Abstract. Many desirable properties have been identified for Boolean functions with cryptographic applications. Obtaining optimal tradeoffs among such properties is hard. In this paper we show how simulated annealing, a search technique inspired by the cooling processes of molten metals, can be used to derive functions with profiles of cryptographicallyrelevant properties as yet unachieved by any other technique.

The design of Boolean functions with properties of cryptographic significance is a hard task. In this paper, we adopt an unorthodox approach to the design of such functions. Our search space is the set of functions that possess the required properties. It is “Boolean-ness ” that is evolved. Key words: Q1 1.

Abstract. This paper outlines a general approach to the iterative incremental improvement of the cryptographic properties of arbitrary Boolean functions. These methods, which are known as hill climbing, offer a fast way to obtain Boolean functions that have properties superior to those of randomly generated functions. They provide a means to improve the attainable compromise between conflicting cryptographic criteria. We give an overview of the different options available, concentrating on reducing the maximum value of the Walsh-Hadamard transform and autocorrelation function. A user selected heuristic allows the methods to be flexible. Thus we obtain Boolean functions that are locally optimal with regard to one or more important cryptographic properties such as nonlinearity and global autocorrelation. 1

Abstract — Boolean function design is at the heart of cryptography, and is the subject of a great deal of theoretical research. We have use a simulated annealing approach to find functions with particular desirable cryptographic properties; for functions of a small numbers of variables, results with properties as good as (and sometimes better than) the best so far have been achieved. The success of this approach is very sensitive to the cost function chosen; here we investigate this property, and describe a metasearch approach to finding the most effective cost function for this class of problems. I.

Abstract. We explore three applications of geometric sequences in constructing cryptographic Boolean functions. First, we construct 1-resilient functions of n Boolean variables with nonlinearity 2 n−1 −2 (n−1)/2, n odd. The Hadamard transform of these functions is 3-valued, which limits the efficiency of certain stream cipher attacks. From the case for n odd, we construct highly nonlinear 1-resilient functions which disprove a conjecture of Pasalic and Johansson for n even. Our constructions do not have a potential weakness shared by resilient functions which are formed from concatenation of linear functions. Second, we give a new construction for balanced Boolean functions with high nonlinearity, exceeding 2 n−1 −2 (n−1)/2, which is not based on the direct sum construction. Moreover, these functions have high algebraic degree and large linear span. Third, we construct balanced vectorial Boolean functions with nonlinearity 2 n−1 − 2 (n−1)/2 and low maximum correlation. They can be used as nonlinear combiners for stream cipher systems with high throughput. 1

Abstract. This paper shows how suitable choice of cost function can significantly affect the power of optimisation methods for the synthesising of Boolean functions. In particular we show how simulated annealing, coupled with a new cost function motivated by Parseval’s Theorem, can be used to drive the search into areas of design from which traditonal techniques, such as hill-climbing, can find then find excellent solutions. 1

Abstract — Substitution boxes are important components in many modern day block and stream ciphers. Their study has attracted a great deal of attention over many years. The development of a variety of cryptosystem attacks over the years has lead to the development of criteria for resilience to such attacks. Some general criteria such as high non-linearity and low autocorrelation have been proposed as useful criteria (providing some protection against attacks such as linear cryptanalysis and differential cryptanalysis). There has been little application of evolutionary search to the development of S-boxes. In this paper we show how a cost function that has found excellent single output Boolean functions can be generalised to provide improved results for small S-boxes. I.

This paper presents an efficient approach for classification of the affine equivalence classes of cosets of the first order Reed-Muller code with respect to cryptographic properties such as correlation-immunity, resiliency and propagation characteristics. First, we apply the method to completely classify all the 48 classes into which the general affine group AGL(2, 5) partitions the cosets of RM(1, 5). Second, after distinguishing the 34 affine equivalence classes of cosets of RM(1, 6) in RM(3, 6) we perform the same classification for these classes. We also study the algebraic immunity of the corresponding affine equivalence classes. Moreover, several relations are derived between the algebraic immunity and other cryptographic properties. Finally, we introduce two new indicators which can be used to distinguish affine inequivalent Boolean functions when the known criteria are not sufficient. From these indicators a method can be derived for finding the affine relation between two functions (if such exists). The efficiency of the method depends on the structure of the Walsh or autocorrelation spectrum.

Abstract. 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

Cryptography is an indispensable component of much modern-day system security. It has also been an attractive application domain for researchers in non-standard computation. In this paper I shall identify what I believe to be