J. Seberry, X. Zhang, and Y. Zheng. Nonlinearly Balanced Boolean Functions and Their Propagation Characteristics. In D. Stinson, editor, Advances in Cryptology - CRYPTO '93, 13th Annual International Cryptology Conference, Santa Barbara, California, USA, August 22-26, 1993. Proceedings, volume 773 of Lecture Notes in Computer Science, pages 49--60, Berlin Heidelberg, 1994. Springer-Verlag.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... The Hamming distance d(f, l) between two boolean functions f and l with n inputs is the number of di#erent outputs over all 2 inputs: d(f, l) x#GF(2) n (f(x) l(x) where the sum is taken over GF(2) For balanced boolean functions f with n inputs the following bound for N f holds ([SZZ93]) N f N max,bal (n) 2, if n even, if n odd, where y = ##x## denotes the biggest even integer y with y x. We consider the normalised nonlinearity p e,f = 2 n N f of a boolean function f : GF(2) GF(2) We call a boolean function f highly nonlinear if p e,f 0.45. In ....

Jennifer Seberry, Xian-Mo Zhang, and Yuliang Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Douglas R. Stinson, editor, Advances in Cryptology, CRYPTO'93, LNCS 773, pages 49--60. Springer-Verlag, 1993.

.... balance, probability of coinciding with an affine function, balance of a directional derivative (the propagation criteria) are investigated under the assumption that function s arguments are independent binary random variables with the uniform probability distribution ( 2] 3] 4] 7] 8] [9]) This paper brings attention to the case when the arguments distributions differ from the uniform distribution. The necessity of such an investigation can be explained by adducing the task of combining pseudorandom binary sequences. Let x 1t ; x nt be n binary pseudorandom sequences ....

.... that it is determined by the minimum sensitivity of f(x) 2 2 Preliminaries We will define the algebraic normal form, the Walsh Hadamard transform, the classes of balanced and correlation immune Boolean functions, bent functions, and adduce their well known properties (see e.g. 2] 4] 8] [9]) The algebraic normal form of a Boolean function f(x) is its representation as a polynomial modulo 2. The (nonlinearity) order of f(x) is defined as the degree of this polynomial. The Walsh Hadamard transform of a real valued function f(x) is a function F (w) X x2B n f(x) Gamma1) x;w) ....

J.Seberry, X.Zhang, Y.Zheng, Nonlinearly Balanced Boolean Functions and Their Propagation Characteristics, Advances in Cryptology - Eurocrypt'93, Proceedings, pp.49-60, Springer-Verlag, 1994

....achievable for any Boolean function is 2 n 1 2 n 2 1 and the functions having this nonlinearity are called bent functions. However, bent functions are not balanced. Construction of balanced Boolean functions on even number of variables with very high nonlinearity has been considered in [24, 4, 21]. Dobbertin [4] has conjectured that, for even n, nlb(n) 2 n 1 2 n 2 nlb( n 2 ) where nlb(n) is the maximum possible nonlinearity for an n variable balanced function. 1 The nonlinearity question is open for functions on odd number of variables. It is known that [1, 14, 7] for odd ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced Boolean Functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, pages 49{ 60. Springer-Verlag, 1994.

....completely solved for balanced and resilient functions on n variables for n 5. Now we consider the cases n = 6 to n = 10 separately. Case n = 6: A bent function on 6 variables has nonlinearity 28. It is possible to construct balanced functions on 6 variables having maximum nonlinearity 26 (see [21]) In [16] a computer search was carried out on 6 variable resilient functions and the maximum nonlinearities for 1, 2 and 3 resilient functions were shown to be 24, 24, 16 respectively. These results follow very easily from Corollary 3.1 and Theorem 3.2. Also it is possible to construct (6; 1; ....

....However, the construction of (7; 2; 56) function seems to be a dicult one. 11 Case n = 8: A bent function on 8 variables has nonlinearity 120. The maximum possible nonlinearity of balanced functions is 118. It is possible to construct balanced functions on 8 variables having nonlinearity 116 [21]. The problem of constructing an 8 variable balanced function with nonlinearity 118 has been open for quite some time. Here we present a result which could be an important step in solving this problem. Theorem 5.1 Let if possible f be a (8; 0; 118) function. Then one can write f = 1 X 8 )f 1 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced Boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, pages 49-60. Springer-Verlag, 1994.

....by 1. Theorem 17. Let F be a differentially 2 uniform quadratic permutation on Vn . Then M , the difference trait matrix of F , is a Sylvester Hadamard matrix if the row order is ignored. Proof. From Theorem 16, the 2 n rows of M comprise all the linear sequences of length 2 n . By Lemma 1 of [16], each linear sequence of length 2 n is a row of Hn . Thus M can be changed to Hn by re ordering its rows. Obviously, W ff , ff and M can be defined for any permutation on Vn , not restricted to quadratic ones. Theorem 18. Let F be a differentially 2 uniform quadratic permutation on Vn and M ....

Seberry, J., Zhang, X. M., Zheng, Y.: Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93 (1994) vol. 773, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 49--60

....i input bits are complemented for 1 i n. It so happens that the notion of perfect nonlinear functions coincides with the idea of bent functions [112] These have already been well researched in other areas of mathematics and have been connected with functions used in the design of S boxes [118]. A second issue of interest in the field of S box design is the so called Strict Avalanche Criterion (SAC) 34, 73] A Boolean function f(x) satisfies SAC if the output changes with probability 1=2 whenever exactly one of the input bits changes. This property is useful both in the design of ....

J. Seberry, X.M. Zhang, and Y. Zheng. Nonlinearly balanced Boolean functions and their propogation characteristics. In D.R. Stinson, editor, Advances in Cryptology --- Crypto '93, pages 49--60, SpringerVerlag, New York, 1994.

.... a function f : Sigma n Sigma is defined as the Hamming distance between the function and the set of all affine functions [44] The concept of nonlinearity can be extended to measure nonlinearity of arbitrary functions f : Sigma n Sigma m including permutations (n = m) see [47] 38] [58]) Strict Avalanche Criterion or SAC was introduced by Webster and Tavares [65] A function f : Sigma n Sigma m satisfies the SAC if f(x Phi ff) is balanced for all x 2 Sigma n and for all ff whose weight is 1 (wt(ff) 1) In other words, it characterizes the number of output bits ....

.... Note that the nonlinearity of balanced functions is always smaller than the nonlinearity of bent functions which attain the maximum nonlinearity and satisfy SAC [53] The tradeoff between nonlinearity and the propagation criterion (including the SAC) for balanced functions is discussed in [57] and [58]. Charnes and Pieprzyk [10] studied the relation between the nonlinearity and the linear nonequivalence. They showed that it is not possible to select five balanced, SAC satisfying, linear nonequivalent functions in five boolean variables without reduction of nonlinearity. Nyberg [37] discussed ....

J. Seberry, X.M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, Lecture Notes in Computer Science (D.R Stinson (Ed)), volume 773, pages 49--60, New York, 1994. Springer Verlag.

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, Lecture Notes in Computer Science, vol. 773, pp. 49-60, Springer-Verlag, Berlin, Heidelberg, New York, 1994.

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, volume 773, Lecture Notes in Computer Science, pages 49--60. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

....respectively. When it is not the case, we can always find a nonsingular n Theta n matrix A whose entries are from GF (2) such that the subspaces W 0 and U 0 associated with f 0 (x) f(xA) have the required forms. f 0 and f have the same algebraic degree and nonlinearity (see Lemma 10 of [18]) This shows that the following theorem is true. Theorem 1 For any function on V n , the nonlinearity of f satisfies N f = 2 n Gamma1 Gamma 2 n Gamma 1 2 ae Gamma1 , where ae is the maximum dimension of the linear subspaces in f0g [ c . Theorem 1 indicates that the nonlinearity ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, volume 773 of Lecture Notes in Computer Science, pages 49--60. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

.... 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 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93. Springer-Verlag, Berlin, Heidelberg, New York, 1993. to appear.

....by 1. Theorem 5 Let F be a differentially 2 uniform quadratic permutation on V n . Then M , the difference trait matrix of F , is a Sylvester Hadamard matrix if the row order is ignored. Proof. From Theorem 4, the 2 n rows of M comprise all the linear sequences of length 2 n . By Lemma 1 of [16], each linear sequence of length 2 n is a row of H n . Thus M can be changed to H n by re ordering its rows. ut Obviously, W ff , ff and M can be defined for any permutation on V n , not restricted to quadratic ones. Theorem 6 Let F be a differentially 2 uniform (not necesarrily quadratic) ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, volume 773, Lecture Notes in Computer Science, pages 49--60. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

.... 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 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93. Springer-Verlag, Berlin, Heidelberg, New York, 1993. to appear.

....of cryptographic significance, although the concept itself seems interesting from a combinatorial point of view. In contrast, the other generalization of the SAC, namely the propagation criterion, has well established its position in cryptographic design. This can be seen from work represented by [1, 16, 15, 5, 20, 21]. A function satisfying the propagation criterion of degree k shows the perfect avalanche characteristic with respect to vectors of Hamming weight not larger than k. This property, however, does not rule out the possibility that the function can have vectors of Hamming weight larger than k as its ....

....k. This property, however, does not rule out the possibility that the function can have vectors of Hamming weight larger than k as its linear structures. For instance, all currently known methods for constructing functions satisfying higher degree propagation criteria, including those presented in [15, 5, 20, 21], yield functions having undesirable linear structures. Therefore the propagation criterion, though being an extension of the SAC, is merely another indicator for local properties. On the other hand, the criterion is too strict in the sense that it requires that f(x) Phi f(x Phi ff) be 100 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, volume 773, Lecture Notes in Computer Science, pages 49--60. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

....on V n . Then N f = min i=0; 2 n 1 Gamma1 d(f; i ) is called the nonlinearity of f . It is well known that the nonlinearity of f on V n satisfies N f = 2 n Gamma1 Gamma 2 1 2 n Gamma1 . An extensive investigation of highly nonlinear balanced functions has been carried out in [23]. Let ff = a 1 ; a n ) 2 V n and fi = b 1 ; b n ) 2 V n . Then the scalar product of ff and fi, denoted by hff; fii, is defined by hff; fii = L n j=1 a j b j , where the addition and the multiplication are over GF (2) A function f on V n is said to be bent if 2 Gamma n 2 ....

....first proved in [22] the problem can be circumvented by a suitable nondegenerate linear transformation on the coordinates of the mapping. Note that the balancedness, the nonlinearity and the algebraic degree of a function are not affected by a nondegenerate linear transformation on coordinates [23]. Lemma 4 Let f 1 , f 2 , fm be functions on V n . Suppose that A is an n Theta n nondegenerate matrix on GF (2) with the property that for each row ff i of A, 1 = i = n, and for each function f j , 1 = j = m, f j (x) Phi f j (x Phi ff i ) is balanced. Then f 1 (xA) ....

Seberry, J., Zhang, X. M., and Zheng, Y. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93 (1993), Springer-Verlag, Berlin, Heidelberg, New York. to appear.

....although the concept itself seems interesting from a combinatorial point of view. In contrast, the other generalization of the SAC, namely the propagation criterion, has well established its position in cryptographic design. This can be seen from work represented by [AT90, PLL 91, PGV91, DT93, SZZ94b, SZZ95] A function satisfying the propagation criterion of degree k shows the perfect avalanche characteristic with respect to vectors of Hamming weight not larger than k. This property, however, does not rule out the possibility that the function can have vectors of Hamming weight larger than k ....

....however, does not rule out the possibility that the function can have vectors of Hamming weight larger than k as its linear structures. For instance, all currently known methods for constructing functions satisfying higher degree propagation criteria, including those presented in [PGV91, DT93, SZZ94b, SZZ95] yield functions having undesirable linear structures. Therefore the propagation criterion, though being an extension of the SAC, is merely another indicator for local properties. On the other hand, the criterion is too strict in the sense that it requires that f(x) Phi f(x Phi ff) be ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, volume 773, Lecture Notes in Computer Science, pages 49--60. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

....When this is not the case, we can always find a nonsingular n Theta n matrix A whose entries are from GF (2) such that the sub spaces W 0 and U 0 associated with f 0 (x) f(xA) have the required forms. f 0 and f have the same algebraic degree and nonlinearity (see Lemma 10 of [SZZ94b] This shows that the following theorem is true. Theorem 1 For any function on V n , the nonlinearity of f satisfies N f = 2 n Gamma1 Gamma 2 n Gamma 1 2 ae Gamma1 , where ae is the maximum dimension of the linear sub spaces in f0g [ c . Theorem 1 indicates that the nonlinearity ....

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, Lecture Notes in Computer Science, vol. 773, pp. 49-60, Springer-Verlag, Berlin, Heidelberg, New York, 1994.

....ffi i ) runs through the values zero and one an equal number of times while x runs through V n . That is, x) satisfies the SAC. ut Note that the algebraic degree, the nonlinearity and the balancedness of a function is unchanged under a nondegenerate linear transformation of coordinates [MS90, SZZ93a] In addition the number of nonzero vectors with respect to which the function satisfies the propagation criterion is also invariant under the transformation [SZZ93a] In the case of S boxes (tuples of functions) the profile of its difference distribution table, which measures the strength ....

.... the nonlinearity and the balancedness of a function is unchanged under a nondegenerate linear transformation of coordinates [MS90, SZZ93a] In addition the number of nonzero vectors with respect to which the function satisfies the propagation criterion is also invariant under the transformation [SZZ93a] In the case of S boxes (tuples of functions) the profile of its difference distribution table, which measures the strength against the differential cryptanalysis [BS91, BS93] also remains invariant under such a transformation [SZZ93c] Thus Theorem 1 provides us with a very useful tool to ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93. Springer-Verlag, Berlin, Heidelberg, New York, 1993. to appear.

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J. Seberry, X. Zhang, and Y. Zheng. Nonlinearly Balanced Boolean Functions and Their Propagation Characteristics. In D. Stinson, editor, Advances in Cryptology - CRYPTO '93, 13th Annual International Cryptology Conference, Santa Barbara, California, USA, August 22-26, 1993. Proceedings, volume 773 of Lecture Notes in Computer Science, pages 49--60, Berlin Heidelberg, 1994. Springer-Verlag.

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93. Springer-Verlag, Berlin, Heidelberg, New York, 1993. to appear.

No context found.

Seberry, J., Zhang, X. M., and Zheng, Y. Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93 (1994), vol. 773, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, New York, pp. 49--60.

No context found.

Seberry, J., Zhang, X. M., Zheng, Y.: Nonlinearly balanced boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93 (1994) vol. 773, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 49--60

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced Boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, pages 49-60. Springer-Verlag, 1994.

No context found.

J. Seberry, X.M. Zhang, Y. Zheng, \Nonlinearly Balanced Boolean Functions and their Propagation Characteristics ", LNCS 773, Crypto'93, pp. 49-60, Springer-Verlag, 1993.

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearly balanced Boolean functions and their propagation characteristics. In Advances in Cryptology - CRYPTO'93, pages 49-60. Springer-Verlag, 1994.

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