Construction and Analysis of Boolean Functions of 2t + 1 Variables with Maximum Algebraic Immunity
|
|
- Loraine Preston
- 5 years ago
- Views:
Transcription
1 Constuction and Analysis of Boolean Functions of 2t + 1 Vaiables with Maximum Algebaic Immunity Na Li and Wen-Feng Qi Depatment of Applied Mathematics, Zhengzhou Infomation Engineeing Univesity, Zhengzhou, , China mylina 1980@yahoo.com.cn, wenfeng.qi@263.net Abstact. In this pape, we study the constuction of (2t + 1-vaiable Boolean functions with maximum algebaic immunity, and we also analyze some othe cyptogaphic popeties of this kind of functions, such as nonlineaity, esilience. We fist identify seveal classes of this kind of functions. Futhe, some necessay conditions of this kind of functions which also have highe nonlineaity ae obtained. In this way, a modified constuction method is poposed to possibly obtain (2t + 1-vaiable Boolean functions which have maximum algebaic immunity and highe nonlineaity, and a class of such functions is also obtained. Finally, we pesent a sufficient and necessay condition of (2t + 1-vaiable Boolean functions with maximum algebaic immunity which ae also 1-esilient. Keywods: Algebaic attack, algebaic immunity, Boolean functions, balancedness, nonlineaity, esilience. 1 Intoduction The ecent pogess in eseach elated to algebaic attacks [1,2,5,6] seems to theaten all LFSR-based steam ciphes. It is known that Boolean functions used in steam ciphes should have high algebaic degee [11]. Howeve, a Boolean function may have low degee multiples even if its algebaic degee is high. By this fact it is possible to obtain an ove-defined system of multivaiate equations of low degee whose unknowns ae the bits of the initialization of the LFSR(s. Then the secet key can be discoveed by solving the system. To measue the esistance to algebaic attacks, a new cyptogaphic popety of Boolean functions called algebaic immunity (AI has been poposed by W. Meie et al. [16]. When used in a cyptosystem, a Boolean function should have high AI. Now, it is known that the AI of an n-vaiable Boolean function is uppe bounded by n 2 [6,16]. Balancedness, nonlineaity and coelation-immunity ae thee othe impotant cyptogaphic citeia. In some sense, algebaic immunity is compatible with the fome two citeia: a Boolean functions with low nonlineaity will have low AI [7,14], a Boolean function of an odd numbe of vaiables with maximum AI must be balanced [7]. The existence of links between algebaic immunity and coelation-immunity emains open. Constuctions of Boolean functions with maximum AI ae obviously impotant. Futhe, it is moe impotant to constuct these functions which also satisfy This wok was suppoted by National Natue Science Foundation of China unde Gant numbe
2 2 Na Li and Wen-Feng Qi some othe citeia (such as balancedness, a high nonlineaity, a high coelationimmunity ode,.... Some classes of symmetic Boolean functions with maximum AI wee obtained in [3] and [9], and it was shown in [12] that thee is only one such symmetic function (besides its complement when the numbe of input vaiables is odd. A constuction keeping in mind the basic theoy of algebaic immunity was pesented in [9], which also povided some functions with maximum AI. In [4], Calet intoduced a geneal method (fo any numbe of vaiables and an algoithm (fo an even numbe of vaiables fo constucting balanced functions with maximum AI. In [13], a method was poposed fo constucting functions of an odd numbe of vaiables with maximum AI, which convet the poblem of constucting such a function to the poblem of finding an invetible submatix of a 2 n 1 2 n 1 matix. And it was stated that any such function can be obtained by this method. In this pape, we study the constuction of (2t +1-vaiable Boolean functions with maximum AI, and we also analyze some othe cyptogaphic popeties of this kind of functions. Fom the chaacteistic of the matix used in the constuction poposed in [13], we obtain some necessay o sufficient conditions of (2t + 1-vaiable Boolean functions with maximum AI. Futhe, by studying the Walsh specta of this kind of functions, we obtain some necessay conditions of this kind of functions which also have highe nonlineaity and thus we popose a modified constuction to obtain such functions. We finally pesent a sufficient and necessay condition of (2t + 1-vaiable Boolean functions with maximum AI which ae also 1-esilient. 2 Peliminaies Let F n 2 be the set of all n-tuples of elements in the finite field F 2. To avoid confusion with the usual sum, we denote the sum ove F 2 by. A Boolean function of n vaiables is a function fom F n 2 into F 2. Any n-vaiable Boolean function f can be uniquely expessed by a polynomial in F 2 [x 1,..., x n ]/(x 2 1 x 1,..., x 2 n x n, which is called its algebaic nomal fom (ANF. The algebaic degee of f, denoted by deg(f, is the degee of this polynomial. Boolean function f can also be identified by a binay sting of length 2 n, called its tuth table, which is defined as (f(0, 0,..., 0, f(1, 0,..., 0, f(0, 1,..., 0,..., f(1, 1,..., 1. Let 1 f = X F n 2 f(x = 1}, 0 f = X F n 2 f(x = 0}. The set 1 f (esp. 0 f is called the on set (esp. off set. The cadinality of 1 f, denoted by wt(f, is called the Hamming wight of f. We say that an n-vaiable Boolean function f is balanced if wt(f = 2 n 1. The Hamming distance between two functions f and g, denoted by d(f, g, is the Hamming weight of f g. Let S = (s 1, s 2,..., s n F n 2, the Hamming weight of S, denoted by wt(s, is the numbe of 1 s in s 1, s 2,..., s n }.
3 Constuction and Analysis of Boolean Functions 3 Walsh specta is an impotant tool fo studying Boolean functions. Let X = (x 1,..., x n and S = (s 1,..., s n both belonging to F n 2 and thei inne poduct X S = x 1 s 1... x n s n. Let f be a Boolean function of n vaiables. Then the Walsh tansfom of f is an intege valued function ove F n 2 which is defined as W f (S = X F n 2 ( 1 f(x X S. Affine functions ae those Boolean functions of degee at most 1. The nonlineaity of an n-vaiable Boolean function f is its Hamming distance fom the set of all n-vaiable affine functions, i.e., nl(f = mind(f, g g is an affine function}. The nonlineaity of f can be descibed by its Walsh specta as nl(f = 2 n max S F n 2 W f (S. Coelation immune functions and esilient functions ae two impotant classes of Boolean functions. A function is mth ode coelation immune (esp. m-esilient if and only if its Walsh specta satisfies W f (S = 0, fo 1 wt(s m (esp. 0 wt(s m. Definition 1. [16] Fo a given n-vaiable Boolean function f, a nonzeo n- vaiable Boolean function g is called an annihilato of f if f g = 0, and the algebaic immunity of f, denoted by AI(f, is the minimum value of d such that f o f 1 admits an annihilating function of degee d. Fo convenience, two odeings on vectos and monomials ae defined as follows. Definition 2. A vecto odeing < v on F n 2 is defined as: let (a 1,..., a n, (b 1,..., b n F n 2, then (a 1,..., a n < v (b 1,..., b n if and only if n i=1 a i < n i=1 b i, o n i=1 a i = n i=1 b i and thee exists 1 i < n such that a i > b i, a j = b j fo 1 j < i. Example 1. If n = 3, then (0, 0, 0 < v (1, 0, 0 < v (0, 1, 0 < v (0, 0, 1 < v (1, 1, 0 < v (1, 0, 1 < v (0, 1, 1 < v (1, 1, 1. Definition 3. A monomial odeing < m on F 2 [x 1,..., x n ]/(x 2 1 x 1,..., x 2 n x n is defined as: let x a xan n, x b x bn n F 2 [x 1,..., x n ]/(x 2 1 x 1,..., x 2 n x n, then x a xan n < m x b x bn n if and only if (a 1,..., a n < v (b 1,..., b n. It is clea that < v and < m ae both total odeings. Let A be an l l matix, and integes 1 i 1, i 2..., i k l, 1 j 1, j 2..., j k l. Denoted by A (i1,...,i k the k l matix with the th (1 k ow vecto equal to the i th ow vecto of A, and A (i1,...,i k ;j 1,...,j k the k k matix with the th (1 k column vecto equal to the j th column vecto of A (i1,...,i k.
4 4 Na Li and Wen-Feng Qi 3 Constuction of Boolean functions with maximum AI In this section, we biefly eview the method to constuct Boolean functions with maximum AI poposed in [13]. Let n be a positive intege, X = (x 1,..., x n F n 2. Let v(x =(1, x 1,..., x n, x 1 x 2,..., x n 1 x n,......, P n x 1 x n 2 1,..., x n 2 +2 x 2 1 i=0 ( n F n i 2, whee the monomials ae odeed accoding to the odeing < m. It is clea that n 2 1 ( n i=0 i = 2 n 1 when n is odd. Let f be an n-vaiable Boolean function, ( n i matix with the set of ow vectos let V (1 f denote the wt(f n 2 1 i=0 v(x X 1 f }, and V (0 f denote the (2 n wt(f n 2 1 the set of ow vectos v(x X 0 f }. i=0 ( n i matix with Lemma 1. [3,9] Let odd n = 2t + 1 and f be an n-vaiable Boolean function which satisfies a fo wt(x t f(x = a 1 fo wt(x > t, whee a F 2, then AI(f = t + 1. When a = 1, the function descibed in Lemma 1 is called the majoity function, and we denote it by F n. It is clea that F n is balanced. We aange the vectos in 1 Fn (esp. 0 Fn accoding to the ode < v, and denote them by X 1,..., X 2 n 1 (esp. Y 1,..., Y 2 n 1, i.e. X 1 < v... < v X 2 n 1 (esp. Y 1 < v... < v Y 2 n 1. Let X j = (x j,1,..., x j,n (esp. Y i = (y i,1,..., y i,n. The ith ow vecto of V (1 Fn (esp. V (0 Fn is v(x i (esp. v(y i. The idea of the constuction poposed in [13] is to obtain a new function by changing the values of the majoity function at some vectos. The poblem of finding out the appopiate vectos is conveted to the poblem of finding out a k k invetible submatix of the 2 n 1 2 n 1 invetible matix W = V (0 Fn V (1 Fn 1. Theoem 1. [13] Let n = 2t + 1, and f an n-vaiable Boolean function. Then, AI(f= t + 1 if and only if thee exist integes 1 i 1 <... < i k 2 n 1, 1 j 1 <... < j k 2 n 1, such that f = f (i1,...,i k ;j 1,...,j k and W (i1,...,i k ;j 1,...,j k is invetible, whee f (i1,...,i k ;j 1,...,j k is defined as Fn (X 1 if X X f (i1,...,i k ;j 1,...,j k (X = j1,..., X jk, Y i1,..., Y ik } F n (X else. (1 Constuction 1 [13] Let n = 2t + 1. The following method can geneate a Boolean function of n vaiables with maximum AI. Step1: Select andomly an intege 1 k 2 n 2 and k integes 1 i 1 <... < i k 2 n 1.
5 Constuction and Analysis of Boolean Functions 5 Step2: Find out k integes 1 j 1 <... < j k 2 n 1, such that the j 1 th,..., j k th column vectos of W (i1,...,i k ae linealy independent. Then, the Boolean function f (i1,...,i k ;j 1,...,j k defined by (1 has AI t + 1. Remak 1. 1 Fo any fixed 1 k 2 n 2 and any k integes 1 i 1 <... < i k 2 n 1, thee always exist k integes 1 j 1 <... < j k 2 n 1 such that W (i1,...,i k ;j 1,...,j k is invetible. 2 Any Boolean function of 2t + 1 vaiables with maximum AI can be constucted by this method. Fo the est of this pape, we always suppose n = 2t Popeties of W and seveal classes of n-vaiable Boolean functions with maximum AI In this section, we fist show some impotant popeties of the matix W = V (0 Fn V (1 Fn 1, then use these conclusions to obtain some necessay o sufficient conditions of n-vaiable Boolean function achieving maximum AI. Let A be a 2 n 1 2 n 1 matix, and divide A into (t+1 2 submatixes, denoted by A i,j, 1 i t + 1, 1 j t + 1, defined as whee l = A i,j = A (i 1 +1, i , i ;s j 1 +1,s j ,s j, 0 if l = 0, s if l > 0 l = l k=1 ( n t+k 0 if l = 0. if l > 0 l 1 k=0 ( n k It is clea that the ow (esp. column vectos of W i,j coespond to the vectos in F n 2 with Hamming weight i + t (esp. j 1. Poposition 1. [10] V (1 Fn 1 =V (1 Fn. Poposition 2. Let W = V (0 Fn V (1 Fn 1, then 0 if W i,j = V (0 Fn i,j t j+1 ( t+i j+1 = 0 else whee 0 denotes the matix with all enties 0., fo 1 i, j t + 1, Poof. By Poposition 1, W = V (0 Fn V (1 Fn 1 = V (0 Fn V (1 Fn. Let Y = (y 1,..., y n 0 Fn and wt(y = i > t, x 1 x j be a monomial of degee j(0 j t. Denote the tanspose of the column vecto of V (1 Fn coesponding to x 1 x j by u(x 1 x j. That is, u(x 1 x j is the evaluation of x 1 x j at the vectos belonging to 1 Fn. We can epesent u(x 1 x j as (g(1, g(x 1,..., g(x n, g(x 1 x 2, g(x 1 x 3,..., g(x n 1 x n,..., g(x 1 x t,..., g(x t+2 x n, (2
6 6 Na Li and Wen-Feng Qi whee g is a function on the monomials of degee at most t, which satisfies g(x a 1 1 if 1 x1 x xan n = j x a 1 1 xan n 0 else On the othe hand, we can also epesent v(y as. (3 (h(1, h(x 1,..., h(x n, h(x 1 x 2, h(x 1 x 3,..., h(x n 1 x n,..., h(x 1 x t,..., h(x t+2 x n, (4 whee h is a function on the monomials of degee at most t, which satisfies h(x a 1 1 if x a 1 1 xan n = 1 xan n x y 1 1 xyn n. (5 0 else Denote the inne poduct of v(y and u(x 1 x j by c. If y 1,..., y j ae not all 1, by (2, (3, (4 and (5, we have c = 0 = h(x 1 x j. If y 1,..., y j ae all 1, we have h(x 1 x j = 1 and c = x 1 x j x a 1 1 xan n, x a 1 1 xan n x y 1 1 xyn n wt(a 1,...,a n t t j ( i j 1 = It is clea that the ow (esp. column vectos of W i,j coespond to the vectos in F n 2 with Hamming weight i + t (esp. j 1. Theefoe, we complete the poof. Coollay 1. 1 Fo any 2 i t + 1, W i,t+2 i = 0. 2 Fo any 1 j t + 1, W 1,j = V (0 Fn 1,j. 3 Fo any 1 i t + 1, W i,t+1 = V (0 Fn i,t+1. Poof. 1 If 2 i t + 1 and j = t + 2 i, then t j+1 2 If i = 1, then t j+1. ( t + i j + 1 i 1 ( 2i 1 = = 2 2i 2 mod 2 = 0. ( t + i j + 1 = t j+1 ( t j + 2 = 2 t j+2 1 mod 2 = 1. 3 If j = t + 1, then t j+1 ( t + i j + 1 = 1. We can obtain some necessay conditions of n-vaiable Boolean functions with maximum AI.
7 Constuction and Analysis of Boolean Functions 7 Theoem 2. Let 1 k 2 n 1, 1 i 1 <... < i k 2 n 1, 1 j 1 <... < j k 2 n 1 t j. If thee exist 0 j t, t + 1 i n such that = 0, and #X X j1,..., X jk } wt(x = j} + #Y Y i1,..., Y ik } wt(y = i} > k, then, AI(f (i1,...,i k ;j 1,...,j k < t + 1. Poof. By Theoem 1, it is sufficient to show that W (i1,...,i k ;j 1,...,j k is not invetible. By Poposition 2 and the fist condition, we have that W i t,j+1 = 0. Then the second condition implies that W (i1,...,i k ;j 1,...,j k has a submatix with the numbe of ows and columns geate than k whose enties ae all 0. Theefoe, W (i1,...,i k ;j 1,...,j k is not invetible. Coollay 2. Let 1 k 2 n 1, 1 i 1 <... < i k 2 n 1, 1 j 1 <... < j k 2 n 1. If thee exists 0 t 1 such that #X X j1,..., X jk } wt(x = } + #Y Y i1,..., Y ik } wt(y = n } > k, then, AI(f (i1,...,i k ;j 1,...,j k < t + 1. In the following of this section, seveal classes of n-vaiable Boolean functions with maximum AI ae povided. Theoem 3. Let 1 k 2 n 1, 1 i 1 <... < i k 2 n 1, 1 j 1 <... < j k 2 n 1. If the following conditions ae both satisfied, then AI(f (i1,...,i k ;j 1,...,j k = t Thee exist 1 a 1 <... < a s n, such that x j,a 1 =... = x j,a s = 0 fo 1 k. 2 Fo any X j (1 k, thee exists coespondingly Y i Y i1,..., Y ik }, such that y i,a = x j,a fo a / a 1,..., a s }, and t wt(x j l=0 ( wt(yi wt(x j = 1. l Poof. If X j1,..., X jk and Y i1,..., Y ik satisfy the two conditions, then by Poposition 2, W (i1,...,i k ;j 1,...,j k is in the fom of lowe tiangula with all enties on the diagonal equal to 1. Theefoe W (i1,...,i k ;j 1,...,j k is invetible, which implies that W (i1,...,i k ;j 1,...,j k is invetible, and the esult holds by Theoem 1. Example 2. Let n = 7, L 1 = (1, 0, 0, 0, 0, 0, 0, (0, 1, 1, 0, 0, 0, 0, (0, 0, 1, 1, 0, 0, 0, (1, 1, 1, 0, 0, 0, 0, } 1 Fn, L 2 = (1, 0, 0, 0, 1, 1, 1, (0, 1, 1, 0, 1, 1, 0, (0, 0, 1, 1, 0, 1, 1, (1, 1, 1, 0, 1, 1, 1, } 0 Fn. Then the function ( i j has AI 4. Fn (X 1 if X L f(x = 1 L 2 F n (X else
8 8 Na Li and Wen-Feng Qi Theoem 4. Let 1 2k 2 n 1, 1 i 1 <... < i 2k 2 n 1, 1 j 1 <... < j 2k 2 n 1. wt(x j = w 1, wt(y i = w 1 fo 1 k, and wt(x j = w 2, wt(y i = w 2 fo k + 1 2k. If one of the following two conditions is satisfied, then AI(f (i1,...,i 2k ;j 1,...,j 2k = t + 1. t w 2 and ae not both 1, and 1 t w 1 2 t w 1 ( w 2 w 1 ( w 1 w 2 AI(f (i1,...,i k ;j 1,...,j k = AI(f (ik+1,...,i 2k ;j k+1,...,j 2k = t + 1. ( w 1 w 1 t w 2 and ( w 2 w 2 ae not both 1, and AI(f (i1,...,i k ;j k+1,...,j 2k = AI(f (ik+1,...,i 2k ;j 1,...,j k = t + 1. Poof. Let M denote the 2k 2k matix W (i1,...,i 2k ;j 1,...,j 2k. The fist condition implies that M (1,...,k;1,...,k and M (k+1,...,2k;k+1,...,2k ae both invetible, and at least one of M (1,...,k;k+1,...,2k and M (k+1,...,2k;1,...,k is 0. Then, M is invetible, and the esult holds by Theoem 1. If the second condition is satisfied, the esult can be poved in the same way. Example 3. Let n = 7, L 1 =(0, 0, 0, 0, 1, 1, 0, (0, 0, 0, 0, 1, 0, 1, (0, 0, 0, 0, 0, 1, 1, (1, 1, 0, 0, 1, 0, 0, (1, 1, 0, 0, 0, 1, 0, (1, 1, 0, 0, 0, 0, 1}, L 2 =(1, 1, 0, 0, 1, 1, 0, (1, 1, 0, 0, 1, 0, 1, (1, 1, 0, 0, 0, 1, 1, (1, 1, 1, 1, 1, 0, 0, (1, 1, 1, 1, 0, 1, 0, (1, 1, 1, 1, 0, 0, 1}. Then the function Fn (X 1 if X L f(x = 1 L 2 F n (X else has AI 4. Theoem 5. Let 1 k n, Y i1,..., Y ik belong to 0 Fn and thei Hamming weight ae w 1,..., w k, espectively. If = 1 fo 1 i k, and 1 t 1 ( wi 1 2 thee exist 1 j 1 <... < j k n, such that the j 1 th,..., j k th column of Y i1 the matix... ae linealy independent, Y ik then, AI(f (i1,...,i k ;j 1 +1,...,j k +1 = t + 1. Poof. By Poposition 2, W (i1,...,i k ;j 1 is invetible if the two conditions ae +1,...,j k +1 both satisfied, then, and the esult holds by Theoem 1. Example 4. Let n = 7, L 1 =(1, 0, 0, 0, 0, 0, 0, (0, 1, 0, 0, 0, 0, 0, (0, 0, 1, 0, 0, 0, 0}, L 2 =(1, 0, 1, 0, 1, 1, 1, (0, 1, 1, 0, 1, 0, 1, (1, 1, 1, 1, 0, 1, 0}. Then the function Fn (X 1 if X L f(x = 1 L 2 F n (X else has AI 4.
9 Constuction and Analysis of Boolean Functions 9 5 Nonlineaity and esilience of Boolean functions with maximum AI At fist, we give the Walsh specta of majoity functions. Note that although the fist item and the case of wt(s = 1 in the second item in the following lemma have been given in [9], we still give the poof fo completeness. Lemma 2. Let S F n 2. 1 If wt(s is even, then W Fn (S = 0. 2 If wt(s is odd, then Poof. Since ( n 1 W Fn (S = ( 1 (wt(s+1/2 2 t wt(x=i (wt(s 1/2 ( 1 S X = K i (wt(s, n, we have W Fn (S = n i=t+1 K i (wt(s, n i=1 2i 1 n 2i. t K i (wt(s, n, (6 whee K i (k, n is the so-called Kawtchouk polynomial [15, Page 151, Pat I] defined by i ( ( k n k K i (k, n = ( 1 j, i = 0, 1,..., n. j i j j=0 Kawtchouk polynomials also have popeties [15, Page 153, Pat I] as follows. P1. K i (k, n = ( 1 k K n i (k, n. P2. e i=0 K i(k, n = K e (k 1, n 1. P3. (n kk i (k + 1, n = (n 2iK i (k, n kk i (k 1, n fo nonnegative integes i and k. If wt(s is even, then by (6 and P1, we have W Fn (S = 0. If wt(s is odd, then by (6, P1 and P2, we have W Fn (S = 2 i=0 t K i (wt(s, n = 2K t (wt(s 1, n 1. i=0 By the definition of Kawtchouk polynomials, we have K t (k, n 1 = 0 if k is odd. Thus by P3, we have (wt(s 1/2 W Fn (S = ( 1 (wt(s 1/2+1 2i 1 2K t (0, n 1 n 2i ( n 1 = ( 1 (wt(s+1/2 2 t i=1 (wt(s 1/2 i=1 2i 1 n 2i.
10 10 Na Li and Wen-Feng Qi Lemma 3. Let S, T F n 2. 1 If wt(s + wt(t = n + 1, then W Fn (S = ( 1 t W Fn (T. 2 If both wt(s and wt(t ae odd, and 0 < wt(s < wt(t t + 1, then W Fn (S > W Fn (T. Poof. 1 Since Kawtchouk polynomials have the following popety, we have that K i (k, n = ( 1 i K i (n k, n, W Fn (S = 2K t (wt(s 1, n 1 = 2( 1 t K t (n 1 (wt(s 1, n 1 = 2( 1 t K t (wt(t 1, n 1 = ( 1 t W Fn (T. 2 It is obvious fom the second item of Lemma 2. Remak 2. By Lemma 3, we have max W T F n Fn (T = W Fn (S 1 = W Fn (S n = 2 2 ( n 1 whee wt(s 1 = 1, wt(s n = n. Theefoe, nl(f n = 2 n 1 ( n 1 t [9]. And ( n 1 max T F n 2,wt(T 1,n W F n (T = W Fn (S 3 = W Fn (S n 2 = 2 n 2 whee wt(s 3 = 3, wt(s n 2 = n 2. We note that the diffeence between the maximal and the secondaily maximal absolute value of Walsh specta is quite geat, which is 2 n 3 ( n 1. n 2 t Algebaic immunity has the following elationship with nonlineaity. Lemma 4. [14] Let f be an n-vaiable Boolean function, AI(f = k, then and this bound is tight. nl(f 2 n 1 n k i=k 1 ( n 1 Remak 3. Lemma 4 togethe with Remak 2 implies that F n has the wost nonlineaity among all n-vaiable Boolean functions with maximum AI. Theoem 6. The Walsh specta of f = f (i1,...,i k ;j 1,...,j k is given by W f (S = W Fn (S 4( i S X j, t S Y i., t,
11 Constuction and Analysis of Boolean Functions 11 Poof. W f (S = = 2 n 1 2 n 1 ( 1 f(x+s X + 1,...,2 n 1 }\j 1,...,j k } 1,...,2 n 1 }\i 1,...,i k } = W Fn (S 2( = W Fn (S 2( = W Fn (S 2( = W Fn (S 4( ( 1 f(y+s Y ( 1 Fn(X+S X + ( 1 Fn(Y+S Y + ( 1 Fn(X j +S X j + ( 1 1+S X j + ( 1 Fn(X j +1+S X j + ( 1 Fn(Y i +1+S Y i ( 1 Fn(Y i +S Y i ( 1 S Y i (2S X j 1 + S X j (1 2S Y i S Y i. Fom the above analysis in this section, some necessay conditions of Boolean functions with maximum AI and these functions which also have highe nonlineaity than that of F n can be obtained. Theoem 7. Let 1 k 2 n 1, 1 i 1 <... < i k 2 n 1, 1 j 1 <... < j k 2 n 1. If one of the following conditions is satisfied, then AI(f (i1,...,i k ;j 1,...,j k < t Thee exists 1 n, such that x j1, x jk, > y i1, y ik,. 2If n 1 mod 4, #X X j1,..., X jk } wt(x is odd} > #Y Y i1,..., Y ik } wt(y is odd}; if n 3 mod 4, #X X j1,..., X jk } wt(x is odd} < #Y Y i1,..., Y ik } wt(y is odd}. Poof. By Theoem 6, the fist condition means that W (S > f(i1,...,i k ;j 1,...,j k W Fn (S fo S = (0,..., 0, 1, 0,..., 0. Thus, we have nl(f }} (i1,...,i k ;j 1,...,j k < nl(f n 1 by Remak 2. Theefoe, by Remak 3, we have AI(f (i1,...,i k ;j 1,...,j k < t + 1. If the second condition is satisfied, then W (S > W f(i1,...,i k ;j 1,...,j k F n (S fo S = (1, 1,..., 1. In the same way, the esult can be poved. Theoem 8. Let f = f (i1,...,i k ;j 1,...,j k be an n-vaiable Boolean function with AI t + 1. If one of the following conditions is satisfied, then f has the wost nonlineaity among all n-vaiable Boolean functions with maximum AI.
12 12 Na Li and Wen-Feng Qi 1 Thee exists 1 n, such that x j1, x jk, = y i1, y ik,. 2 #X X j1,..., X jk } wt(x is odd} = #Y Y i1,..., Y ik } wt(y is odd}. Poof. By Theoem 6, the fist condition means that W (S = f(i1,...,i k ;j 1,...,j k W Fn (S fo S = (0,..., 0, 1, 0,..., 0. Thus, we have nl(f }} (i1,...,i k ;j 1,...,j k nl(f n 1 by Remak 2. Theefoe, by Remak 3, we have nl(f (i1,...,i k ;j 1,...,j k = nl(f n, and the esult is poved. If the second condition is satisfied, then W (S = W f(i1,...,i k ;j 1,...,j k F n (S fo S = (1, 1,..., 1. In the same way, the esult can be poved. Coollay 3. Fo any 1 i, j 2 n 1, if AI(f (i;j = t + 1 then f (i;j has the wost nonlineaity among all n-vaiable Boolean functions with maximum AI. Poof. Fom Theoem 8, it is sufficient to conside the case of i = 2 n 1, j = 1, i.e. X = (0, 0,..., 0, Y = (1, 1,..., 1. In this case, fom the fist item of Coollay 1 we have AI(f (i;j < t + 1 which contadicts the assumption. Theoem 9. If 1 k n 3 4(n 2 ( n 1 2 n 1 + 2min min t ( y i,s 1 s n ( n 1 t, then nl(f(i1,...,i k ;j 1,...,j k is given by x j,s, ( 1 t (N 1 N 2 }, whee N 1 = #Y Y i1,..., Y ik } wt(y is odd }, N 2 = #X X j1,..., X jk } wt(x is odd }. Poof. Denote f (i1,...,i k ;j 1,...,j k ; by f. Fom Theoem 6 we have, W Fn (S 4k W f (S W Fn (S + 4k. Let S, T F n 2, and wt(s = 1 o n, wt(t / 1, n}. If 1 k n 3 by Remak 2, 4(n 2 W f (S W Fn (S 4k W Fn (T + 4k W f (T. ( n 1 t, then Theefoe, we have max T F n 2 W f (T = max wt(s=1,n W f (S. Case 1. wt(s = 1 and S = (0,..., 0, 1, 0,..., 0. By Theoem 6 we have }} s 1 ( n 1 W f (S = 2 4( t y i,s x j,s. Case 2. wt(s = n. By Theoem 6 we have ( n 1 W f (S = 2 4(( 1 t (N 1 N 2. t Hence the esult follows fom nl(f = 2 n max S F n 2 W f (S.
13 Constuction and Analysis of Boolean Functions 13 Now, we modify Constuction 1 to constuct n-vaiable Boolean functions with maximum AI and possibly having highe nonlineaity. Constuction 2 Step1: Select andomly an intege 1 k 2 n 2 and k integes 1 i 1 <... < i k 2 n 1, which satisfy i min y i,s is as lage as possible; 1 s n ii if n 1 mod 4, #Y Y i1,..., Y ik } wt(y is odd } is as lage as possible; if n 3 mod 4, #Y Y i1,..., Y ik } wt(y is even } is as lage as possible. Step2: Find out k integes 1 j 1 <... < j k 2 n 1, which satisfies i the j 1 th,..., j k th column vectos of W (i1,...,i k ae linealy independent; ii a = min ( k y i,s k x j,s is as lage as possible; 1 s n iii if n 1 mod 4, b = #Y Y i1,..., Y ik } wt(y is odd } #X X j1,..., X jk } wt(x is odd } is as lage as possible; if n 3 mod 4, c = #X X j1,..., X jk } wt(x is odd } #Y Y i1,..., Y ik } wt(y is odd } is as lage as possible. Then, the Boolean function f (i1,...,i k ;j 1,...,j k defined by (1 has AI t + 1 and has possibly a highe nonlineaity. Remak 4. Fom Theoem 9, the function obtained by Constuction 2 will has a highe nonlineaity than that of F n if 1 k n 3 ( n 1 4(n 2 t and a > 0, b > 0 (if n 1 mod 4 o c > 0 (if n 3 mod 4, and it possibly has a nonlineaity equal to that of F n if k > n 3 4(n 2. Futhe, the following theoem povides a class of n-vaiable Boolean functions with maximum AI which also have highe nonlineaity than that of F n. ( n 3 Theoem 10. Let n 3 mod 4, 1 k minn, n 1 4(n 2 t }, Yi1,..., Y ik belong to 0 Fn and thei Hamming weights ae w 1,..., w k, espectively. If 1 t 1 = 1, i = 1,..., k; and ( wi 1 2 w 1,..., w k ae not all odd; and 3 thee exist 1 j 1 <... < j k n, such that the j 1 th,..., j k th columns of Y i1 the matix... ae linealy independent; and Y ik 4 fo any s / j 1,..., j k }, y i1,s y ik,s 1; and fo any s j 1,..., j k }, y i1,s y ik,s 2. then, AI(f (i1,...,i k ;j 1 +1,...,j k +1 = t+1 and nl(f (i1,...,i k ;j 1 +1,...,j k +1 nl(f n +2. Example 5. The Boolean function defined in Example 4 has AI 4. And nl(f = nl(f n + 2.
14 14 Na Li and Wen-Feng Qi Finally, we obtain the following sufficient and necessay condition of Boolean functions with maximum AI which ae also esilient functions. Theoem 11. Let f = f (i1,...,i k ;j 1,...,j k be an n-vaiable Boolean function. Then, f is 1-esilient function if and only if fo s = 1,..., n. y i,s x j,s = 1 ( n 1, 2 t Coollay 4. Let f = f (i1,...,i k ;j 1,...,j k be an n-vaiable Boolean function. Then, f is 1-esilient function and has AI t + 1 if and only if y i,s x j,s = 1 ( n 1, 2 t fo s = 1,..., n, and W (i1,...,i k ;j 1,...,j k is invetible. 6 Conclusion Possessing a high algebaic immunity is a necessay condition fo Boolean functions used in steam ciphes against algebaic attacks. In this pape, some classes of (2t + 1-vaiable Boolean functions with maximum AI ae obtained. Futhe, some necessay conditions of this kind of functions which also have highe nonlineaity ae pesented and thus a modified constuction method is poposed to obtain such functions. Finally, a sufficient and necessay condition of (2t + 1- vaiable Boolean functions with maximum AI which ae also 1-esilient is pesented. Howeve, it is still open that what is the highest nonlineaity of Boolean functions with maximum AI and how to constuct Boolean functions which have maximum AI and the highest nonlineaity. Refeences 1. F. Amknecht. Impoving fast algebaic attacks. In FSE 2004, volume 3017 of Lectue Notes in Compute Science, pages Spinge-Velag, F. Amknecht and M. Kause. Algebaic attacks on combines with memoy. In Advances in Cyptology CRYPTO 2003, volume 2729 of Lectue Notes in Compute Science, pages Spinge-Velag, A. Baeken and B. Peneel. On the algebaic immunity of symmetic Boolean functions. In INDOCRYPT 2005, volume 3797 of Lectue Notes in Compute Science, pages Spinge-Velag, C. Calet. A method of constuction of balanced functions with optimum algebaic immunity. Available at 5. N. Coutois. Fast algebaic attacks on steam ciphes with linea feedback. In Advances in Cyptology CRYPTO 2003, volume 2729 of Lectue Notes in Compute Science, pages Spinge-Velag, N. Coutois and W. Meie. Algebaic attacks on steam ciphes with linea feedback. In Advances in Cyptology EUROCRYPT 2003, volume 2656 of Lectue Notes in Compute Science, pages Spinge-Velag, 2003.
15 Constuction and Analysis of Boolean Functions D. K. Dalai, K. C. Gupta and S. Maita. Results on algebaic immunity fo cyptogaphically significant Boolean functions. In INDOCRYPT 2004, volume 3348 of Lectue Notes in Compute Science, pages Spinge-Velag, D. K. Dalai, K. C. Gupta and S. Maita. Cyptogaphically significant Boolean functions: constuction and analysis in tems of algebaic immunity. In FSE 2005, volume 3557 of Lectue Notes in Compute Science, pages Spinge-Velag, D. K. Dalai, S. Maita and S. Saka. Basic theoy in constuction of Boolean functions with maximum possible annihilato immunity. Designs, Codes and Cyptogaphy, 40:41-58, D. K. Dalai and S. Maita. Reducing the Numbe of Homogeneous Linea Equations in Finding Annihilatos. Available at C. Ding, G. Xiao and W. Shan. The stability theoy of steam ciphes. Spinge-Velag, N. Li and W. F. Qi. Symmetic Boolean functions depending on an odd numbe of vaiables with maximum algebaic immunity. IEEE Tansaction on Infomation Theoy, 52(5: , May N. Li and W. F. Qi. Constuction and count of Boolean functions of an odd numbe of vaiables with maximum algebaic immunity. Available at M. Lobanov. Tight bound between nonlineaity and algebaic immunity. Available at F. J. MacWilliams and N. J. A. Sloane. The theoy of eo-coecting codes. Elsevie, Noth-Holland, W. Meie, E. Pasalic and C. Calet. Algebaic attacks and decomposition of Boolean functions. In Advances in Cyptology EUROCRYPT 2004, volume 3027 of Lectue Notes in Compute Science, pages Spinge-Velag, 2004.
New problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationarxiv: v1 [math.co] 1 Apr 2011
Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and
More informationConstruction and Count of Boolean Functions of an Odd Number of Variables with Maximum Algebraic Immunity
arxiv:cs/0605139v1 [cs.cr] 30 May 2006 Construction and Count of Boolean Functions of an Odd Number of Variables with Maximum Algebraic Immunity Na Li, Wen-Feng Qi Department of Applied Mathematics, Zhengzhou
More informationOn the Quasi-inverse of a Non-square Matrix: An Infinite Solution
Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationMultiple Criteria Secretary Problem: A New Approach
J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and
More informationA STUDY OF HAMMING CODES AS ERROR CORRECTING CODES
AGU Intenational Jounal of Science and Technology A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES Ritu Ahuja Depatment of Mathematics Khalsa College fo Women, Civil Lines, Ludhiana-141001, Punjab, (India)
More informationSOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES
italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationSOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS
Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationThe Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr
The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid
More informationTHE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee
Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationNew Finding on Factoring Prime Power RSA Modulus N = p r q
Jounal of Mathematical Reseach with Applications Jul., 207, Vol. 37, o. 4, pp. 404 48 DOI:0.3770/j.issn:2095-265.207.04.003 Http://jme.dlut.edu.cn ew Finding on Factoing Pime Powe RSA Modulus = p q Sadiq
More informationSecret Exponent Attacks on RSA-type Schemes with Moduli N = p r q
Secet Exponent Attacks on RSA-type Schemes with Moduli N = p q Alexande May Faculty of Compute Science, Electical Engineeing and Mathematics Univesity of Padebon 33102 Padebon, Gemany alexx@uni-padebon.de
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationGradient-based Neural Network for Online Solution of Lyapunov Matrix Equation with Li Activation Function
Intenational Confeence on Infomation echnology and Management Innovation (ICIMI 05) Gadient-based Neual Netwok fo Online Solution of Lyapunov Matix Equation with Li Activation unction Shiheng Wang, Shidong
More informationProbablistically Checkable Proofs
Lectue 12 Pobablistically Checkable Poofs May 13, 2004 Lectue: Paul Beame Notes: Chis Re 12.1 Pobablisitically Checkable Poofs Oveview We know that IP = PSPACE. This means thee is an inteactive potocol
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
More informationSyntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)
Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationLifting Private Information Retrieval from Two to any Number of Messages
Lifting Pivate Infomation Retieval fom Two to any umbe of Messages Rafael G.L. D Oliveia, Salim El Rouayheb ECE, Rutges Univesity, Piscataway, J Emails: d746@scaletmail.utges.edu, salim.elouayheb@utges.edu
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationWeighted least-squares estimators of parametric functions of the regression coefficients under a general linear model
Ann Inst Stat Math (2010) 62:929 941 DOI 10.1007/s10463-008-0199-8 Weighted least-squaes estimatos of paametic functions of the egession coefficients unde a geneal linea model Yongge Tian Received: 9 Januay
More informationOn decompositions of complete multipartite graphs into the union of two even cycles
On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationCONSTRUCTION OF EQUIENERGETIC GRAPHS
MATCH Communications in Mathematical and in Compute Chemisty MATCH Commun. Math. Comput. Chem. 57 (007) 03-10 ISSN 0340-653 CONSTRUCTION OF EQUIENERGETIC GRAPHS H. S. Ramane 1, H. B. Walika * 1 Depatment
More informationHua Xu 3 and Hiroaki Mukaidani 33. The University of Tsukuba, Otsuka. Hiroshima City University, 3-4-1, Ozuka-Higashi
he inea Quadatic Dynamic Game fo Discete-ime Descipto Systems Hua Xu 3 and Hioai Muaidani 33 3 Gaduate School of Systems Management he Univesity of suuba, 3-9- Otsua Bunyo-u, oyo -0, Japan xuhua@gssm.otsua.tsuuba.ac.jp
More informationRelating Branching Program Size and. Formula Size over the Full Binary Basis. FB Informatik, LS II, Univ. Dortmund, Dortmund, Germany
Relating Banching Pogam Size and omula Size ove the ull Binay Basis Matin Saueho y Ingo Wegene y Ralph Wechne z y B Infomatik, LS II, Univ. Dotmund, 44 Dotmund, Gemany z ankfut, Gemany sauehof/wegene@ls.cs.uni-dotmund.de
More informationEnumerating permutation polynomials
Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem
More informationQUANTUM ALGORITHMS IN ALGEBRAIC NUMBER THEORY
QUANTU ALGORITHS IN ALGEBRAIC NUBER THEORY SION RUBINSTEIN-SALZEDO Abstact. In this aticle, we discuss some quantum algoithms fo detemining the goup of units and the ideal class goup of a numbe field.
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More informationAvailable online through ISSN
Intenational eseach Jounal of Pue Algeba -() 01 98-0 Available online though wwwjpainfo ISSN 8 907 SOE ESULTS ON THE GOUP INVESE OF BLOCK ATIX OVE IGHT OE DOAINS Hanyu Zhang* Goup of athematical Jidong
More informationIn statistical computations it is desirable to have a simplified system of notation to avoid complicated formulas describing mathematical operations.
Chapte 1 STATISTICAL NOTATION AND ORGANIZATION 11 Summation Notation fo a One-Way Classification In statistical computations it is desiable to have a simplified system of notation to avoid complicated
More informationPearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms
Peason s Chi-Squae Test Modifications fo Compaison of Unweighted and Weighted Histogams and Two Weighted Histogams Univesity of Akueyi, Bogi, v/noduslód, IS-6 Akueyi, Iceland E-mail: nikolai@unak.is Two
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationA pathway to matrix-variate gamma and normal densities
Linea Algeba and its Applications 396 005 317 38 www.elsevie.com/locate/laa A pathway to matix-vaiate gamma and nomal densities A.M. Mathai Depatment of Mathematics and Statistics, McGill Univesity, 805
More informationarxiv: v1 [math.ca] 12 Mar 2015
axiv:503.0356v [math.ca] 2 Ma 205 AN APPLICATION OF FOURIER ANALYSIS TO RIEMANN SUMS TRISTRAM DE PIRO Abstact. We develop a method fo calculating Riemann sums using Fouie analysis.. Poisson Summation Fomula
More informationCERFACS 42 av. Gaspard Coriolis, Toulouse, Cedex 1, France. Available at Date: April 2, 2008.
ON THE BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES IAIN S. DUFF and BORA UÇAR Technical Repot: No: TR/PA/08/26 CERFACS 42 av. Gaspad Coiolis, 31057 Toulouse, Cedex 1, Fance. Available at http://www.cefacs.f/algo/epots/
More informationarxiv: v2 [math.ag] 4 Jul 2012
SOME EXAMPLES OF VECTOR BUNDLES IN THE BASE LOCUS OF THE GENERALIZED THETA DIVISOR axiv:0707.2326v2 [math.ag] 4 Jul 2012 SEBASTIAN CASALAINA-MARTIN, TAWANDA GWENA, AND MONTSERRAT TEIXIDOR I BIGAS Abstact.
More informationDuality between Statical and Kinematical Engineering Systems
Pape 00, Civil-Comp Ltd., Stiling, Scotland Poceedings of the Sixth Intenational Confeence on Computational Stuctues Technology, B.H.V. Topping and Z. Bittna (Editos), Civil-Comp Pess, Stiling, Scotland.
More informationChannel matrix, measurement matrix and collapsed matrix. in teleportation
Channel matix, measuement matix and collapsed matix in telepotation XIN-WEI ZHA, JIAN-XIA QI and HAI-YANG SONG School of Science, Xi an Univesity of Posts and Telecommunications, Xi an, 71011, P R China
More informationUsing Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu
Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of
More informationAnalytical Solutions for Confined Aquifers with non constant Pumping using Computer Algebra
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) Analytical Solutions fo Confined Aquifes with non constant Pumping using Compute Algeba
More informationBounds for Codimensions of Fitting Ideals
Ž. JOUNAL OF ALGEBA 194, 378 382 1997 ATICLE NO. JA966999 Bounds fo Coensions of Fitting Ideals Michał Kwiecinski* Uniwesytet Jagiellonski, Instytut Matematyki, ul. eymonta 4, 30-059, Kakow, Poland Communicated
More informationVanishing lines in generalized Adams spectral sequences are generic
ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationHOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS?
6th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS? Cecília Sitkuné Göömbei College of Nyíegyháza Hungay Abstact: The
More informationMatrix Colorings of P 4 -sparse Graphs
Diplomabeit Matix Coloings of P 4 -spase Gaphs Chistoph Hannnebaue Januay 23, 2010 Beteue: Pof. D. Winfied Hochstättle FenUnivesität in Hagen Fakultät fü Mathematik und Infomatik Contents Intoduction iii
More informationSTUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 104467/20843828AM170027078 542017, 15 32 STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS
More informationDonnishJournals
DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş
More informationMean Curvature and Shape Operator of Slant Immersions in a Sasakian Space Form
Mean Cuvatue and Shape Opeato of Slant Immesions in a Sasakian Space Fom Muck Main Tipathi, Jean-Sic Kim and Son-Be Kim Abstact Fo submanifolds, in a Sasakian space fom, which ae tangential to the stuctue
More informationLecture 25: Pairing Based Cryptography
6.897 Special Topics in Cyptogaphy Instucto: Ran Canetti May 5, 2004 Lectue 25: Paiing Based Cyptogaphy Scibe: Ben Adida 1 Intoduction The field of Paiing Based Cyptogaphy has exploded ove the past 3 yeas
More informationTight Upper Bounds for the Expected Loss of Lexicographic Heuristics in Binary Multi-attribute Choice
Tight Uppe Bounds fo the Expected Loss of Lexicogaphic Heuistics in Binay Multi-attibute Choice Juan A. Caasco, Manel Baucells Except fo fomatting details and the coection of some eata, this vesion matches
More information15.081J/6.251J Introduction to Mathematical Programming. Lecture 6: The Simplex Method II
15081J/6251J Intoduction to Mathematical Pogamming ectue 6: The Simplex Method II 1 Outline Revised Simplex method Slide 1 The full tableau implementation Anticycling 2 Revised Simplex Initial data: A,
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
More informationSemicanonical basis generators of the cluster algebra of type A (1)
Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationAn Exact Solution of Navier Stokes Equation
An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in
More informationMoment-free numerical approximation of highly oscillatory integrals with stationary points
Moment-fee numeical appoximation of highly oscillatoy integals with stationay points Sheehan Olve Abstact We pesent a method fo the numeical quadatue of highly oscillatoy integals with stationay points.
More informationTHE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS
THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS ESMERALDA NĂSTASE MATHEMATICS DEPARTMENT XAVIER UNIVERSITY CINCINNATI, OHIO 4507, USA PAPA SISSOKHO MATHEMATICS DEPARTMENT ILLINOIS STATE UNIVERSITY
More informationMeasure Estimates of Nodal Sets of Polyharmonic Functions
Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of
More informationCALCULATING THE NUMBER OF TWIN PRIMES WITH SPECIFIED DISTANCE BETWEEN THEM BASED ON THE SIMPLEST PROBABILISTIC MODEL
U.P.B. Sci. Bull. Seies A, Vol. 80, Iss.3, 018 ISSN 13-707 CALCULATING THE NUMBER OF TWIN PRIMES WITH SPECIFIED DISTANCE BETWEEN THEM BASED ON THE SIMPLEST PROBABILISTIC MODEL Sasengali ABDYMANAPOV 1,
More informationOn Polynomials Construction
Intenational Jounal of Mathematical Analysis Vol., 08, no. 6, 5-57 HIKARI Ltd, www.m-hikai.com https://doi.og/0.988/ima.08.843 On Polynomials Constuction E. O. Adeyefa Depatment of Mathematics, Fedeal
More informationEQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS
EQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS ATHULA GUNAWARDENA AND ROBERT R MEYER Abstact A d-dimensional gid gaph G is the gaph on a finite subset in the intege lattice Z d in
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationOn Computing Optimal (Q, r) Replenishment Policies under Quantity Discounts
Annals of Opeations Reseach manuscipt No. will be inseted by the edito) On Computing Optimal, ) Replenishment Policies unde uantity Discounts The all - units and incemental discount cases Michael N. Katehakis
More informationLocalization of Eigenvalues in Small Specified Regions of Complex Plane by State Feedback Matrix
Jounal of Sciences, Islamic Republic of Ian (): - () Univesity of Tehan, ISSN - http://sciencesutaci Localization of Eigenvalues in Small Specified Regions of Complex Plane by State Feedback Matix H Ahsani
More informationPerturbation to Symmetries and Adiabatic Invariants of Nonholonomic Dynamical System of Relative Motion
Commun. Theo. Phys. Beijing, China) 43 25) pp. 577 581 c Intenational Academic Publishes Vol. 43, No. 4, Apil 15, 25 Petubation to Symmeties and Adiabatic Invaiants of Nonholonomic Dynamical System of
More informationModel and Controller Order Reduction for Infinite Dimensional Systems
IT J. Eng. Sci., Vol. 4, No.,, -6 Model and Contolle Ode Reduction fo Infinite Dimensional Systems Fatmawati,*, R. Saagih,. Riyanto 3 & Y. Soehayadi Industial and Financial Mathematics Goup email: fatma47@students.itb.ac.id;
More informationq i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by
CSISZÁR f DIVERGENCE, OSTROWSKI S INEQUALITY AND MUTUAL INFORMATION S. S. DRAGOMIR, V. GLUŠČEVIĆ, AND C. E. M. PEARCE Abstact. The Ostowski integal inequality fo an absolutely continuous function is used
More informationFailure Probability of 2-within-Consecutive-(2, 2)-out-of-(n, m): F System for Special Values of m
Jounal of Mathematics and Statistics 5 (): 0-4, 009 ISSN 549-3644 009 Science Publications Failue Pobability of -within-consecutive-(, )-out-of-(n, m): F System fo Special Values of m E.M.E.. Sayed Depatment
More informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More informationBoundedness for Marcinkiewicz integrals associated with Schrödinger operators
Poc. Indian Acad. Sci. (Math. Sci. Vol. 24, No. 2, May 24, pp. 93 23. c Indian Academy of Sciences oundedness fo Macinkiewicz integals associated with Schödinge opeatos WENHUA GAO and LIN TANG 2 School
More informationA scaling-up methodology for co-rotating twin-screw extruders
A scaling-up methodology fo co-otating twin-scew extudes A. Gaspa-Cunha, J. A. Covas Institute fo Polymes and Composites/I3N, Univesity of Minho, Guimaães 4800-058, Potugal Abstact. Scaling-up of co-otating
More informationWeb-based Supplementary Materials for. Controlling False Discoveries in Multidimensional Directional Decisions, with
Web-based Supplementay Mateials fo Contolling False Discoveies in Multidimensional Diectional Decisions, with Applications to Gene Expession Data on Odeed Categoies Wenge Guo Biostatistics Banch, National
More informationDEMONSTRATING THE INVARIANCE PROPERTY OF HOTELLING T 2 STATISTIC
ISSN: 319-8753 Intenational Jounal of Innovative Reseach in Science, Engineeing and echnology Vol., Issue 7, July 013 DEMONSRAING HE INVARIANCE PROPERY OF HOELLING SAISIC Sani Salihu Abubaka 1, Abubaka
More informationLecture 16 Root Systems and Root Lattices
1.745 Intoduction to Lie Algebas Novembe 1, 010 Lectue 16 Root Systems and Root Lattices Pof. Victo Kac Scibe: Michael Cossley Recall that a oot system is a pai (V, ), whee V is a finite dimensional Euclidean
More information6 Matrix Concentration Bounds
6 Matix Concentation Bounds Concentation bounds ae inequalities that bound pobabilities of deviations by a andom vaiable fom some value, often its mean. Infomally, they show the pobability that a andom
More informationOn the Number of Rim Hook Tableaux. Sergey Fomin* and. Nathan Lulov. Department of Mathematics. Harvard University
Zapiski Nauchn. Seminaov POMI, to appea On the Numbe of Rim Hook Tableaux Segey Fomin* Depatment of Mathematics, Massachusetts Institute of Technology Cambidge, MA 0239 Theoy of Algoithms Laboatoy SPIIRAN,
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More informationSolving Some Definite Integrals Using Parseval s Theorem
Ameican Jounal of Numeical Analysis 4 Vol. No. 6-64 Available online at http://pubs.sciepub.com/ajna///5 Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationAlternative Tests for the Poisson Distribution
Chiang Mai J Sci 015; 4() : 774-78 http://epgsciencecmuacth/ejounal/ Contibuted Pape Altenative Tests fo the Poisson Distibution Manad Khamkong*[a] and Pachitjianut Siipanich [b] [a] Depatment of Statistics,
More informationarxiv: v1 [math.nt] 12 May 2017
SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking
More informationThis is a very simple sampling mode, and this article propose an algorithm about how to recover x from y in this condition.
3d Intenational Confeence on Multimedia echnology(icm 03) A Simple Compessive Sampling Mode and the Recovey of Natue Images Based on Pixel Value Substitution Wenping Shao, Lin Ni Abstact: Compessive Sampling
More information