = x 2 + xy satisfies the following: b 6 (x) W b6 (x) {2, 2, 5} {2, 2, 5} {2, 2, 5} {2, 5, 2} {2, 2, 5} {5, 2, 2} {2, 5, 2} {5, 2, 2} {2, 2, 5} The edge-weighted adjacency matrix is given by A 6 = A 2, and therefore the corresponding Laplacian matrix is given by L 6 = L 2. The eigenvalues of the Laplacian matrix are: λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, 20

21 with respective multiplicities m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 6 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 6 is presented in the calculation below: K(Γ 6 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 6 is shown below: K(Γ 6 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 6 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 6 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 25. The bent function b 7 (x, y) = xy + y 2 satisfies the following: b 7 (x) W b7 (x) {2, 2, 5} {2, 5, 2} {2, 5, 2} {2, 2, 5} {2, 2, 5} {5, 2, 2} {2, 2, 5} {5, 2, 2} {2, 2, 5} The edge-weighted adjacency matrix is given by A 7 = A 2, and therefore the corresponding Laplacian matrix is given by L 7 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 7 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 7 is presented in the calculation below: K(Γ 7 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 7 is shown below: K(Γ 7 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 7 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 7 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. 21

22 Example 26. The bent function b 8 (x, y) = x 2 + xy satisfies the following: b 8 (x) W b8 (x) {2, 2, 5} {2, 2, 5} {2, 2, 5} {5, 2, 2} {2, 5, 2} {2, 2, 5} {5, 2, 2} {2, 2, 5} {2, 5, 2} The edge-weighted adjacency matrix is given by A 8 = A 2, and therefore the corresponding Laplacian matrix is given by L 8 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 8 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 8 is presented in the calculation below: K(Γ 8 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 8 is shown below: K(Γ 8 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 8 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 8 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 27. The bent function b 9 (x, y) = xy y 2 satisfies the following: b 9 (x) W b9 (x) {2, 2, 5} {5, 2, 2} {5, 2, 2} {2, 2, 5} {2, 5, 2} {2, 2, 5} {2, 2, 5} {2, 2, 5} {2, 5, 2} The edge-weighted adjacency matrix is given by A 9 = A 2, and therefore the corresponding Laplacian matrix is given by L 9 = L 2. The eigenvalues of the Laplacian matrix are: λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, 22

23 with respective multiplicities m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 9 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 9 is presented in the calculation below: K(Γ 9 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 9 is shown below: K(Γ 9 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 9 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 9 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 28. The bent function b 10 (x, y) = x 2 y 2 satsifies the following: b 10 (x) W b10 (x) {4, 4, 1} {4, 1, 4} {4, 1, 4} {4, 1, 4} {1, 4, 4} {1, 4, 4} {4, 1, 4} {1, 4, 4} {1, 4, 4} The edge-weighted adjacency matrix is given by A 10 = A 1, and therefore the corresponding Laplacian matrix is given by L 10 = L 1. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 10 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 10 is presented in the calculation below: K(Γ 10 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 10 is shown below: K(Γ 10 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 10 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 10 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. 23

24 Example 29. The bent function b 11 (x, y) = xy satisfies the following: b 11 (x) W b11 (x) {2, 2, 5} {2, 2, 5} {2, 2, 5} {2, 2, 5} {2, 5, 2} {5, 2, 2} {2, 2, 5} {5, 2, 2} {2, 5, 2} The edge-weighted adjacency matrix is given by A 11 = A 2, and therefore the corresponding Laplacian matrix is given by L 11 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 11 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 11 is presented in the calculation below: K(Γ 11 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 11 is shown below: K(Γ 11 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 11 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 11 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 30. The bent function b 12 (x, y) = x 2 xy + y 2 following: satisfies the b 12 (x) W b12 (x) {4, 4, 1} {1, 4, 4} {1, 4, 4} {4, 1, 4} {1, 4, 4} {4, 1, 4} {4, 1, 4} {4, 1, 4} {1, 4, 4} The edge-weighted adjacency matrix is given by A 12 = A 1, and therefore the corresponding Laplacian matrix is given by L 12 = L 1. The eigenvalues of the Laplacian matrix are: λ 1 = 0, λ 2 = 12, λ 3 = 15, 24

25 with respective multiplicities m 1 = 1, m 2 = 4, m 3 = 4. K(Γ 12 ) = (Z/60Z) 2 (Z/180Z) 2. The verification of (8) in the case of b 12 is presented in the calculation below: K(Γ 12 ) = = = λ m 2 3 p m. The verification of (9) in the case of b 12 is shown below: and K(Γ 12 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/12Z) 3 K(Γ 12 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/15Z) 3. Example 31. The bent function b 13 (x, y) = x 2 xy y 2 satisfies the following: b 13 (x) W b13 (x) {4, 4, 1} {4, 1, 4} {4, 1, 4} {1, 4, 4} {1, 4, 4} {4, 1, 4} {1, 4, 4} {4, 1, 4} {1, 4, 4} The edge-weighted adjacency matrix is given by A 13 = A 1, and therefore the corresponding Laplacian matrix is given by L 13 = L 1. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 12, λ 3 = 15, m 1 = 1, m 2 = 4, m 3 = 4. K(Γ 13 ) = (Z/60Z) 2 (Z/180Z) 2. The verification of (8) in the case of b 13 is presented in the calculation below: K(Γ 13 ) = = = λ m 2 3 p m. The verification of (9) in the case of b 13 is shown below: and K(Γ 13 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/12Z) 3 K(Γ 13 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/15Z) 3. 25

26 Example 32. The bent function b 14 (x, y) = x 2 xy satisfies the following: b 14 (x) W b14 (x) {2, 2, 5} {2, 2, 5} {2, 2, 5} {2, 5, 2} {5, 2, 2} {2, 2, 5} {2, 5, 2} {2, 2, 5} {5, 2, 2} The edge-weighted adjacency matrix is given by A 14 = A 2, and therefore the corresponding Laplacian matrix is given by L 14 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 14 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 14 is presented in the calculation below: K(Γ 14 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 14 is shown below: K(Γ 14 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 14 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 14 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 33. The bent function b 15 (x, y) = xy+y 2 satisfies the following: b 15 (x) W b15 (x) {2, 2, 5} {2, 5, 2} {2, 5, 2} {2, 2, 5} {5, 2, 2} {2, 2, 5} {2, 2, 5} {2, 2, 5} {5, 2, 2} The edge-weighted adjacency matrix is given by A 15 = A 2, and therefore the corresponding Laplacian matrix is given by L 15 = L 2. The eigenvalues of the Laplacian matrix are: λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, 26

27 with respective multiplicities m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 15 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 15 is presented in the calculation below: K(Γ 15 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 15 is shown below: K(Γ 15 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 15 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 15 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. Example 34. The bent function b 16 (x, y) = xy satisfies the following: b 16 (x) W b16 (x) {2, 2, 5} {2, 2, 5} {2, 2, 5} {2, 2, 5} {5, 2, 2} {2, 5, 2} {2, 2, 5} {2, 5, 2} {5, 2, 2} The edge-weighted adjacency matrix is given by A 16 = A 2, and therefore the corresponding Laplacian matrix is given by L 16 = L 2. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, m 1 = 1, m 2 = 2, m 3 = 2, m 4 = 4. K(Γ 16 ) = (Z/9Z) 2 (Z/54Z) 2. The verification of (8) in the case of b 16 is presented in the calculation below: K(Γ 16 ) = = = λ m 2 3 λ m 4 4 p m. The verification of (9) in the case of b 16 is shown below: K(Γ 16 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/3Z), K(Γ 16 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/6Z), K(Γ 16 ) = (Z/9Z) 2 (Z/54Z) 2 (Z/9Z) 3. 27

28 Example 35. The bent function b 17 (x, y) = x 2 + xy + y 2 following: satisfies the b 17 (x) W b17 (x) {4, 4, 1} {1, 4, 4} {1, 4, 4} {4, 1, 4} {4, 1, 4} {1, 4, 4} {4, 1, 4} {1, 4, 4} {4, 1, 4} The edge-weighted adjacency matrix is given by A 17 = A 1, and therefore the corresponding Laplacian matrix is given by L 17 = L 1. The eigenvalues of the Laplacian matrix are: with respective multiplicities λ 1 = 0, λ 2 = 12, λ 3 = 15, m 1 = 1, m 2 = 4, m 3 = 4. K(Γ 17 ) = (Z/60Z) 2 (Z/180Z) 2. The verification of (8) in the case of b 17 is presented in the calculation below: K(Γ 17 ) = = = λ m 2 3 p m. The verification of (9) in the case of b 17 is shown below: and K(Γ 17 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/12Z) 3 K(Γ 17 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/15Z) 3. Example 36. The bent function b 18 (x, y) = x 2 + xy y 2 satisfies the following: b 18 (x) W b18 (x) {4, 4, 1} {4, 1, 4} {4, 1, 4} {1, 4, 4} {4, 1, 4} {1, 4, 4} {1, 4, 4} {1, 4, 4} {4, 1, 4} The edge-weighted adjacency matrix is given by A 18 = A 1, and therefore the corresponding Laplacian matrix is given by L 18 = L 1. The eigenvalues of the Laplacian matrix are: λ 1 = 0, λ 2 = 12, λ 3 = 15, 28

29 with respective multiplicities m 1 = 1, m 2 = 4, m 3 = 4. K(Γ 18 ) = (Z/60Z) 2 (Z/180Z) 2. The verification of (8) in the case of b 18 is presented in the calculation below: K(Γ 18 ) = = = λ m 2 3 p m. The verification of (9) in the case of b 18 is shown below: K(Γ 18 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/12Z) 3 and K(Γ 18 ) = (Z/60Z) 2 (Z/180Z) 2 (Z/15Z) 3. 29

30 Conclusion and Conjectures An analysis of the even bent functions f : GF (3) 2 GF (3) such that f(0) = 0 shows that there are two classes of functions. Class 1 has three eigenvalues λ 1 = 0, λ 2 = 12, λ 3 = 15, and a critical group K(Γ f ) = (Z/60Z) 2 (Z/180Z) 2. The functions that belong to this class are b 1, b 12, b 13, b 17, and b 18. Class 2 has four eigenvalues and a critical group λ 1 = 0, λ 2 = 3, λ 3 = 6, λ 4 = 9, K(Γ f ) = (Z/9Z) 2 (Z/54Z) 2. The functions that belong to this class are b 2, b 3, b 4, b 5, b 6, b 7, b 8, b 9, b 10, b 11, b 14, b 15, and b 16. It is interesting to note that all functions within each class have the same size support, leading to the following conjecture. Conjecture 37. The critical group of the Cayley graph of a bent function f only depends on the size of the support, supp(f). We note that Class 1 has the property that for all f in Class 1 supp(f) = 8, while Class 2 has the property that for all f in Class 2 supp(f) = 4. Because the classes can be categorized in the same way by critical group or supp(f), Conjecture 37 holds for all even bent functions f : GF (3) 2 GF (3) such that f(0) = 0. We also tested this conjecture in the p = 2, m = 4 case and saw that it held for the bent functions we tested. This test was not exhaustive in the p = 2, m = 4 case, so that would be an interesting 30

31 topic to explore in future research. We next introduce some more questions that arose from our analysis of the even bent functions f : GF (3) 2 GF (3) such that f(0) = 0. Notation: Over GF (p), let where p 1 W f = ( n j,k (f)ζp) j 0 k p m 1, j=0 n j,k (f) = {x GF (p) m f(x) r k, x = j}, where r k denotes the binary representation of k. Question 1. Is the multiset S f = {n j,k (f) 0 j p 1} independent of k? Question 2. Does S f only depend on the type or signature of f, where f is bent? Question 3. For each permutation of S f, is the number of times it occurs in W f an n j,k (f)? References [CJMPW] C. Celerier, D. Joyner, C. Melles, D. Phillips, S. Walsh, Edgeweighted Cayley graphs and p-ary bent functions, to appear in INTE- GERS, [J] T. Judson, Abstract Algebra: Theory and Applications available for download at at [L] D. Lorenzini, Smith normal form and Laplacians at math.uga.edu/~lorenz/paperaccepted.pdf [R] J. Rushanan, Eigenvalues and the Smith normal form, Linear Algebra Appl. 216 (1995), [Sp] E. Spence, webpage Strongly Regular Graphs on at most 64 vertices at 31

32 [S] P. Stanica, Graph eigenvalues and Walsh spectrum of Boolean functions, INTEGERS 7(2) (2007), #A32 32