A Note on Symmetric Boolean Functions With Maximum Algebraic Immunity in Odd Number of Variables
Nat. Univ. of Defense Technol., ChangShaIEEE Transactions on Information Theory (Impact Factor: 2.62). 09/2007; DOI:10.1109/TIT.2007.901189
Source: IEEE Xplore
ABSTRACT In this note, it is proved that for each odd positive integer n there are exactly two n-variable symmetric Boolean functions with maximum algebraic immunity.
Conference Proceeding: Results on Algebraic Immunity for Cryptographically Significant Boolean Functions.[show abstract] [hide abstract]
ABSTRACT: Recently algebraic attack has received a lot of attention in cryptographic literature. It has been observed that a Boolean function f, interpreted as a multivariate polynomial over GF(2), should not have low degree multiples when used as a cryptographic primitive. In this paper we show that high nonlinearity is a necessary condition to resist algebraic attack and explain how the Walsh spectra values are related to the algebraic immunity (resistance against algebraic attack) of a Boolean function. Next we present enumeration results on linearly independent annihilators. We also study certain classes of highly nonlinear resilient Boolean functions for their algebraic immunity.Progress in Cryptology - INDOCRYPT 2004, 5th International Conference on Cryptology in India, Chennai, India, December 20-22, 2004, Proceedings; 01/2004
Conference Proceeding: Algebraic Attacks Over GF(q).[show abstract] [hide abstract]
ABSTRACT: Recent algebraic attacks on LFSR-based stream ciphers and S-boxes have generated much interest as they appear to be extremely powerful. Theoretical work has been developed focusing around the Boo- lean function case. In this paper, we generalize this theory to arbitrary finite fields and extend the theory of annihilators and ideals introduced at Eurocrypt 2004 by Meier, Pasalic and Carlet. In particular, we prove that for any function f in the multivariate polynomial ring over GF(q), f has a low degree multiple precisely when two low degree functions appear in the same coset of the annihilator of f q − − 1 – 1. In this case, many such low degree multiples exist.Progress in Cryptology - INDOCRYPT 2004, 5th International Conference on Cryptology in India, Chennai, India, December 20-22, 2004, Proceedings; 01/2004
Conference Proceeding: On the Algebraic Immunity of Symmetric Boolean Functions.[show abstract] [hide abstract]
ABSTRACT: In this paper, we analyse the algebraic immunity of symmetric Boolean functions. We identify a set of lowest degree annihilators for symmetric functions and propose an ecient algorithm for computing the algebraic immunity of a symmetric function. The existence of several symmetric functions with maximum algebraic immunity is proven. In this way, a new class of function which have good implementation properties and maximum algebraic immunity is found. We also investigate the existence of symmetric functions with high nonlinearity and reasonable order of algebraic immunity. Finally, we give suggestions how to use symmetric functions in a stream cipher. 1 Introdution Symmetric functions have the property that the function value is determined by the weight of the vector. Therefore, a symmetric function in n variables can be defined by a vector of length n+1 which represents the function values of the dierent weights of the vectors. For this reason, symmetric functions are very interesting functions in order to obtain low memory in software. Also in hardware implementation, only a low number of gates is required (15). Properties such as balancedness and resiliency, propagation characteristics and nonlinearity are studied in (1). It is shown that these functions do not behave very good in general with respect to a combination of the properties nonlinearity, degree, and resiliency, which are important properties for resisting distinguishing and correlation attacks. In 2002, several successfull algebraic attacks on stream ciphers were proposed. The success of these attacks do not mainly depend on the classical properties of nonlinearity or resiliency, but mainly on the weak behaviour with respect to the property of algebraic immunity. In this paper we study the resistance against algebraic attacks for the symmetric functions. We identify a set of lowest degree annihilators of a symmetric function. Since the size of this set is very small in comparison with the general case, the algorithm for computing the algebraic immunity of a symmetric function becomes much more ecient. We prove the existence of several symmetric functions with optimal algebraic immunity. The idea is then to use these functions which have good algebraic immunity in combination with highly nonlinear functions as building block in the design of a stream cipher. First, Sect. 2 deals with some background on Boolean functions and more in particular on symmetric Boolean functions. In Sect. 3, we investigate the algebraic immunity of homogeneous symmetric functions. Based on the identification of a set of lowest degree annihilators of a symmetric function, we propose an algorithm for computing the algebraic immunity of symmetric functions in Sect. 4. Sect. 5 presents the proofs on several symmetric functions which possess maximum algebraic immunity. In Sect. 6, we investigate the existence of symmetric functions with reasonable AI and better nonlinearity as the sym- metric functions with maximum AI. Finally, we conclude in Sect. 7 by summerizing the good and bad properties of symmetric functions when used in a concrete design. We also present some open problems.Progress in Cryptology - INDOCRYPT 2005, 6th International Conference on Cryptology in India, Bangalore, India, December 10-12, 2005, Proceedings; 01/2005
Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.