Enumeration of Balanced Symmetric Functions over GF (p)

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1 Enumeration of Balanced Symmetric Functions over GF () Shaojing Fu 1, Chao Li 1 and Longjiang Qu 1 Ping Li 1 Deartment of Mathematics and System Science, Science College of ational University of Defence Technology, Changsha, China, shaojing1984@yahoocn Abstract It is roved that the construction and enumeration of the number of balanced symmetric functions over GF () are equivalent to solving an equation system and enumerating the solutions Furthermore, we give an lower bound on number of balanced symmetric functions over GF (), and the lower bound rovides best known results Key words: symmetric functions, balanced functions, Permutation 1 Introduction Since symmetry guarantees that all of the inut bits have equal status in a very strong sense, symmetric Boolean functions dislay some interesting roerties A lot of research about symmetry in characteristic has been reviously done YXYang and BGuo [1] studied the balanced symmetric functions and correlation immune symmetric functions SMaitra and PSarkar [] studied the maximum nonlinearity of symmetric Boolean function on odd number of variables ACanteaut and MVideau [3] established the link between the eriodicity of simlified value vector of an symmetric Boolean functions and its degree ABraeken, BPreneel [4] studied Algebraic immunity of Symmetric Boolean function On the other hand, it is natural to extend various crytograhic ideas from GF() to other finite fields of characteristic >, GF () or GF( n ), being a rime number For examle, [5] and [6] studied the correlation immune and resilient functions on GF () Also, [7] and [8] investigated the generalized bent functions on GF( ) Li and Cusick [9] first introduced the strict avalanche criterion over GF () In [10], they generalized most results of [11] and determined all the linear structures of symmetric functions over GF () Recently, Cusick and Li [1] give a lower bound for the number of balanced symmetric functions over GF (), and they show the existence of nonlinear balanced symmetric functions In [13], Pinhui Ke imroved the lower bound, then he showed that the number of n-variable balanced symmetric functions over GF() is not less than the number of solutions of an given equation systems over Z In this aer, we rove that enumeration of the number of balanced symmetric functions over GF () is equivalent to solve the equation system in [13] and we give formulas to count balanced symmetric functions over GF () Then based on our formulas and Cusick s method, the lower bound on the number

2 Shaojing Fu et al of n-variable balanced symmetric functions over finite fields GF () resented in [13] is imroved Preliminaries In this aer, Z is the set of ositive integers, Z is the set of ositive integers, and is a odd rime number Let GF () be the finite field of elements, and GF() n be the vector sace of dimension n over GF () An n-variable function f(x 1, x,, x n ) can be seen as a multivariate olynomial over GF (), that is, f(x 1, x,, x n ) = n k 1,k,,k n=0 a k1,k,,k n x k1 1 xk xkn n where the coefficients a k1,k,,k n are a constant in GF () This reresentation of f is called the algebraic normal form (AF) of f The number k 1 + k + + k n is defined as the degree of term with nonzero coefficient The greatest degree of all the terms of f is called the Algebraic degree of f, denoted by deg(f) The function f(x) is called an affine function if f(x) = a 1 x 1 + a x + + a n x n + a 0 If a 0 = 0, f(x) is also called a linear functionwe will denote by F n the set of all functions of n variables and by L n the set of affine ones We will call a function nonlinear if it is not in L n Definition 1 f: GF() n GF() is balanced if #{x GF () n f(x) = k} = n 1 for any k = 1,,, 1 Definition f:gf() n GF() is a symmetric if for any ermutation π on {1,,, n}, we have f(x 1, x,, x n ) = f(x π(1), x π(n),, x π(n) ) Denote the set of ermutations on {1,,, n} by S n Then we define the following equivalence relation on GF() n : for any x = (x 1,, x n ), y = (y 1,, y n ) GF () n we say x and y are equivalent, if there exists a ermutation π S n such that (y 1,, y n ) = (x π(1),, x π(n) ) (by abuse of notation we write y = π(x)) Let x = {y π S n, π(x) = y}, Let x = ( x 1, x,, x n ) be the reresentative of x, where 0 x 1 x x n 1 Obviously, we have x = ỹ x = ȳ It is easy to show that f: GF() n GF() is symmetric if f(x) = f(y) whenever x = ỹ Definition 3 Let x = (0,, 0, 1,, 1,, 1,, 1) be reresentative of the classes x, then the vector (i 0, i 1,, i 1 ) is called the corresonding }{{}}{{}}{{} i 0 i 1 i 1 vector of x Let the corresonding vector of x be (i 0, i 1,, i 1 ), then # x = n! i 0!i 1! i 1! The number of different equivalence classes x is the number of solutions of the linear equation i 0 +i 1 ++i 1 = n, where i k is the number of times k aears in x We know that the number of solutions to this linear diohantine equation is the same as the number of n-combinations of a set with elements, that is C(n + 1, n)

3 Enumeration of Balanced Symmetric Functions over GF () 3 3 Enumeration of Balanced Symmetric Functions From the definition of symmetric function, we know that symmetric function has the same value for any n-tule in the same equivalence classes So in order to get symmetric function,we should artition the vectors in the C(n + 1, n) equivalence classes into grous, and the vectors in the same equivalence class must be in the same grou We start with the definition of similar equivalence class Definition 4 x and ỹ are called the similar equivalence class, if there exists π S n, such that x s corresonding vector (i 0, i 1,, i 1 ) and ỹ s corresonding vector (j 0, j 1,, j 1 ) satisfies (i 0, i 1,, i 1 ) = (j π(0), j π(1),, j π( 1) ) The class-sets of x is the set constituted by all the similar equivalence class of x Let all the C(n+ 1, n) classes are divided into class-sets (Ω 1, Ω,, Ω ) by using the similar equivalence relation, M i = #Ω i and T i = #{ x x Ω i }(Because any equivalence classes in a class-sets have same cardinality, T i is denoted without misarehending) For a fixed solution x Ω i, let (i 0, i 1,, i 1 ) be the corresonding vector of x, and m l be the number of times that l aears in {i 0, i 1,, i 1 } Then! it is easy to show that M i = m, T n! 0!m 1! m n! i = i 0!i 1! i 1! Consider the equation system Φ: i=0 x i,j T i = n 1, 1 j Φ : i=0 x i,j = M i, 1 i x i,j Z, x i,j 0 In [13], Pinhui Ke resented the following result Theorem 1 [13]The number of n-variable balanced symmetric functions over GF() is not less than the number of solutions of equation system Φ From theorem 1, we know that Pinhui Ke only rove that the number of solutions of equation system Φ is a lower bound ow we will give a method to construct balanced symmetric functions by using the solutions of Φ, and give formulas to count balanced symmetric functions over GF () First, let M Φ be the number of solutions of Φ, and the solutions be X 1, X,, X MΦ, where X k = x (k) 1,1 x(k) x (k) 1,1 x(k) 1, x(k) 1, 1, x(k) 1,, 1 k M Φ x (k),1 x(k), x(k), Theorem Let n be the number of n-variable balanced symmetric functions over GF(), then n = M Φ k=1! j=1 x(k) i,j

4 4 Shaojing Fu et al Proof To construct balanced symmetric functions, basically, we try to divide n vectors into grous (on each grou, the function has the same value), such that each grou has n 1 vectors Of course, the vectors in the same equivalence class must be in the same grou since the function is symmetric For a fixed solution X k, we construct grous A 1, A,, A as follow, (1) Select x (k) 1,1 equivalence classes from Ω 1, x (k),1 equivalence classes from Ω,, x (k),1 equivalence classes from Ω, regard the vectors in these equivalence classes as the vectors of A 1 () Select x (k) 1, equivalence classes from the rest x (k) 1,1 equivalence classes of Ω 1, x (k), equivalence classes from the rest x (k),1 equivalence classes of Ω,, x (k), equivalence classes from the rest x (k),1 equivalence classes of Ω, regard the vectors in these equivalence classes as the vectors of A (t) Select x (k) 1,t equivalence classes from the rest t 1 j=1 x(k) 1,j equivalence j=1 x(k) classes in Ω 1, x (k),t equivalence classes from the rest t 1 j=1 x(k),j equivalence classes in Ω,, x (k),t equivalence classes from the rest t 1,j equivalence classes in Ω, regard the vectors in these equivalence classes as the vectors of A t ( 1) Select x (k) 1, 1 equivalence classes from the rest j=1 x(k) 1,j classes in Ω 1, x (k), 1 equivalence classes from the rest j=1 x(k),j equivalence classes in Ω,, x (k), 1 equivalence classes from the rest j=1 x(k),j equivalence classes in Ω, regard the vectors in these equivalence classes as the vectors of A 1 () there are x (k) 1, equivalence classes in the rest of Ω 1, x (k), equivalence classes in the rest of Ω,, x (k), equivalence classes in the rest of Ω, regard the vectors in these equivalence classes as the vectors of A Let f(x): {x f(x) = j 1} = A j, 1 j It is obvious that f(x) is a balanced symmetric function, and the number of balanced symmetric function constructed by the solution equals the ways to select A 1, A,, A, that is j=1 x(k) i,j! Given two solutions X k1 X k, without lost of generality, Let x (k1) 1,1 x(k) 1,1, the corresonding A 1 constructed byx k1, X k are different, so the balanced symmetric functions are different Hence, the total number of symmetric functions constructed by the construction above is M Φ k=1 j=1 x(k) i,j! ow we show that any n-variable balanced symmetric functions can be constructed by the construction above Given a balanced symmetric function f(x), let A j = {x f(x) = j 1} and A j = { x x A j}, and x i,j = #{A j Ωi }, then It is easy to show that x i,j (1 i, 1 j ) is a solution of the equation system Φ So we get the count

5 Enumeration of Balanced Symmetric Functions over GF () 5 Examle 1 Let n = 3, = 5, we can get = 3, = 5, = 0, M 3 = 10, and M Φ = 81 Then the number of 3-variable balanced symmetric functions over GF (5) is When and n become larger, it is hard to solve the equation system Φ In this case, we solve another easier equation, and we can obtain an imroved lower bound of [13] by solving this equation when is not a roer divisor of n Consider the equation with restricted conditions: Let Ψ = {(y (1) 1, y(1) Θ : i=0 z i T i = 0, z i Z, z i M i ), (y() 1, y() ), (y(w ) 1, y (W ),, y (W ) )},, y(1),, y() be the set of the solutions of Θ whose most right nonzero comonent is ositive Then we can get the following lower bound on the number of n-variable balanced symmetric Functions Theorem 3 If is not a roer divisor of n, then n A(, t) 1 t=1 1 k 1,k,,k t W where A(, t) = n! (n k)! (( M i )!) + (( Mi )!) t t j=1 ( Mi y(kj) 1 )!( Mi + y(kj) 1 )! Proof If n, let m l be the number of times that l aears in {i 0, i 1,, i 1 } Then { m 0 + m m n = 0 m m n m n = n! so is a roer divisor of M i = m 0!m 1! m n! for any 1 i Then It is easy to show that the matrix M 1 M = M a solution of Φ For any k = 1,,, W y(k) 1 y(k) y (k) + y(k) 1 + y(k) + y (k)

6 6 Shaojing Fu et al is also a solution of Φ Select t vectors (y (k1) 1, y (k1),, y (k1) ), (y(k) 1, y (k),, y (k) ), (y(kt) 1, y (kt),,y (kt) ) from Ψ, then add these t vectors to any t columns of the matrix resectively At the same time, subtract (y (k1) 1, y (k1),, y (k1) ),, y(k) ),, (y (kt) 1, y (kt),, y (kt) ) to another t columns of the matrix By the selecting we can obtain A(, ) #Ψ A(, ) #Ψ new matrices, these matrices are also solutions of Φ ow We distinguish +1 case (0)If we change 0 columns of the matrix, then we can construct balanced symmetric functions (1)If we change columns of the matrix, then we can construct 1 k 1 W ( 1) (( Mi )!) t ( Mi y(k1) 1 )!( Mi + y(k1) 1 )! balanced symmetric functions (t)if we change t columns of the matrix, then we can construct 1 k 1,k,,k t W A(, t) (( Mi )!) t t j=1 ( Mi y(kj) 1 )!( Mi + y(kj) 1 )! (( M i )!) balanced symmetric functions ) If we change 1 columns of the matrix, then we can construct ( +1 1 k 1,k,,k 1 W A(, t) balanced symmetric functions Hence we get the lower bound ( Mi 1 )! j=1 ( Mi y(kj) 1 )!( Mi + y(kj) 1 )! At the end of this section, for n = 3 and = 5, we comare the lower bound obtained by [1], [13], and Theorem 3 in the following table: Table 1 the number of 3-variable balanced symmetric functions over GF(5) [1] [13] Theorem 3 uer bound

7 4 Conclusion Enumeration of Balanced Symmetric Functions over GF () 7 In this aer, we obtain some counting results about balanced symmetric functions over finite field GF(), and we get a lower bound by finding solutions of an equation system With an examle, we show that this bound is better than the known results, but when is a roer divisor of n, it is still an oen roblem to get a lower bound on the number of balanced symmetric functions Besides, Finding more solutions to imrove this lower bound may be another interesting roblem for the future research Acknowledgments The work in this aer is suorted by the ational atural Science Foundation of China (O: ) References 1 YXYang, BGuo Further enumerating Boolean functions of crytograhic signifiance JCrytology, 8(3),115-1, 1995 SMaitra, PSarkar Maximum nonlinearity of symmetric Boolean functions on odd number of variables IEEE Transon Infortheory, 48(9), , 00 3 ACanteaut, MVideau Symmetric Boolean functions IEEE Transon Infor theory, 51(8), , ABraeken, BPreneel On the algebraic immunity of symmetric Boolean functions in IDOCRYPT005,LCS 3797, Berlin, Germay:Sringer-Verlag, 35-48, Mulan Liu, Peizhong Lu and GL Mullen Correlation-Immune Functions over Finite Fields IEEE Trans on Information Theory 44 (1998), Yuu Hu and Guozhen Xiao Resilient Functions Over Finite Fields IEEE Trans on Information Theory 49 (003), Keqin Feng and Fengmei Liu ew Results On The onexistence of Generalized Bent Functions IEEE Trans on Information Theory 49 (003), PV Kumar, RA Scholtz, and LR Welch Generalized Bent Functions and Their Proerties J Combinatorial Theory (A) 40 (1985), Yuan Li and TW Cusick Strict Avalanche Criterion over Finite Fields J Math Cryt 1(007): Yuan Li and TW Cusick Linear Structures of Symmetric Functions over Finite Fields Information Processing Letters 97 (006), Ed Dawson and Chuan-kun Wu On the Linear Structure of Symmetric Boolean Functions Australasian Journal of Combinatorics 16 (1997), TW Cusick, Yuan Li and P Stanica Balanced symmetric functions over GF() IEEE Trans on Information Theory 54(3), , PHKe Imrove lower bound on the number of balanced symmetric functions over GF() htt://erintiarcorg/008/165df