The Experts below are selected from a list of 138 Experts worldwide ranked by ideXlab platform
Melek D Yucel - One of the best experts on this subject based on the ideXlab platform.
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9 variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class
Information & Computation, 2010Co-Authors: Selcuk Kavut, Melek D YucelAbstract:We give a new lower bound to the covering radius of the first order Reed-Muller Code RM(1,n), where n@?{9,11,13}. Equivalently, we present the n-variable Boolean functions for n@?{9,11,13} with maximum nonlinearity found till now. In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the generalized classes of ''k-RSBFs and k-DSBFs (k-Dihedral Symmetric Boolean Functions)'', where k is a positive integer dividing n. Secondly, utilizing a steepest-descent like iterative heuristic search algorithm, we have found 9-variable Boolean functions with nonlinearity 242 within the classes of both 3-RSBFs and 3-DSBFs. Thirdly, motivated by the fact that RSBFs are invariant under a special permutation of the input vector, we have classified all possible permutations up to the linear equivalence of Boolean functions that are invariant under those permutations.
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9 variable boolean functions with nonlinearity 242 in the generalized rotation class
arXiv: Cryptography and Security, 2008Co-Authors: Selcuk Kavut, Melek D YucelAbstract:In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rota- tion Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the "generalized classes of k-RSBFs and k-DSBFs (k-Dihedral Symmet- ric Boolean Functions)", where k is a positive integer dividing n and k-RSBFs is a subset of l-RSBFs if k < l. Secondly, utilizing the steepest-descent like iterative heuristic search algorithm used previously to identify the 9-variable RSBFs with non- linearity 241, we have made a search within the classes of 3-RSBFs and 3-DSBFs. The search has accomplished to find 9-variable Boolean functions with nonlinearity 242 in both of these classes. It should be emphasized that although the class of 3- RSBFs contains functions with nonlinearity 242; 1-RSBFs or simply RSBFs, which is a subset of 3-RSBFs, does not contain any. This result also shows that the covering radius of the first order Reed-Muller Code R(1, 9) is at least equal to 242. Thirdly, motivated by the fact that RSBFs are invariant under a special permutation of the input vector, we have classified all possible permutations up to the linear equivalence of Boolean functions that are invariant under those permutations. Specifically, for 9-variable Boolean functions, 9! possible permutations are classified into 30 classes; and the search algorithm identifies some of these classes as rich. The rich classes yield new Boolean functions with nonlinearity 242 having different autocorrelation spectra from those of the functions found in the generalized 3-RSBF and 3-DSBF classes. However, there is no zero value in the Walsh spectra of these functions; hence, none of them can be linearly transformed to 9-variable balanced functions with nonlinearity 242.
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generalized rotation symmetric and dihedral symmetric boolean functions 9 variable boolean functions with nonlinearity 242
Applicable Algebra in Engineering Communication and Computing, 2007Co-Authors: Selcuk Kavut, Melek D YucelAbstract:Recently, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs and Dihedral Symmetric Boolean Functions (DSBFs). These functions do not have any zero in the Walsh spectrum values, hence they cannot be made balanced easily. This result also shows that the covering radius of the first order Reed-Muller Code R(1, 9) is at least 242.
Qichun Wang - One of the best experts on this subject based on the ideXlab platform.
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the covering radius of the reed muller Code rm 2 7 is 40
Discrete Mathematics, 2019Co-Authors: Qichun WangAbstract:Abstract It was proved by J. Schatz that the covering radius of the second order Reed–Muller Code R M ( 2 , 6 ) is 18 (Schatz (1981)). However, the covering radius of R M ( 2 , 7 ) has been an open problem for many years. In this paper, we prove that the covering radius of R M ( 2 , 7 ) is 40, which is the same as the covering radius of R M ( 2 , 7 ) in R M ( 3 , 7 ) . As a corollary, we also find new upper bounds for the covering radius of R M ( 2 , n ) , n = 8 , 9 , 10 .
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the covering radius of the reed muller Code rm 2 7 is 40
arXiv: Information Theory, 2018Co-Authors: Qichun WangAbstract:It was proved by J. Schatz that the covering radius of the second order Reed--Muller Code $RM(2, 6)$ is 18 (IEEE Trans Inf Theory 27: 529--530, 1985). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for $RM(2,n)$, $n=8,9,10$.
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on the covering radius of the third order reed muller Code rm 3 7
Designs Codes and Cryptography, 2018Co-Authors: Qichun Wang, Chik How Tan, Theo Fanuela PrabowoAbstract:The covering radius of the third order Reed–Muller Code of length 128 has been an open problem for many years. The best upper bound of it is known to be 22. In this paper, we give a sufficient and necessary condition for the covering radius of RM(3, 7) to be equal to 22. Using this condition, we prove that the covering radius of RM(3, 7) in RM(4, 7) is 20. Therefore, if the third-order nonlinearity of a 7-variable Boolean function is greater than 20, then its algebraic degree is at least 5. As a corollary, we conclude that the covering radius of RM(3, 7) in the set of 2-resilient Boolean functions is at most 20 which improves the bound given by Borissov et al. (IEEE Trans Inf Theory 51:1182–1189, 2005).
Bart Preneel - One of the best experts on this subject based on the ideXlab platform.
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on the covering radius of second order binary reed muller Code in the set of resilient boolean functions
Lecture Notes in Computer Science, 2003Co-Authors: Yuri L Borissov, An Braeken, Svetla Nikova, Bart PreneelAbstract:Let \(\mathcal R_{t,n}\) denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary Reed-Muller Code RM(2,6) in the sets \(\mathcal R_{t,6}\), t=0,1,2 is 16. Second, we show that the covering radius of the binary Reed-Muller Code RM(2,7) in the set \(\mathcal R_{3,7}\) is 32. We derive a new lower bound for the covering radius of the Reed-Muller Code RM(2,n) in the set \(\mathcal R_{n-1,4}\). Finally, we present new lower bounds in the sets \(\mathcal R_{t,7}\), t=0,1,2.
Kaoru Kurosawa - One of the best experts on this subject based on the ideXlab platform.
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new covering radius of reed muller Codes for t resilient functions
Selected Areas in Cryptography, 2001Co-Authors: Tetsu Iwata, Takayuki Yoshiwara, Kaoru KurosawaAbstract:In stream ciphers, we should use a t-resilient Boolean function f(X) with large nonlinearity to resist fast correlation attacks and linear attacks. Further, in order to be secure against an extension of linear attacks, we wish to find a t-resilient function f(X) which has a large distance even from low degree Boolean functions. From this point of view, we define a new covering radius p(t, r, n) as the maximum distance between a t-resilient function f(X) and the r-th order Reed-Muller Code RM(r, n). We next derive its lower and upper bounds. Finally, we present a table of numerical bounds for p(t, r, n).
Selcuk Kavut - One of the best experts on this subject based on the ideXlab platform.
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9 variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class
Information & Computation, 2010Co-Authors: Selcuk Kavut, Melek D YucelAbstract:We give a new lower bound to the covering radius of the first order Reed-Muller Code RM(1,n), where n@?{9,11,13}. Equivalently, we present the n-variable Boolean functions for n@?{9,11,13} with maximum nonlinearity found till now. In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the generalized classes of ''k-RSBFs and k-DSBFs (k-Dihedral Symmetric Boolean Functions)'', where k is a positive integer dividing n. Secondly, utilizing a steepest-descent like iterative heuristic search algorithm, we have found 9-variable Boolean functions with nonlinearity 242 within the classes of both 3-RSBFs and 3-DSBFs. Thirdly, motivated by the fact that RSBFs are invariant under a special permutation of the input vector, we have classified all possible permutations up to the linear equivalence of Boolean functions that are invariant under those permutations.
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9 variable boolean functions with nonlinearity 242 in the generalized rotation class
arXiv: Cryptography and Security, 2008Co-Authors: Selcuk Kavut, Melek D YucelAbstract:In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rota- tion Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the "generalized classes of k-RSBFs and k-DSBFs (k-Dihedral Symmet- ric Boolean Functions)", where k is a positive integer dividing n and k-RSBFs is a subset of l-RSBFs if k < l. Secondly, utilizing the steepest-descent like iterative heuristic search algorithm used previously to identify the 9-variable RSBFs with non- linearity 241, we have made a search within the classes of 3-RSBFs and 3-DSBFs. The search has accomplished to find 9-variable Boolean functions with nonlinearity 242 in both of these classes. It should be emphasized that although the class of 3- RSBFs contains functions with nonlinearity 242; 1-RSBFs or simply RSBFs, which is a subset of 3-RSBFs, does not contain any. This result also shows that the covering radius of the first order Reed-Muller Code R(1, 9) is at least equal to 242. Thirdly, motivated by the fact that RSBFs are invariant under a special permutation of the input vector, we have classified all possible permutations up to the linear equivalence of Boolean functions that are invariant under those permutations. Specifically, for 9-variable Boolean functions, 9! possible permutations are classified into 30 classes; and the search algorithm identifies some of these classes as rich. The rich classes yield new Boolean functions with nonlinearity 242 having different autocorrelation spectra from those of the functions found in the generalized 3-RSBF and 3-DSBF classes. However, there is no zero value in the Walsh spectra of these functions; hence, none of them can be linearly transformed to 9-variable balanced functions with nonlinearity 242.
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generalized rotation symmetric and dihedral symmetric boolean functions 9 variable boolean functions with nonlinearity 242
Applicable Algebra in Engineering Communication and Computing, 2007Co-Authors: Selcuk Kavut, Melek D YucelAbstract:Recently, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs and Dihedral Symmetric Boolean Functions (DSBFs). These functions do not have any zero in the Walsh spectrum values, hence they cannot be made balanced easily. This result also shows that the covering radius of the first order Reed-Muller Code R(1, 9) is at least 242.