Jennifer Seberry, Xian-Mo Zhang, Yuliang Zheng "Improving the Strict Avalanche Characteristics of Cryptographic Functions", Information Processing Letters 50 (1994), pp. 37-41.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

....presented in [SZZ93a] or by expanding smaller S boxes according to [KMI90] Another approach, introduced in [P91, N93] yields cryptographically strong S boxes which do not satisfy SAC. But the resulting S boxes can be modified by transforming their inputs by a suitable linear transformation [SZZ93b] into SAC satisfying S boxes. Both previous constructions yield large but complex cryptographically strong S boxes. Youssef and Tavares [YT96] presented simplier S blocks, so called Substitution Permutation Networks (SPN) with cryptographically good properties. The SPN s constructed by their ....

Seberry J., Zhang X.-M., Zheng Y., Improving the Strict Avalanche Characteristics of Cryptographic Functions, Technical Report 93-9, The Centre for Computer Security Research, University of Wollongong, Australia, 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. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

.... 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. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

....2 V s , 2 V p and 6= 0. Note that there exist 2 s p 2 p such vectors as = satisfying 2 V s , 2 V p and 6= 0. This implies that there are at least 2 s p 2 p non zero vectors such that f(x) f(x ) is balanced. Since 2 s p 2 p 2 s p 1 , by using Theorem 7 of [7], we have proved (iv) Next we present a method for constructing nonseparable functions that are highly nonlinear, balanced and correlation immune. Theorem 5 Let p, s and r be integers with 0 s; r p. Set (p; r) p 1 p 2 p r . If 2 p s 1 (p; r) ....

J. Seberry, X. M. Zhang and Y. Zheng, Improving the Strict Avalanche Characteristics of Cryptographic Functions, Information Processing Letters, Vol.50, pp.37-41, 1994.

....of . Hence there exist n linearly independent vectors in c . In other words, there exist n linearly independent vectors with respect to which f satisfies the propagation criterion. Hence we can choose a nonsingular n Theta n matrix A over GF (2) such that g(x) f(xA) satisfies the SAC (see [9]) The nonsingular linear transformation A does not alter any of the properties of f in Example 1 We can further improve the function in Example 2 so as to obtain a rth order plateaued functions on Vn having the highest degree and satisfying all the properties in Example 1. Example 3. Given any ....

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

....ff) is balanced for all ff 2 V n with W (ff) 1, where x = x 1 ; x n ) It is widely accepted that the component functions of an S box employed by a modern block cipher should all satisfy the SAC. A general technique for constructing SAC fulfilling cryptographic functions can be found in [22]. While the SAC measures the avalanche characteristics of a function, the linear structure is a concept that in a sense complements the former, namely, it indicates the straightness of a function. Definition 3 Let f be a function on V n . A vector ff 2 V n is called a linear structure of f if ....

.... 1 g. As the rank of Omega is s, we can choose s functions from Omega Gamma say g j 1 , g js , such that they are all linearly independent. Since s = 2 s Gammat Gamma2 , we have t j 1 Delta Delta Delta t js s Delta 2 n Gammas t 1 = 2 n Gamma1 . By Theorem 2 of [22], there exists a nonsingular matrix A of order n over GF (2) such that all component functions of (g j 1 (xA) g js (xA) satisfy the SAC. Furthermore, as each g j is a nonzero linear combination of f 1 , f s , there is a nonsingular matrix B of order s over GF (2) such that (g j 1 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 1994. (to appear).

....ff) is balanced for all ff 2 Vn with W (ff) 1, where x = x 1 ; xn ) It is widely accepted that the component functions of an S box employed by a modern block cipher should all satisfy the SAC. A general technique for constructing SAC fulfilling cryptographic functions can be found in [22]. While the SAC measures the avalanche characteristics of a function, the linear structure is a concept that in a sense complements the former, namely, it indicates the straightness of a function. Definition 4. Let f be a function on Vn . A vector ff 2 Vn is called a linear structure of f if f(x) ....

....: g 2 s Gamma1 1 g. As the rank of Omega is s, we can choose s functions from Omega , say g j1 , g js , such that they are all linearly independent. Since s 2 s Gammat Gamma2 , we have t j1 Delta Delta Delta t js s Delta 2 n Gammas t 1 2 n Gamma1 . By Theorem 2 of [22], there exists a nonsingular matrix A of order n over GF (2) such that all component functions of (g j1 (xA) g js (xA) satisfy the SAC. Furthermore, as each g j is a nonzero linear combination of f 1 , f s , there is a nonsingular matrix B of order s over GF (2) such that (g j1 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

....criterion is 2 2k Gamma 2 k 1 2 k Gamma1 Gamma 1 which is larger than 2 2k Gamma1 . Hence these vectors contain at least 2k linear independent ones. Let A be the matrix with the 2k linear independent vectors as its rows. Then A is nonsingular and of order 2k. By Theorem 3 of [19], f(zA) satisfies the SAC. All the properties described in Theorem 4 are affected by the nonsingular transform A. 5.2 On V 2k 1 To construct functions on V 2k 1 with good avalanche characteristics, we need a permutation m(u) on V k with a special property that u Phi m(u) is also a permutation ....

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

....that L 2 n Gammak Gamma1 j=0 h 1j (x) 0. Therefore the maximum algebraic degree is always achievable. This proves (iv) v) follows from (i) and Theorem 1. ut A problem with G = g 1 ; g k ) is that it does not satisfy the SAC. Using the following Lemma 4 which was 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 ....

Seberry, J., Zhang, X. M., and Zheng, Y. Improving the strict avalanche characteristics of cryptographic functions. Submitted for publication, 1993.

....criterion is 2 2k Gamma 2 k 1 2 k Gamma1 Gamma 1 which is larger than 2 2k Gamma1 . Hence these vectors contain at least 2k linear independent ones. Let A be the matrix with the 2k linear independent vectors as its rows. Then A is nonsingular and of order 2k. By Theorem 3 of [SZZ94a] f(zA) satisfies the SAC. All the properties described in Theorem 16 are affected by the nonsingular transform A. 5.2 On V 2k 1 To construct functions on V 2k 1 with good avalanche characteristics, we need a permutation m(u) on V k with a special property that u Phi m(u) is also a ....

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37-- 41, 1994.

....is balanced for all ff 2 V n with W (ff) 1, where x = x 1 ; x n ) It is widely accepted that the component functions of an S box employed by a modern block cipher should all satisfy the SAC. A general technique for constructing SAC fulfilling cryptographic functions can be found in [17]. While the SAC measures the avalanche characteristics of a function, the linear structure is a concept that in a sense is complementary to the former, namely, the linear structure indicates the smoothness of a function. Definition 3 Let f be a function on V n . A vector ff 2 V n is called a ....

....s Gamma1 1 g. As the rank of Delta is s, we can choose s functions from Delta, say g j 1 , g js , such that they are all linearly independent. Since s = 2 s Gammat Gamma2 , we have t j 1 Delta Delta Delta t js s Delta 2 n Gammas t 1 = 2 n Gamma1 . By Theorem 2 of [17], there exists a nonsingular matrix A of order n over GF (2) such that all component functions of (g j 1 (xA) g js (xA) satisfy the SAC. Furthermore, as each g j is a nonzero linear combination of f 1 , f s , there is a nonsingular matrix B of order s over GF (2) such that (g j 1 ....

[Article contains additional citation context not shown here]

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

No context found.

Jennifer Seberry, Xian-Mo Zhang, Yuliang Zheng "Improving the Strict Avalanche Characteristics of Cryptographic Functions", Information Processing Letters 50 (1994), pp. 37-41.

No context found.

Seberry J., Zhang X.-M., Zheng Y., Improving the Strict Avalanche Characteristics of Cryptographic Functions, Technical Report 93-9, The Centre for Computer Security Research, University of Wollongong, Australia, 1993.

No context found.

J. Seberry, X. M. Zhang, and Y. Zheng. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters, 50:37--41, 1994.

No context found.

Seberry, J., Zhang, X. M., and Zheng, Y. Improving the strict avalanche characteristics of cryptographic functions. Information Processing Letters 50 (1994), 37--41.

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