J. Seberry and X.M. Zhang. Highly nonlinear 0-1 balanced Boolean functions satisfying strict avalanche criterion. In Advances in Cryptology --- AUSCRYPT '92 Proceedings, Lecture Notes in Computer Science 718. Springer-Verlag, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Phi anxn g fi : N(f) must be large to avoid the linear attack [7] Preneel et al. showed a balanced SAC(n Gamma 2) function for n =odd [11] Lloyd [5] showed a condition such that SAC(n Gamma 3) functions are balanced. Balanced SAC functions with high nonlinearity were constructed by [14]. Recently, other balanced SAC functions were given by [16] However, 1) No general methods are known which design Boolean functions satisfying PC(l) of order k except deg(f) 2. For deg(f) 2, see [11, 12] 2) Balanced SAC(k) functions are not known for 1 k n Gamma 4. 3) Balanced ....

....: xn ) 2.1 Balance and Algebraic Degree We say that f(x) is balanced if ; where x = x 1 ; xn ] Definition 1. We call f(x) c Phi a 1 x 1 Phi Delta Delta Delta Phi anxn an affine function. Proposition 2. A non constant affine function is balanced. Proposition 3. [14] f(x 1 ; x s ) Phi g(y 1 ; y t ) is balanced if f is balanced or g is balanced. The following form is called the algebraic normal form of f . f(x 1 ; xn ) a 0 Phi i=1 a i x i Phi a ij x i x j Phi Delta Delta Delta Phi a 12: n x 1 x 2 : xn : deg(f) ....

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J. Seberry and X.M. Zhang. Highly nonlinear 0-1 balanced Boolean functions satisfying strict avalanche criterion. In Advances in Cryptology --- AUSCRYPT '92 Proceedings, Lecture Notes in Computer Science 718. Springer-Verlag, 1993.

.... Phi f(x Phi ff) is balanced for any ff such that W (ff) 1, where W (ff) denotes the Hamming weight of ff [21] f satisfies SAC(k) if every function obtained from f(x 1 ; xn ) by keeping any k input bits constant satisfies SAC [6] Several researchers studied the properties of SAC(k) [5, 6, 8 10, 15 17, 19]. Especially, Preneel showed that f satisfies SAC(k) if f is quadratic and every variable x i occurs in at least k 1 second order terms of the algebraic normal form [16] Preneel, Govaerts and Vandewalle [15] showed the number of quadratic functions which satisfy SAC(k) for 3 n 7. This ....

J. Seberry, X. M. Zhang, and Y. Zheng. Highly nonlinear balanced Boolean functions satisfying high degree propagation criterion. Technical Report No. 93-1, Department of Computer Science, The University of Wollongong, Australia, 1993.

....x 1 8 1 1 1 8 a n x n g fi fi : N(f) must be large to avoid the linear attack [7] Preneel et al. showed a balanced SAC(n 0 2) function for n =odd [11] Lloyd [5] showed a condition such that SAC(n 0 3) functions are balanced. Balanced SAC functions with high nonlinearity were constructed by [14]. Recently, other balanced SAC functions were given by [16] However, 1) No general methods are known which design Boolean functions satisfying PC(l) of order k except deg(f) 2. For deg(f) 2, see [11, 12] 2) Balanced SAC(k) functions are not known for 1 k n 0 4. 3) Balanced functions ....

....Degree We say that f(x) is balanced if fi fi fx j f(x) 0g fi fi = fi fi fx j f(x) 1g fi fi = 2 n01 ; where x = x 1 ; xn ] Definition 1. We call f(x) c 8 a 1 x 1 8 1 1 1 8 an xn an affine function. Proposition 2. A non constant affine function is balanced. Proposition 3. [14] f(x 1 ; x s ) 8 g(y 1 ; y t ) is balanced if f is balanced or g is balanced. The following form is called the algebraic normal form of f . f(x 1 ; x n ) a 0 8 n M i=1 a i x i 8 M 1i jn a ij x i x j 8 1 1 1 8 a 12. n x 1 x 2 . x n : deg(f) denotes the degree ....

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J. Seberry and X.M. Zhang. Highly nonlinear 0-1 balanced Boolean functions satisfying strict avalanche criterion. In Advances in Cryptology --- AUSCRYPT '92 Proceedings, Lecture Notes in Computer Science 718. Springer-Verlag, 1993.

.... Phi f(x Phi ff) is balanced for any ff such that W (ff) 1, where W (ff) denotes the Hamming weight of ff [21] f satisfies SAC(k) if every function obtained from f(x 1 ; xn ) by keeping any k input bits constant satisfies SAC [6] Several researchers studied the properties of SAC(k) [5, 6, 8 10, 15 17, 19]. Especially, Preneel showed that f satisfies SAC(k) if f is quadratic and every variable x i occurs in at least k 1 second order terms of the algebraic normal form [16] Preneel, Govaerts and Vandewalle [15] showed the number of quadratic functions which satisfy SAC(k) for 3 = n = 7. ....

J. Seberry, X. M. Zhang, and Y. Zheng. Highly nonlinear balanced Boolean functions satisfying high degree propagation criterion. Technical Report No. 93-1, Department of Computer Science, The University of Wollongong, Australia, 1993.

....all but two or three elements in f0; 1g n 0f(0; 0)g. Then, we show a method that constructs, for any odd n 3, n input Boolean functions which satisfy the PC with respect to sets of all but one elements in f0; 1g n 0 f(0; 0)g. This method is a generalized version of the one in [8] and relates the numbers of the Boolean functions and of the balanced Boolean functions in B n satisfying the PC of degree n 0 1 with that of the perfectly nonlinear Boolean functions in B n01 for odd n. Secondly, concerned with the SAC of higher orders, we prove that the upper bound of the ....

....n 0 f(0; 0)g such that b 1 6= b 2 , and A = f0; 1g n 0 f(0; 0)g 0 fb 1 ; b 2 g. If f 2 B n satisfies the PC with respect to A, then, ffl for every even n 2, f 2 PC n (n) ffl for every odd n 3, f satisfies the PC with respect to A [ fb 1 g or A [ fb 2 g. 2 Seberry, et al.[8] presented a method that, for any even n 2, generates balanced Boolean functions in B n satisfying the propagation characteristics with respect to sets of all but three elements in f0; 1g n 0 f(0; 0)g. The above theorem shows that their construction is optimal in the sense that the ....

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J. Seberry, X.-M. Zhang, Y. Zhang: "Highly nonlinear balanced Boolean functions satisfying high degree propagation criterion," Tech. Rep. The Univ. Wollongong, tr--93--1 (1993).

....f(x) is balanced if jfx j f(x) 0gj = jfx j f(x) 1gj = 2 n Gamma1 , where x = x 1 ; x n ] Definition 2.4 We call f(x) a 0 Phi a 1 x 1 Phi Delta Delta Delta Phi a n x n an affine function. Proposition 2.1 A non constant affine function f is balanced. Proposition 2. 2 [7] A Boolean function f(x 1 ; x n ) Phi g(y 1 ; y k ) is balanced if f is balanced or g is balanced. 2.3 PC(l) SAC and Bent function Definition 2.5 [5] f(x 1 ; x n ) satisfies PC(l) if f(x) Phi f(x Phi ff) is balanced for any ff 2 f0; 1g n such that 1 W (ff) l. We ....

J. Seberry and X.M. Zhang. Highly nonlinear 0-1 balanced Boolean functions satisfying strict avalanche criterion. In Advances in Cryptology --- AUSCRYPT '92 Proceedings, Lecture Notes in Computer Science 718. Springer-Verlag, 1993.

....identified. Finally, compatibility of the PC and the self duality is discussed. For every odd n 3, the existence of self dual functions with n variables satisfying the PC of degree n 0 1 is shown and an exact characterization of these functions is achieved. These results are shown implicitly in [SZZ93] and [HI95b] It is also shown that, for every n 2, The degree of the PC of every self dual function with n variables is at most n=2 0 1. The optimality of the result is also shown. Section 2 gives basic concepts and discusses Boolean functions satisfying the PC. Relationships between the PC and ....

J. Seberry, X. M. Zhang, Y. Zheng: "Highly nonlinear balanced Boolean functions satisfying high degree propagation criterion," Tech. Rep. The Univ. Wollongong, tr--93--1 (1993).

.... ff) is balanced for any ff such that W (ff) 1, where W (ff) denotes the Hamming weight of ff [21] f satisfies SAC(k) if every function obtained from f(x 1 ; Delta Delta Delta ; xn ) by keeping any k input bits constant satisfies SAC [6] Several researchers studied the properties of SAC(k) [5, 6, 8 10, 15 17, 19]. Especially, Preneel showed that f satisfies SAC(k) if f is quadratic and every variable x i occurs in at least k 1 second order terms of the algebraic normal form [16] Preneel, Govaerts and Vandewalle [15] showed the number of quadratic functions which satisfy SAC(k) for 3 n 7. This paper ....

J. Seberry, X. M. Zhang, and Y. Zheng. Highly nonlinear balanced Boolean functions satisfying high degree propagation criterion. Technical Report No. 93-1, Department of Computer Science, The University of Wollongong, Australia, 1993.

....digests of variable lengths this is exactly what Haval does. Haval is a one way hashing algorithm that can produce digests of 128, 160, 192, 224 and 256 bits. The Haval algorithm uses some of the ideas behind the design of the MD5 algorithm. It also uses five boolean functions as presented in [SZ93]. Each of the functions have the following properties which are considered desirable for a one way hashing algorithm : 1. The functions are 0 1 balanced. 2. They are highly non linear. 3. They satisfy the Strict Avalanche Criterion. 4. They cannot be transformed into one another by applying linear ....

J. Seberry and X.-M. Zhang. Highly nonlinear 0-1 balanced boolean functions satisfying strict avalanche criterion, 1993.

....the PC with respect to all but one elements in f0; 1g n 0 f(0; 0)g if 3 is odd. Secondly, we discuss the construction of Boolean functions with n variables that satisfy the PC with respect to all but one or three elements in f0; 1g n 0 f(0; 0)g. Seberry, Zhang and Zheng[SZZ93] presented methods for the construction of balanced Boolean functions satisfying the PC of high degrees. For odd n 3, they proposed a method for constructing balanced Boolean functions with n variables satisfying the PC with respect to all but one elements in f0; 1g n 0f(0; 0)g and ....

....Boolean functions satisfying the PC with respect to all but one or three nonzero elements This section gives an exact characterization of Boolean functions with the odd number of inputs that satisfy the PC with respect to all but one nonzero vectors. The motivation of this research is a method in [SZZ93] to construct balanced Boolean functions with the odd number of inputs that satisfy the PC with respect to all but one nonzero vectors and that to construct balanced Boolean functions with the even number of inputs that satisfy the PC with respect to all but three nonzero vectors. 4.1 Boolean ....

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J. Seberry, X. M. Zhang, Y. Zheng: "Highly nonlinear balanced Boolean functions satisfying high degree propagation criterion," Tech. Rep. The Univ. Wollongong, tr--93--1 (1993).

....lengths for fingerprints provide practical applications with a broad spectrum of choices. The algorithm, which we call HAVAL, uses some of the principles behind the design of the MD family. In addition, HAVAL makes an elegant use of Boolean functions recently discovered by Seberry and Zhang [SZ92]. These functions have nice properties which include 1. they are 0 1 balanced, 2. they are highly non linear, 3. they satisfy the Strict Avalanche Criterion (SAC) 4. they can not be transformed into one another by applying linear transformation to the input coordinates and 5. they are not ....

....by a cryptographic algorithm bears no resemblance in structure (with respect to linear transformation of coordinates and complementation of functions. Finally, P5 ensures that the sequences of the functions are not mutually correlated either via linear functions or via the bias in output bits. In [SZ92], Seberry and Zhang presented a novel method for constructing Boolean functions that have the properties P1, P2 and P3. In particular, they showed that given a bent function from V 2k to GF (2) where k = 1, one can obtain a Boolean function from V 2k 1 to GF (2) that has the properties P1, P2 ....

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J. Seberry and X.-M. Zhang. Highly nonlinear 0-1 balanced boolean functions satisfying strict avalanche criterion, 1992. AusCrypt'92, Gold Coast.

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