J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. Presented at EUROCRYPT'94, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... extensions of some cryptographic criteria of scalar output Boolean functions to vector output Boolean functions have been studied recently [3, 4, 13, 18] For example, it is known that F is uniformly distributed if and only if all nonzero linear combinations of component functions f i are balanced [4, 11, 20]. f satisfies perfect nonlinear if f(x) Phi f(x Phi ff) is balanced for any ff j = 0. F = f 1 ; fm ) satisfies perfect nonlinear if all nonzero linear combinations of ff i g satisfy perfect nonlinear. Nyberg showed an upper bound on m such that m n=2 for F which satisfies perfect ....

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In EUROCRYPT '94 Proceedings, LNCS 950, pp. 376--388. Springer-Verlag, 1995.

.... extensions of some cryptographic criteria of scalar output Boolean functions to vector output Boolean functions have been studied recently [3, 4, 13, 18] For example, it is known that F is uniformly distributed if and only if all nonzero linear combinations of component functions f i are balanced [4, 11, 20]. f satisfies perfect nonlinear if f(x) Phi f(x Phi ff) is balanced for any ff j = 0. F = f 1 ; fm ) satisfies perfect nonlinear if all nonzero linear combinations of ff i g satisfy perfect nonlinear. Nyberg showed an upper bound on m such that m = n=2 for F which satisfies perfect ....

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In EUROCRYPT '94 Proceedings, LNCS 950, pp. 376--388. Springer-Verlag, 1995.

.... extensions of some cryptographic criteria of scalar output Boolean functions to vector output Boolean functions have been studied recently [3, 4, 13, 18] For example, it is known that F is uniformly distributed if and only if all nonzero linear combinations of component functions f i are balanced [4, 11, 20]. f satisfies perfect nonlinear if f(x) Phi f(x Phi ff) is balanced for any ff 6= 0. F = f 1 ; fm ) satisfies perfect nonlinear if all nonzero linear combinations of ff i g satisfy perfect nonlinear. Nyberg showed an upper bound on m such that m n=2 for F which satisfies perfect ....

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology --- EUROCRYPT '94 Proceedings, Lecture Notes in Computer Science 950, pages 376--388. Springer-Verlag, 1995.

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J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. Presented at EUROCRYPT'94, 1994.

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J. Seberry, X. M. Zhang, and Y. Zheng, "Relationships among nonlinearity criteria," in Advances in Cryptology - EUROCRYPT '94. 1995, vol. 950, Lecture Notes in Computer Science, pp. 376--388, Springer-Verlag, Berlin, Heidelberg, New York.

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J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Science, pages 376--388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Science, pages 376--388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

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Seberry, J., Zhang, X. M., and Zheng, Y. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94 (1995), vol. 950, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, New York, pp. 376--388.

....are high nonlinearity, high degree of propagation, few linear structures, high algebraic degree etc. These properties are often called nonlinearity criteria. An important topic is to investigate relationships among the various nonlinearity criteria. Progress in this direction has been made in [21], where connections have been revealed among the strict avalanche characteristic, differential characteristics, linear structures and nonlinearity, of quadratic functions. 1 In this paper we carry on the investigation initiated in [21] and bring together nonlinearity and propagation ....

....criteria. Progress in this direction has been made in [21] where connections have been revealed among the strict avalanche characteristic, differential characteristics, linear structures and nonlinearity, of quadratic functions. 1 In this paper we carry on the investigation initiated in [21] and bring together nonlinearity and propagation characteristic of a Boolean function (quadratic or non quadratic) These two cryptographic criteria are seemly quite separate, in the sense that the former indicates the minimum distance between a Boolean function and all the affine functions ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94, volume 950 of Lecture Notes in Computer Science, pages 376--388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

....are high nonlinearity, high degree of propagation, few linear structures, high algebraic degree etc. These properties are often called nonlinearity criteria. An important topic is to investigate relationships among the various nonlinearity criteria. Progress in this direction has been made in [SZZ95b] where connections have been revealed among the strict avalanche characteristic (SAC) differential characteristics, linear structures and nonlinearity, of quadratic functions. In this paper we carry on the investigation initiated in [SZZ95b] and bring together nonlinearity and propagation ....

....Progress in this direction has been made in [SZZ95b] where connections have been revealed among the strict avalanche characteristic (SAC) differential characteristics, linear structures and nonlinearity, of quadratic functions. In this paper we carry on the investigation initiated in [SZZ95b] and bring together nonlinearity and propagation characteristic of a Boolean function (quadratic or non quadratic) These two cryptographic criteria are seemly quite separate, in the sense that the former indicates the minimum distance between a Boolean function and all the affine functions ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94, volume 950, Lecture Notes in Computer Science, pages 376-- 388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

....each vector in V s 2 n Gammas times while x runs through V n once. S boxes employed by a block cipher must be regular, since otherwise the cipher would be prone to statistical attacks. For a regular n Theta s S box, its differential uniformity is larger than 2 n Gammas (see also Lemma 2 of [17]) The robustness of the S box is further determined by the number of nonzero entries in the first column of the table. We are particularly interested in n Theta s S boxes that have the following property: for any nonzero vector ff 2 V n , F (x) Phi F (x Phi ff) runs through half of the vectors ....

....other half contain a value zero. For simplicity, we say such a difference distribution table to be uniformly half occupied . Clearly an n Theta s S box with a UHODDT or uniformly half occupied difference distribution table achieves the differential uniformity of 2 n Gammas 1 . In Theorem 3 of [17], it has been proved that for quadratic S boxes, 2 n Gammas 1 is the lower bound on differential uniformity. Note that a differentially 2 uniform permutation is also a permutation with a UHODDT, and vice versa. These permutations have many nice properties [13, 2, 9, 10, 11, 12] In particular, ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94, volume 950, Lecture Notes in Computer Science, pages 376-- 388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

....nonlinearity, high degree of propagation, few linear structures, high algebraic degree etc. These properties are often called nonlinearity criteria. An important topic is to investigate relationships among the various nonlinearity criteria. Progress in this direction has been made in [2] 8] [14], where connections have been revealed among the strict avalanche characteristic (SAC) differential characteristics, linear structures and nonlinearity, of quadratic functions. In this paper we carry on the investigation initiated in [14] and bring together nonlinearity and propagation ....

....Progress in this direction has been made in [2] 8] 14] where connections have been revealed among the strict avalanche characteristic (SAC) differential characteristics, linear structures and nonlinearity, of quadratic functions. In this paper we carry on the investigation initiated in [14] and bring together nonlinearity and propagation characteristic of a function (quadratic or non quadratic) These two cryptographic criteria seem to be quite different, in the sense that the former indicates the minimum distance between a function and all the affine functions whereas the latter ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94, volume 950, Lecture Notes in Computer Science, pages 376--388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

....V n . In the rest of the section we report the result we have obtained on the lower bound of Delta f of cubic functions. This result can be regarded as the first step towards fully answering the question about Delta f . The following two results (see for instance Lemma 9 of [18] and Lemma 5 of [22] respectively) will be employed in the discussions of cubic functions. Lemma 4 f(x 1 ; x n ) x 1 ; x r ) Phi h(x r 1 ; x n ) is balanced on V n if is balanced on V r or h is balanced on V n Gammar . Lemma 5 If f is a quadratic function and does not have nonzero ....

....g is not bent. Note that g is quadratic. By Lemma 5, g has nonzero linear structures. it is easy to see [14] that all the linear structures of a function on V n form a linear subspace of V n . Denote by W the linear subspace formed by the linear structures of g, and by r the dimension of W . From [22], there exists a nonsingular matrix A of order n on GF(2) such that g (x) g(xA) can be expressed as g (x 1 ; x n ) x 1 ; x r ) Phi h(x r 1 ; x n ) where is a linear function on W while h is a function on V n Gammar that does not have nonzero linear ....

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94, volume 950, Lecture Notes in Computer Science, pages 376--388. Springer-Verlag, Berlin, Heidelberg, New York, 1995.

....for all fl = fi; ff) with W (fi) 6= 0, where fi 2 V s and ff 2 V t . A mapping (tuple of functions) f 1 ; f s ) where each f i is a function on V n and n = s, is said to be regular if for each vector y 2 V s there are exactly 2 n Gammas vectors in V n that are mapped to y. In [26], the following result is proved: Theorem 1 A mapping (f 1 ; f s ) where each f i is a function on V n and n = s, is regular if and only if all nonzero linear combinations of f 1 , f s are balanced. A good S box must be a regular mapping. Otherwise some output vectors appear ....

Seberry, J., Zhang, X. M., and Zheng, Y. Relationships among nonlinearity criteria. In preparation, 1993.

..... In the rest of the section we report the result we have obtained on the lower bound of Delta f of cubic functions. This result can be regarded as the first step towards fully answering the question about Delta f . The following two results (see for instance Lemma 9 of [SMZ93] and Lemma 5 of [SZZ94c] respectively) will be employed in the discussions of cubic functions. Lemma 9. f(x 1 ; xn ) x 1 ; x r ) Phi h(x r 1 ; xn ) is balanced on Vn if is balanced on V r or h is balanced on Vn Gammar . Lemma 10. If f is a quadratic function and does not have nonzero ....

....is not bent. Note that g is quadratic. By Lemma 10, g has nonzero linear structures. it is easy to see [Nyb93] that all the linear structures of a function on Vn form a linear subspace of Vn . Denote by W the linear subspace formed by the linear structures of g, and by r the dimension of W . From [SZZ94c] there exists a nonsingular matrix A of order n on GF(2) such that g (x) g(xA) can be expressed as g (x 1 ; xn ) x 1 ; x r ) Phi h(x r 1 ; xn ) where is a linear function on W while h is a function on Vn Gammar that does not have nonzero linear ....

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. Presented at EUROCRYPT'94, 1994.

....lemma is helpful in understanding the relationship between a resilient function and its component functions. It has been called XOR Lemma and expressed in terms of independence of random variables in [1] 2] It also appears as Corollary 7. 39 in [13] Here we follow the version described in [14]. Lemma 1: Let F = f 1 ; fm ) be a function from Vn to Vm , where n and m are integers with n m 1 and each f j is a function on Vn . Then F is unbiased, namely, it runs through all the vectors in Vm each 2 n Gammam times while x runs through Vn once, if and only if each nonzero ....

J. Seberry, X. M. Zhang, and Y. Zheng, "Relationships among nonlinearity criteria," in Advances in Cryptology - EUROCRYPT '94. 1995, vol. 950, Lecture Notes in Computer Science, pp. 376--388, Springer-Verlag, Berlin, Heidelberg, New York.

....are high nonlinearity, high degree of propagation, few linear structures, high algebraic degree etc. These properties are often called nonlinearity criteria. An important topic is to investigate relationships among the various nonlinearity criteria. Progress in this direction has been made in [SZZ94d] where connections have been revealed among the strict avalanche characteristic, differential characteristics, linear structures and nonlinearity, of quadratic functions. In this paper we carry on the investigation initiated in [SZZ94d] and bring together nonlinearity and propagation ....

....criteria. Progress in this direction has been made in [SZZ94d] where connections have been revealed among the strict avalanche characteristic, differential characteristics, linear structures and nonlinearity, of quadratic functions. In this paper we carry on the investigation initiated in [SZZ94d] and bring together nonlinearity and propagation characteristic of a function (quadratic or non quadratic) These two cryptographic criteria are seemly quite separate, in the sense that the former indicates the minimum distance between a Boolean function and all the affine functions whereas the ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. Presented at EUROCRYPT'94, 1994.

....come back to this issue shortly. The following lemma is helpful in understanding the relationship between a resilient function and its component functions. It has been called XOR Lemma and expressed in terms of independence of random variables in [5, 1] Here we follow the version described in [18]. Lemma 1 A function (f 1 ; f m ) where each f i is a function on V n and n = m, is unbiased, namely, it runs through all the vectors in Vm each 2 n Gammam times while x runs through V n once, if and only if each nonzero linear combinations of f 1 , f m are balanced. ....

Seberry, J., Zhang, X. M., and Zheng, Y. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94 (1995), vol. 950, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, New York, pp. 376--388.

....P 2 n Gammak i=1 c i g i = g i 0 is n Gamma k 1. ut Corollary 2 Psi(z) 1 (z) k (z) a mapping from V n to V k , where each j is defined as in Theorem 3, runs through all the 2 k vectors in V n each 2 n Gammak times while z runs through V n . Proof. By Theorem 1 of [12], this corollary is equivalent to (i) of Theorem 3. ut Since any matrix obtained by permuting the columns of a group Hadamard matrix is still a group Hadamard matrix, we can obtain an extremely large number of boolean function sets with the cryptographic properties mentioned in Theorem 3 and ....

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In preparation, 1993.

....robustness is considered when more complete information about the strength is needed. An n Theta s S box F = f 1 ; f s ) is said to be regular if F runs through each vector in V s 2 n Gammas times while x runs through Vn once. The following lemma is exactly the same as Theorem 1 of [SZZ95b]. Lemma 4. Let F = f 1 ; f s ) be a mapping from Vn to V s , where each f j is a function on Vn . Then F is regular if and only if each nonzero linear combination of f 1 ; f s is balanced. S boxes employed by a block cipher must be regular, since otherwise the cipher would be ....

....if each nonzero linear combination of f 1 ; f s is balanced. S boxes employed by a block cipher must be regular, since otherwise the cipher would be prone to statistical attacks. For a regular n Theta s S box, its differential uniformity is larger than 2 n Gammas (see also Lemma 2 of [SZZ95b]) The robustness of the S box is further determined by the number of nonzero entries in the first column of the table. We are particularly interested in n Theta s S boxes that have the following property: for each nonzero vector ff 2 Vn , F (x) Phi F (x Phi ff) runs through half of the vectors ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94, volume 950, Lecture Notes in Computer Science, pages 376--388. Springer-Verlag, Berlin, Heidelberg, New York, 1995. 162 Zhang X., Zheng Y.: On the Difficulty of Constructing Cryptographically Strong ...

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Seberry, J., Zhang, X. M., and Zheng, Y. Relationships among nonlinearity criteria. In Advances in Cryptology - EUROCRYPT'94 (1995), vol. 950, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, New York, pp. 376--388.

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