關於組合設計理論及其相關應用的研究(III)

行政院國家科學委員會專題研究計畫 成果報告

關於組合設計理論及其相關應用的研究(3/3)

中 華 民 國 95 年 1 月 17 日

Bent 函數及其相關的強正則圖 黃大原、游貴弘 布林函數 Walsh 質譜的計算的複雜度，一般而言十分困難。然而某些特定 族類的該函數的值譜所具有的特殊性質，能有效降低其複雜度。值得一提 的是布林函數 f 的值譜分析可轉化為凱氏圖 Cayley Graphs 的質譜相關問 題，因而代數圖論在 Bent 函數的研究裏，伴演一個十分積極的角色，由 其是相異係數不大時為然。我們據以得到其所對應的強正則圖的參數及其 值譜；並且列出點數不超過 280 的所有可能情形。

The complexity of computing the Walsh spectrum of Boolean functions is difficult in general, however several interesting classes of such functions have a very special spectrum, whose ad hoc computation can be carried out

significantly faster than in the general case. It is worth to note that the spectral analysis of Boolean functions can be viewed as a Cayley graph eigenvalue problem, this observations allow the using of tools from algebraic graph theory for investigations related to the spectral coefficients of Boolean functions, especially when the number of distinct coefficients is small. The main motivation for introducing the graph Gf is that its spectrum coincides

with the Walsh spectrum of its associated Boolean function f(x). This brings the problem of analyzing the spectral coefficients of Boolean functions into the framework of spectral analysis of graphs, i.e., it makes it possible to use techniques from graph spectra for the evaluation of spectral coefficients. More precisely, the results from algebraic graph theory can be applied to analyze Boolean functions with a few distinct spectral coefficients, the fewer is the number of distinct coefficients, the stronger are the algebraic properties of the function; this leads to a nice interpretation for the well-known class of bent functions in terms of strongly regular graphs.

Strongly Regular Graphs associated with Bent Functions

Tayuan Huang and Kuei You Department of Applied Mathematics

National Chiao-Tung University Hsinchu, Taiwan [email protected]

The complexity of computing the Walsh spectrum of Boolean functions is difficult in general, however several interesting classes of such functions have a very special spectrum, whose ad hoc computation can be carried out significantly faster than in the general case. It is worth to note that the spectral analysis of Boolean functions can be viewed as a Cayley graph eigenvalue problem, this observations allow the using of tools from algebraic graph theory for investigations related to the spectral coefficients of Boolean functions, especially when the number of distinct coefficients is small. The main motivation for introducing the graph Gf

is that its spectrum coincides with the Walsh spectrum of its associated Boolean function f(x). This brings the problem of analyzing the spectral coefficients of Boolean functions into the framework of spectral analysis of graphs, i.e., it makes it possible to use techniques from graph spectra for the evaluation of spectral coefficients. More precisely, the results from algebraic graph theory can be applied to analyze Boolean functions with a few distinct spectral coefficients, the fewer is the number of distinct coefficients, the stronger are the algebraic properties of the function; this leads to a nice interpretation for the well-known class of bent functions in terms of strongly regular graphs.

1. Bent Functions

The Fourier transform of a Boolean function_{g x Z}( ) : ₂n _→Z is defined to be

2

( )

∑

∑

( )

λ . A Boolean function is called a bent function if 2 2 :Z Z f n _→

(

)

(₋ ( ) ∗ _{λ =± for any , the term of bent was coined by}

Rothaus [9]. n Z₂ ∈ λ Theorem [9]

If f(x) is a bent function on with , then n = 2k must be even, and the degree of f(x) is at most k; moreover f(x) is irreducible whenever deg(f(x)) = k .

n

Z₂ n≥3

3 ≥

Some basic properties of bent functions together with their relationships with some

combinatorial structures are summarized in the following theorem. The Boolean function f(x) is bent if and only if the matrix _{[( 1)−} f x y( + )_{] is a Hadamard matrix. The Fourier}

transform of a bent function is again a bent function. 2. The Cayley Graphs associated with Bent Functions

The Cayley graph Gf = (V(f), Ef) associated with a Boolean function

is defined on the vertex set V(f) =

2 2 :Z Z f n _→ 2n Z , with u, w_∈Z2n_{adjacent if} f w u⊕ ∈Ω = , or equivalently . The graph G

1₍₁₎

f−

1

( )

f w u⊕ = f isΩ - regular with _f 2n−dimΩf connected

components, the graph Gf is connected ifdim Ωf =n. The spectrum of Gf is usually

denoted by Spec(Gf ) =

(

)

( )

2 ), ( _{f b i} x f n Z x x i b i n ∗ ∈ ∀ ⋅ = − =

∑

Upper and lower bounds on the rank (over the real field) of the adjacency matrix Af of Gf

i.e., the number of nonzero spectral coefficients of the function f , are given in terms of degrees of polynomials representation of f. Some properties of the Fourier coefficients and its associated Cayley graphs are given in the following.

Theorem [1] If f :Z₂n →Z₂, and λ_i,0≤ ≤i 2n − are the eigenvalues of the graph G1, f , then

a. 2n f (b(i)) for ;

i

∗

=

λ 0≤_i≤2n −1

b. the multiplicity of its largest eigenvalue f∗(b(0))is₂n−dimΩf

(which implies the graph Gf

isΩ - regular with _f n−dimΩf

2 connected components and the graph Gf is connected

ifdim Ω_f =n);

A Boolean functions is characterized by its spectrum if it is possible to identify its associated graph (i.e., determine all the details of its topology) only on the basis of the knowledge of its distinct eigenvalues, i.e., without using any information regarding their eigenvectors. It is interesting to note that the fewer the number of distinct spectral coefficients are, the stronger are the algebraic properties of the setΩ ; for instance, it is _f well-known that if a connected graph has exactly m distinct eigenvalues, then its diameter d satisfiesd ≤ m−1.

A k-regular graph G is strongly regular if there exist nonnegative integers a and c such that for all vertices u, v, the number of vertices adjacent to both u and v is a if u and v are adjacent, and c otherwise. A k- regular connected graph is strongly regular if and only if it has exactly three distinct eigenvalues

1 1

|G u( )∩G v( )|

0 k

λ = ,λ1,λ2. A rephrase of Parseval’s identity

gives that ∗_{( (0))}=

∑

i f b i b f 1 2 (k−λ)(k−λ ) ₌2 (r _{k+λ λ1 2}) _where f

k= Ω , and r must be replaced bydim Ω_f if G is not connected. If G is strongly regular, then a k rs r s= + + + andc k rs= + . It was also observed that the class of bent functions is associated to a very special class of strongly regular graphs indeed exactly identifies the bent functions.

Theorem [1]

If Gf is a connected strongly regular graph, then there exists v∈Ωf such that

for each , and there exist h elements

f v u⊕ ∈Ω ₂n \ f u Z∈ Ω w∈Ω_f such that , where f w

v⊕ ∈Ω h e= if v∈Ω and if _f, d v∉Ω for each_f. _{v Z∈ 2}n,

In order to find a complete characterization of the class of functions with three distinct nonzero spectral coefficients and with the additional property a = c, we are then left with the problem of understanding whether or not there exists other integer solutions to

. It was proved in [2] that the equation has integer solutions in x and y only if . As a consequence, bent functions can be characterized as binary functions with a certain class of strongly regular graphs.

2 ₂n ₍₂n ₁₎ x − x+ − y2 =₀ 2 2 _{0, 1, 2}n y ₌ − Theorem [1,2]

a. The associated Caley graph Gf of a bent function is a strongly regular

graphSRG v k( , , , )λ λ .

b. The bent functions are the only binary functions f whose associated graph Gf is a strongly

regular graph SRG v k( , , , )λ λ .

Those graphs Gf with small numbers of distinct eigenvalues are considered: if Gf has a

single eigenvalue, then G_f = K₂n₋₁; if Gf has two distinct eigenvalues, then either

Gf = 2 ₁ 1 f n f K_{Ω +} Ω + whenb(0)∉Ωf, or Gf = 2 f n f K_Ω

Ω with loops otherwise; if Gf has three eigenvalues, then

a.(λ₀,λ₁,λ₂)=(Ω_f ,0,−Ω_f ) if and only if Gf is the complete bipartite graph between

vertices inΩ_f and in_Zk _\Ωf .

2

b.(λ₀,λ₁,λ₂)=(Ω_f ,0,λ₂)if and only if Gf is a complete multipartite graph with 2 2 ( f 1) f G K _λ λ − Ω

= − + and with Spec(Gf) =

1 1 2 2 2 2 (2 1 ) ( ) 1 (1) 2 ((2 ) ,(0) ,( ) ) n n n n λ λ λ − − − + − − c. if Gf is connected, then Gf is a SRG(2 , f ,e,d) n _{Ω with} Spec(Gf) = ( 2 1) 2 1 2 1 1 ₂ ( ( , ( ( ( ) 4( ))) 2 n f f e d e d d f λ λ λ − − − Ω − Ω − + − − − Ω ), ( 2 1) 1 1 2 ( 2 1 ( ( ( ) 4( ))) 2 n ) f f e d e d d λ λ λ − + Ω − − − − − − Ω Theorem if f is a bent function with connected Gf, then Gf is a strongly regular graph

SRG(v,k, ) with (v,k,l. ) = (2 ,2 −1+22−1,2 −2 +22−1,2 −2 +22−1) n n n n n n n _or ) 2 2 , 2 2 , 2 2 , 2 ( −1 − 2−1 −2 − 2−1 −2 − 2−1 n n n n n n n 3 Bent & SRG 2006/1/17

Spec(Gf) = 1 1 1 2 1 2 2 1 2 1 2 1 1 (1) (2 2 1) (2 2 ((2 2 ) ,(2 ) ,( 2 ) ) n n n n n n n n− ₊ − − −− −− ₋ − −+ −) _or Spec(Gf) = 1 1 1 2 1 2 2 1 2 1 2 1 1 (1) (2 2 ) (2 2 ((2 2 ) ,(2 ) ,( 2 ) ) n n n n n n n n− ₋ − − −− − ₋ − −+ −−1) _.

3. Strongly Regular Graphs SRG(n,k,λ,λ)

The Friendship theorem shows that a connected graph with a unique common neighbor for any pairs of distinct vertices has a vertex adjacent to its all other vertices, and K3 is the unique

such regular graph. We now consider those connected k-regular graphs such that any two distinct vertices has a constant number of λ common neighbors, they are strongly regular graphs SRG(n, k, λ, λ). When λ = 1, then G = K3 as just mentioned. The symplectic graphs

Sp(2m) offer a family of such strongly regular graphs with parameters (22m − 1, 22m − 1, 22m − 2, 22m − 2) for positive integers m, note that K3 is the symplectic graph Sp(2). The Cayley graphs

associated with bent functions provide another family of such graphs.

Theorem: Suppose there exists a SRG(n, k, λ, λ) with λ >1, and with distinct eigenvalues

k> >θ τ , then

1. θ = − =τ k− , λ θτ = − −(k λ) are integers with multiplicities 1 (( 1) ) 2 k m n k θ = − − _λ − , and 2(( 1) ) 1 λ τ − + − = k k n m . 2. θ λ| and ( ,n k)= ₍(θ2 θ λ θ)( 2 θ λ)_,θ2 _λ) λ + + − + ₊ .

Proof: (1). Available in monographs, omitted. (2) Let

λ

− =

k k

t , which is a positive integer by (1). Hence 2 2 4

2

t t t

k = ± − λ , both t and b = _t2 ₋4λ _{are of the same parity;} since_t2_−4λ=b2, it follows that_{4λ= +(t b t b)( − , both )}

t + b = 2 4 k k k λ k λ λ ⎛ ⎞ + ⎜ ⎟ − ⎝ − ⎠ − , and t − b = 2 4 k k k λ k λ λ ⎛ ⎞ − _{⎜ ⎟} − − ⎝ − ⎠

must be even. Let t + b = 2h1 and t − b = 2h2 for some positive integers h1 > h2, hence λ =

h1h2, then t = h1 + h2, b = h1 − h2, and k is either h1 (h1 + h2) or h2(h1 + h2). Note

thatθ = k− is either ( in case λ h₁ k h h= ₁( ₁+h₂)) or (in case h₂ k h h= ₂( ₁+h )₂ ), hence |

θ λ. It follows that n (θ2 θ λ θ)( 2 θ λ)

λ

+ + − +

= in either case as required. Q.E.D.

The above lemma paves a way for studying possible feasible parameters (v, k, λ, λ) for a given λ with a pair ( , )h h_{1 2} =( , / )θ λ θ or ( / , )λ θ θ . The trivial decomposition of λ = 1⋅λ

with ( , ) ( ,1)h h_{1 2} = λ leads to (v, k, λ, λ) = (_λ2(_λ+2),_λ(_λ+1),_λ,_λ)or_{(λ+2,λ+1,λ,λ).} The other extremal cases with h h₁, ₂ close to λare considered for , and

respectively.

2

2 m

λ= 2 (2m m₊1)

If with , then (v, k, λ, λ) = which is

identical with those of the symplectic graphs.

m 2 2 = λ (_h1,_h2)₌(2m,2m) ₍₂2m+2 _−1,22m+1_,22m_,22m₎ If_λ =2m(2m +1), then (v, k, λ, λ) = or (22(2m₊1)2,(2m ₊1)(2m+1 ₊1),2m(2m ₊1),2m(2m ₊1)) 2 1 (2 (2m m+ ), 2 (2m m+ ₊1), 2 (2m m₊1), 2 (2m m₊1))

respectively with respectively. For the symplectic graphs Sp(2(m+1)),

which is a SRG with spectrum

1 2 ( , ) (2_{h h =} m₊1, 2 )m 2 2 2 1 2 2 (2 m+ ₋1, 2 m+, 2 , 2 )m m − Spec(G) = ₍₍₂2_m+1 1_{) , (2 )m} 22m+1₋2m₋1_{,( 2 )m} 22m+1₊2m 1₎, −

some examples with small number of vertices are known already, for example: SRG(3, 2, 1, 1 ) with Spec(G) =_{(2 ,1 ,( 1) )}1 0 ₋ 2 ,

SRG(15, 8, 4, 4 ) with Spec(G) = _{(8 , 2 ,( 2) )}1 5 ₋ 9 ,

SRG(63, 32, 16, 16 ) with Spec(G) = _{(32 , 4 ,( 4) )}1 27 ₋ 35 , and SRG(255, 128, 64, 64 ) with Spec(G) = _{(128 ,8 ,( 8)}1 119 ₋ 135 ₎. References:

[1] A. Bernasconi and B. Codenotti, “Spectral Analysis of Boolean Functions as a Graph Eigenvalue Problem,” IEEE Trans. Computers, vol. 48, no. 3, pp. 345-351, Mar. 1999. [2] A. Bernasconi and B. Codenotti, and J. VanderKam, A Characterization of Bent

Functions in terms of Strongly Regular Graphs, IEEE Transactions on Computers Vol.50 No.9, 984-985 September (2001)

[3] N. Biggs Algebraic Graph Theory, Cambridge University Press, Cambridge 1993 [5 C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC

Press, 1996

[7] C. Godsil and G. Royle, Algebraic Graph Theory, Springer GTM 207, 2001

[8] A. Lubotzky, R. Phillips, and P. Sarnak, “Ramanujan Graphs,” Combinatorical, vol. 8, pp. 261-277, 1988.

[9] O. S. Rothaus, “On Bent Functions,” J. Combinatorial Theory (A), vol. 20, pp. 300-305, 1976.