Seberry J., Zhang X.-M., Zheng Y.: "Nonlinearity and Propagation Characteristics of Balanced Boolean Functions"; Technical Report no. 4, Computer Security Research Centre, University of Wollongong, Australia, (1993).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....degree and the largest number of output bits simultaneously. We next apply this method to construct balanced vectorial Boolean functions which have larger nonlinearities than previously known constructions. 1 Introduction Boolean functions play an important role in block ciphers (for example, see [3, 7, 8, 10, 11]) and stream ciphers [12, 2] The nonlinearity N f of a Boolean function f(x 1 ; xn ) is defined as a distance between f and the set of affine functions fa 0 Phi a 1 x 1 Phi Delta Delta Delta Phi anxn g. N f should be large to resist the linear attack [2, 6] f(x 1 ; xn ) ....

....achieves both equalities of eq. 1) and eq. 2) simultaneously. In this paper, we show the first method to construct (n; m) bent functions which satisfy the both equalities of eq. 1) and eq. 2) simultaneously. It is known that bent functions are not balanced. For m = 1, Seberry, Zhang and Zheng [10] and Dobbertin [3] showed balanced functions which have large nonlinearity. For m = n, Nyberg showed balanced vectorial Boolean functions with high nonlinearity [8] We next apply our method to construct balanced vectorial Boolean functions with high nonlinearity. For 2 m n=2, our balanced ....

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J. Seberry, X.M. Zhang and Y. Zheng. Nonlinearity and propagation characteristics of balanced Boolean functions. In Information and Computation, 119(1):1--13, May 1995.

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J. Seberry and X. M. Zhang and Y. Zheng, Nonlinearity and Propagation Characteristics of Balanced Boolean Functions, Information and Computation, 119 (1) (1995), 1-13.

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. To appear in Information and Computation, 1994. 14

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

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J. Seberry, X. M. Zhang, and Y. Zheng, "Nonlinearity and propagation characteristics of balanced boolean functions," Information and Computation, vol. 119, no. 1, pp. 1--13, 1995.

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

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Seberry, J., Zhang, X. M., Zheng, Y.: Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation 119 (1995) 1--13.

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Seberry, J., Zhang, X. M., and Zheng, Y. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation 119, 1 (1995), 1--13.

....by H n , is generated by the recursive relation H n = H n Gamma1 H n Gamma1 H n Gamma1 GammaH n Gamma1 # ; n = 1; 2; H 0 = 1: Each row (column) of H n is a linear sequence of length 2 . The following two formulas are well known to researchers (for a proof see for instance [14, 23]) Let be the sequence of a function f on V n . Then the nonlinearity of f , N f can be calculated by N f = 2 2 maxfjh; i ij; 0 i Gamma 1g (1) Gamma 1, and ( Delta(ff 0 ) Delta(ff 1 ) Delta(ff 2 n Gamma1 ) H n = h; 0 i ; h; 1 i ; h; 2 n ....

....bent functions. Below is the formal definition of bent functions. Definition 4 Let f be a function on V n and denote the sequence of f . Then f is called a bent function if jh; i ij = 2 Gamma 1, where i denotes the ith row of H n . Bent functions can be characterized in various ways [2, 8, 20, 23, 26]. A characterization of particular interest can be found in [8, 20] which states that bent functions on V n exist only when n is even, and that they achieve the highest possible nonlinearity on V n , namely, 2 . 3 Relationships among Three Tables Now we introduce three more notations, k j ....

Seberry, J., Zhang, X. M., and Zheng, Y. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation 119, 1 (1995), 1--13.

....we have N f = 2 (n 1) Hence (13) holds, where p = n 1. On the other hand, it is obvious that (13) holds whenever (b) does. ut 5 Relationships between Avalanche and Correlation Immunity To prove the main theorems, we introduce two more results. The following lemma is part of Lemma 12 in [15]. Lemma 6. Let f 1 be a function on V s and f 2 be a function on V t . Then f 1 (x 1 ; x s ) f 2 (y 1 ; y t ) is a balanced function on V s t if f 1 or f 2 is balanced. Next we look at the structure of a function on Vn that satis es the avalanche criterion of degree n 1. ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1-13, 1995.

....(p t ) n1 and GF (p t ) n2 respectively. Set f(x) f 1 (y) f 2 (z) where x = y; z) where y 2 GF (p t ) n1 and z 2 GF (p t ) n2 . Then f is balanced if f 1 or f 2 is balanced, The above lemma can be veri ed directly. The special case of p = 2 and t = 1 was given in Lemma 12 of [8]. Using Lemma 2, we can prove Lemma 3. Let f 1 and f 2 be two functions of degree two on GF (p t ) n1 and GF (p t ) n2 respectively. Set f(x) f 1 (y) f 2 (z) where x = y; z) where y 2 GF (p t ) n1 and z 2 GF (p t ) n2 . Then f has the property B(k) if both f 1 and f 2 ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1-13, 1995.

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1-13, 1995.

....x 2 Vm and y 2 V k . Then we say that f is a Maiorana McFarland function. Maiorana McFarland functions play an important role in the design of cryptographic functions that satisfy cryptographically desirable properties such as high nonlinearity, propagation characteristics and correlation immunity[1, 2, 7, 8]. It is known that when k = m and Q is a permutation on V k , f is a bent function on V 2k [3, 4] This provides us with a powerful method for constructing as many as (2 k )2 2 k di erent bent functions on V 2k . If we use nonsingular linear transformations on the variables, we will obtain ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1-13, 1995.

....of order m. A Sylvester Hadamard matrix of order 2 n , denoted by H n , is generated by the following recursive relation H 0 = 1; H n = H n Gamma1 H n Gamma1 H n Gamma1 GammaH n Gamma1 # ; n = 1; 2; 1) Let i , 0 = i = 2 n Gamma 1, be the i row of H n . By Lemma 2 of [20], i is the sequence of a linear function i (x) defined by the scalar product i (x) hff i ; xi, where ff i is the ith vector in V n according to the ascending alphabetical order. Definition 4 Let f be a function on V n . The Walsh Hadamard transform of f is defined as f (ff) 2 ....

....the ascending order, is the sequence of f and H n is the Sylvester Hadamard matrix of order 2 n . 3 Definition 5 A function f on V n is called a bent function if its Walsh Hadamard transform satisfies f(ff) Sigma1 for all ff 2 V n . Bent functions can be characterized in various ways [1, 5, 20, 26]. In particular the following four statements are equivalent: i) f is bent. ii) h; i = Sigma2 1 2 n for any affine sequence of length 2 n , where is the sequence of f . iii) f satisfies the propagation criterion with respect to all non zero vectors in V n . iv) M , the matrix of f , ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

....by H n , is generated by the recursive relation H n = H n Gamma1 H n Gamma1 H n Gamma1 GammaH n Gamma1 # ; n = 1; 2; H 0 = 1: Each row (column) of H n is a linear sequence of length 2 n . The following two formulas are well known to researchers (for a proof see for instance [14, 23]) Let be the sequence of a function f on V n . Then the nonlinearity of f , N f can be calculated by N f = 2 n Gamma1 Gamma 1 2 maxfjh; i ij; 0 = i = 2 n Gamma 1g (1) where i is the ith row of H n , i = 0; 1; 2 n Gamma 1, and ( Delta(ff 0 ) Delta(ff 1 ) ....

....definition of bent functions. 4 Definition 4 Let f be a function on V n and denote the sequence of f . Then f is called a bent function if jh; i ij = 2 n 2 ; i = 0; 1; 2 n Gamma 1, where i denotes the ith row of H n . Bent functions can be characterized in various ways [2, 8, 20, 23, 26]. A characterization of particular interest can be found in [8, 20] which states that bent functions on V n exist only when n is even, and that they achieve the highest possible nonlinearity on V n , namely, 2 n Gamma1 Gamma 2 n 2 Gamma1 . 3 Relationships among Three Tables Now we ....

Seberry, J., Zhang, X. M., and Zheng, Y. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation 119, 1 (1995), 1--13.

.... only by bent functions that have the zero vector as their only linear structure, while 1 can be achieved by functions that have only two linear structures (one is the zero vector and the other is a nonzero vector) Examples of the latter are those obtained by concatenating two bent functions (see [19, 23]) In mathematical terms, an n Theta s S box (i.e. with n input bits and s output bits) can be described as a mapping from V n to V s (n = s) To avoid trivial statistical attacks, an S box F should be regular, namely, F (x) should run through all vectors in V s each 2 n Gammas times ....

....of the component functions. The nonlinearity of a function f on V n has been known to be bounded from the above by 2 n Gamma1 Gamma 2 1 2 n Gamma1 . When n is even, the upper bound is achieved by bent functions. Constructions for highly nonlinear balanced functions can be found in [19, 23]. Nonlinearity has been considered to be an important criterion. Recent advances in Linear cryptanalysis put forward by Matsui [10, 11] have further made it explicit that nonlinearity is not just important, but essential to DES like block encryption algorithms. Linear cryptanalysis exploits the ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. To appear in Information and Computation, 1994.

....m. A Sylvester Hadamard matrix of order 2 n , denoted by H n , is generated by the following recursive relation H 0 = 1; H n = H n Gamma1 H n Gamma1 H n Gamma1 GammaH n Gamma1 # ; n = 1; 2; 1) Let i , 0 = i = 2 n Gamma 1, be the i row (column) of H n . By Lemma 1 of [10], i is the sequence of a linear function i (x) defined by the scalar product i (x) hff i ; xi, where ff i is the ith vector in V n according to the ascending lexicographic order. The Walsh Hadamard transform, also called the discrete Fourier transform, has numerous applications in areas ....

....transform satisfies f(ff) Sigma1 for all ff 2 V n . Bent functions on V n exist only when n is even [9] They achieve the highest possible nonlinearity 2 n Gamma1 Gamma 2 1 2 n Gamma1 . The following lemma will be used in this paper (For a proof see for instance Lemma 6 of [10]. Lemma 1 The nonlinearity of a function f on V n can be calculated by N f = 2 n Gamma1 Gamma 1 2 maxfjh ; i ij; 0 = i = 2 n Gamma 1g where is the sequence of f and 0 , 2 n Gamma1 are the rows of H n , namely, the sequences of the linear functions on V n . 3 ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

....2 Gamma n 2 X x2Vn ( Gamma1) f(x) Phihfi;xi = Sigma1; for all fi 2 V n . Here hfi; xi is the scalar product of fi and x, namely, hfi; xi = P n i=1 b i x i , and f(x) Phi hfi; xi is regarded as a real valued function. Bent functions can be characterized in various ways [AT90, Dil72, SZZ95a, YH89] In particular the following four statements are equivalent: i) f is bent. ii) h ; i = Sigma2 1 2 n for any affine sequence of length 2 n , where is the sequence of f . iii) f satisfies the propagation criterion with respect to all non zero vectors in V n . iv) M , the ....

....of f . Due to a very pretty result by R. L. McFarland (see Theorem 3.3 of [Dil72] M can be decomposed into M = 2 Gamman H n diag(h ; 0 i; Delta Delta Delta ; h ; 2 n Gamma1 i)H n where i is the ith row of H n , a Sylvester Hadamard matrix of order 2 n . By Lemma 2 of [SZZ95a] i is the sequence of a linear function defined by i (x) hff i ; xi, where ff i is the ith vector in V n according to the ascending alphabetical order. Clearly MM T = 2 Gamman H n diag(h ; 0 i 2 ; Delta Delta Delta ; h ; 2 n Gamma1 i 2 )H n : 1) On the other hand, we ....

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

....matrix of order m. A Sylvester Hadamard matrix of order 2 n , denoted by Hn , is generated by the following recursive relation H 0 = 1; Hn = Hn Gamma1 Hn Gamma1 Hn Gamma1 GammaH n Gamma1 ; n = 1; 2; 1) Let i , 0 i 2 n Gamma 1, be the i row (column) of Hn . By Lemma 1 of [11], i is the sequence of a linear function i (x) defined by the scalar product i (x) hff i ; xi, where ff i is the i th vector in Vn according to the ascending lexicographic order. Definition3. Let f be a function on Vn . The Walsh Hadamard transform of f is defined as f(ff) 2 ....

....n Gamma1 (see for instance [4] Hence N f 2 n Gamma1 Gamma 2 1 2 n Gamma1 for any function on Vn . Definition5. A function f on Vn is called a bent function if its Walsh Hadamard transform satisfies f (ff) Sigma1 for all ff 2 Vn . Bent functions can be characterized in various ways [1, 5, 10, 11, 14]. A characterization of particular interest can be found in [5, 10] Lemma 6. The following statements are equivalent: i) f is bent, ii) f satisfies the propagation criterion with respect to all non zero vectors in Vn , iii) M = Gamma1) f (ff i Phiff j ) the matrix of f , is a ....

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

....n 1 d(f; i ) where 1 , 2 , 2 n 1 are all the affine functions on Vn . The nonlinearity of functions on Vn coincides with the covering radius of the first order binary Reed Muller code R(1; n) of length 2 n [2] and it is upper bounded by 2 n Gamma1 Gamma 2 1 2 n Gamma1 [4]. If N f = 2 n Gamma1 Gamma 2 1 2 n Gamma1 then f is called a bent function [3] Bent functions on Vn exist only for even n. Let f be a function on Vn and U be an s dimensional subspace of Vn . The restriction of f to a coset Pi j = fi j Phi U , j = 0; 1; 2 n Gammas Gamma 1, ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Info. and Comp., 119(1):1--13, 1995.

.... only by bent functions that have the zero vector as their only linear structure, while 1 can be achieved by functions that have only two linear structures (one is the zero vector and the other is a nonzero vector) Examples of the latter are those obtained by concatenating two bent functions (see [19, 23]) In mathematical terms, an n Theta s S box (i.e. with n input bits and s output bits) can be described as a mapping from Vn to V s (n s) To avoid trivial statistical attacks, an S box F should be regular, namely, F (x) should run through all vectors in V s each 2 n Gammas times while x ....

....of the component functions. The nonlinearity of a function f on Vn has been known to be bounded from the above by 2 n Gamma1 Gamma 2 1 2 n Gamma1 . When n is even, the upper bound is achieved by bent functions. Constructions for highly nonlinear balanced functions can be found in [19, 23]. Nonlinearity has been considered to be an important criterion. Recent advances in Linear cryptanalysis put forward by Matsui [10, 11] have further made it explicit that nonlinearity is not just important, but essential to DES like block encryption algorithms. Linear cryptanalysis exploits the ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. To appear in Information and Computation, 1994.

....the identity matrix of order m. A Sylvester Hadamard matrix of order 2 n , denoted by Hn , is generated by the following recursive relation H 0 = 1; Hn = Hn Gamma1 Hn Gamma1 Hn Gamma1 GammaH n Gamma1 ; n = 1; 2; Let i , 0 i 2 n Gamma1, be the i row of Hn . By Lemma 2 of [4], i is the sequence of a linear function i (x) defined by the scalar product i (x) hff i ; xi, where ff i is the ith vector in Vn according to the ascending alphabetical order. Definition2. Let f be a function on Vn . For a vector ff 2 Vn , denote by (ff) the sequence of f (x Phi ff) ....

....is n Gammas. It is easy to verify that s dimensional subspace U is a maximal odd weighting subspace of f . ut Finally we note that for s = 2, the value of 2 n Gammas in Theorem 11 is very close to 2 n Gamma1 Gamma 2 1 2 n Gamma1 , the upper bound on the nonlinearity of functions on Vn [4]. However Theorem 11 cannot be further improved by extending s to s = 1, as the condition of s 2 in the proof of the theorem cannot be removed. For example, let f be a function on Vn , whose truth table is given as follows 0110011010011001: It is easy to verify that (0000) 0001) form a maximal ....

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1--13, 1995.

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Seberry J., Zhang X.-M., Zheng Y.: "Nonlinearity and Propagation Characteristics of Balanced Boolean Functions"; Technical Report no. 4, Computer Security Research Centre, University of Wollongong, Australia, (1993).

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J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. To appear in Information and Computation, 1994.

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Seberry, J., Zhang, X. M., and Zheng, Y. Nonlinearity and propagation characteristics of balanced boolean functions. Submitted for Publication, 1993. This article was processed using the L A T E X macro package with LLNCS style

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