Enumeration of 9-variable Rotation Symmetric Boolean Functions
نویسندگان
چکیده
The existence of 9-variable Boolean functions having nonlinearity strictly greater than 240 has been shown very recently (May 2006) by Kavut, Maitra and Yücel. The functions with nonlinearity 241 have been identified by a heuristic search in the class of Rotation Symmetric Boolean Functions (RSBFs). In this paper we efficiently perform the exhaustive search to enumerate the 9-variable RSBFs having nonlinearity > 240 and found that there are such functions with nonlinearity 241 only and there is no RSBF having nonlinearity > 241. Our search enumerates 8×189 many 9-variable RSBFs having nonlinearity 241. We further show that there are only two functions which are different up to the affine equivalence. Towards the end we explain the coding theoretic significance of these functions.
منابع مشابه
Generalized Rotation Symmetric and Dihedral Symmetric Boolean Functions - 9 Variable Boolean Functions with Nonlinearity 242
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 Yücel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs an...
متن کاملCounting rotation symmetric functions using Polya's theorem
Homogeneous rotation symmetric (invariant under cyclic permutation of the variables) Boolean functions have been extensively studied in recent years due to their applications in cryptography. In this paper we give an explicit formula for the number of homogeneous rotation symmetric functions over the finite field GF(pm) using Polya’s enumeration theorem, which completely solves the open problem...
متن کاملNew Construction of Even-variable Rotation Symmetric Boolean Functions with Optimum Algebraic Immunity
The rotation symmetric Boolean functions which are invariant under the action of cyclic group have been used as components of different cryptosystems. In order to resist algebraic attacks, Boolean functions should have high algebraic immunity. This paper studies the construction of even-variable rotation symmetric Boolean functions with optimum algebraic immunity. We construct ( / 4 3) n ...
متن کامل9-variable Boolean Functions with Nonlinearity 242 in the Generalized Rotation Class
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 Yücel. 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-D...
متن کاملConstruction of Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity
In this paper we present a theoretical construction of Rotation Symmetric Boolean Functions (RSBFs) on odd number of variables with maximum possible AI and further these functions are not symmetric. Our RSBFs are of better nonlinearity than the existing theoretical constructions with maximum possible AI . To get very good nonlinearity, which is important for practical cryptographic design, we g...
متن کامل