Boolean Functions: Cryptography and Applications

With cryptographic investigations, the design of Boolean functions is a wide area. The Boolean functions play important role in the construction of a symmetric cryptosystem. In this paper the modified hill climbing method is considered. The method allows using hill climbing techniques to modify bent functions used to design balanced, highly nonlinear Boolean functions with high algebraic degree and low autocorrelation. The experimental results of constructing the cryptographically strong Boolean functions are presented. When designing block and stream ciphers, Boolean functions play an important role and define the cryptographic strength of applications to differential and linear cryptanalysis particularly. Often the resistance of cryptosystems to known types of attacks is discussed in terms of Boolean functions used in them. A lot of attention has been given to construction of Boolean functions with desired cryptographic properties in cryptology [1–6]. The main strength criteria of Boolean functions are balancedness, high nonlinearity, high algebraic degree and low autocorrelation. There are three types of methods of constructing nonlinear Boolean functions: random generation, algebraic (or direct) and heuristic methods. Each of them has its own advantages and drawbacks. Generating nonlinear Boolean function via random generation is too difficult to find functions that possess high cryptographic 1 Nrjetix company. email: [email protected] 2 Nrjetix company. email: [email protected] 3 Nrjetix company. email: [email protected] J-F. Michon, P. Valarcher, J-B. Yunès (Eds.): BFCA’08 2 Y. IZBENKO, V. KOVTUN, A. KUZNETSOV properties due to the vast size of search space, especially for function with n > 8, where n is the space size. The attractiveness of these techniques comes out of the simplicity in their implementation. Algebraic methods allow constructing functions that have a set of desired cryptographic properties with low computation complexity, but these functions can have low algebraic complexity [4]. The heuristic methods [1–6] are the newest techniques capable of effective Boolean functions generation with desired cryptographic properties. Because of some intuitive approaches used in heuristic methods and the fact that heuristic methods are not limited by algebraic constructions, these methods can construct Boolean functions with properties that are close to the maximum attained. The core of all heuristic methods is the hill climbing method (HC) introduced in [1]. The HC method allows increasing the nonlinearity of a Boolean function, particularly of the randomly generated one. The HC method may be effectively used with genetic and simulated annealing methods. In this paper we consider a modification of the HC method, which allows constructing highly nonlinear Boolean functions with low autocorrelation. The main idea of the proposed method is ’inverting’ of the HC’s algorithm. There are two main differences in our method from HC method: 1) we are using a bent function as input data instead of a randomly generated Boolean function, 2) we are decreasing nonlinearity of the bent function to a required value instead of increasing nonlinearity of a randomly generated Boolean function. This paper is structured as follows: Section 2 presents the main definitions and terms, Section 3 describes the modified hill climbing method, Section 4 shows the main results. In the final part we make conclusions of our investigations.