Decision diagrams (DDs) are data structures commonly used for representation of discrete functions with large number of variables. Binary DDs (BDDs) are used for representation and manipulation with Boolean functions. Complexity of a BDD is usually measured by its size, that is defined as the number of non-terminal nodes in the BDD. Minimization of the sizes of DDs is a problem greatly considered in literature and many related algorithms (exact and heuristic) have been proposed. However, there are many functions for which BDDs when minimized are still large and can have even an exponential size in the number of variables. An approach to derive compact decision diagram representations for such functions is transformation of BDDs into Multi-valued DDs (MDDs) and Heterogeneous MDDs (HMDDs). Complexity of MDDs and HMDDs is measured by the cost which is a generalization of the notion of the size by taking into account complexity of nodes in MDDs and HMDDs. This paper presents a method for transformation of BDD into HMDD with minimal cost. The proposed method reduces the time for determination of the type of nodes in HMDDs by introducing a matrix expressing dependency (interconnections) among nodes at different levels. Comparing to other methods for conversion of BDDs into HMDDs, the method reduces the number of traverses of a BDD necessary for collecting enough information to construct an equivalent HMDD. For an experimental verification of its efficiency, the method is applied to construction of HMDDs for some benchmark functions and their arithmetic and Walsh spectra.

In the past twenty years, there were only a few constructions for Boolean functions with nonlinearity exceeding the quadratic bound 2n-1-2(n-1)/2 when n is odd (we shall call them Boolean functions with very high nonlinearity). The first basic construction was by Patterson and Wiedemann in 1983, which produced unbalanced function with very high nonlinearity. But for cryptographic applications, we need balanced Boolean functions. Therefore in 1993, Seberry, Zhang and Zheng proposed a secondary construction for balanced functions with very high nonlinearity by taking the direct sum of a modified bent function with the Patterson-Wiedemann function. Later in 2000, Sarkar and Maitra constructed such functions by taking the direct sum of a bent function with a modified Patterson-Wiedemann function. In this paper, we propose a new secondary construction for balanced Boolean functions with very high nonlinearity by recursively composing balanced functions with very high nonlinearity with quadratic functions. This is the first construction for balanced function with very high nonlinearity not based on the direct sum approach. Our construction also have other desirable properties like high algebraic degree and large linear span.

We consider the classification problem to construct a classifier c:{0,1}n{0,1} from a given set of examples (training set), which (approximately) realizes the hidden oracle y:{0,1}n{0,1} describing the phenomenon under consideration. For this problem, a number of approaches are already known in computational learning theory; e.g., decision trees, support vector machines (SVM), and iteratively composed features (ICF). The last one, ICF, was proposed in our previous work (Haraguchi et al., (2004)). A feature, composed of a nonempty subset S of other features (including the original data attributes), is a Boolean function fS:{0,1}S{0,1} and is constructed according to the proposed rule. The ICF algorithm iterates generation and selection processes of features, and finally adopts one of the generated features as the classifier, where the generation process may be considered as embodying the idea of boosting, since new features are generated from the available features. In this paper, we generalize a feature to an extended Boolean function fS:{0,1,*}S{0,1,*} to allow partial knowledge, where * denotes the state of uncertainty. We then propose the algorithm ICF* to generate such generalized features. The selection process of ICF* is also different from that of ICF, in that features are selected so as to cover the entire training set. Our computational experiments indicate that ICF* is better than ICF in terms of both classification performance and computation time. Also, it is competitive with other representative learning algorithms such as decision trees and SVM.

In this paper, a new class of bent functions is constructed by combining class M and class C bent functions. Using the construction method of the class D bent functions defined on the binary vector space, new p-ary generalized bent functions are also introduced for odd prime p.

We give an explicit formula for the number of n-variable clique function in terms of the parameters based upon the numbers of intersecting antichains of the lower half of the n-cube. We present the numbers of clique functions with up to seven variables through computer evaluation of the parameters.

This paper presents implicit representation of binary decision diagrams (implicit BDDs) as a new effecient data structure for Boolean functions. A well-known method of representing graphs by binary decision diagrams (BDDs) is applied to BDDs themselves. Namely, it is a BDD representation of BDDs. Regularity in the structure of BDDs representing certain Boolean functions contributes to significant reduction in size of the resulting implicit BDD repersentation. Since the implicit BDDs also provide canonical forms for Boolean functions, the equivalence of the two implicit BDD forms is decided in time proportional to the representation size. We also show an algorithm to maniqulate Boolean functions on this implicit data structure.

For symmetric cryptosystems, their transformations should have nonlinear elements to be secure against various attacks. Several nonlinearity criteria have been defined and their properties have been made clear. This paper focuses on, among these criteria, the propagation criterion (PC) and the strict avalanche criterion (SAC), and makes a further investigation of them. It discusses the sets of Boolean functions satisflying the PC of higher degrees, the sets of those satisfying the SAC of higher orders and their relationships. We give a necessary and sufficient condition for an n-input Boolean function to satisfy the PC with respect to a set of all but one or two elements in {0,1}n{(0,...,0)}. From this condition, it follows that, for every even n 2, an n-input Boolean function satisfies the PC of degree n 1 if and only if it satisfies the PC of degree n. We also show a method that constructs, for any odd n 3, n-input Boolean functions that satisfy the PC with respect to a set of all but one elements in {0,1}n{(0,...,0)}. This method is a generalized version of a previous one. Concerned with the SAC of higher orders, it is shown that the previously proved upper bound of the nonlinear order of Boolean functions satisfying the criterion is tight. The relationships are discussed between the set of n-input Boolean functions satisfying the PC and the sets of those satisfying the SAC.

This paper discusses Boolean functions satisfying the propagation criterion (PC) and their balancedness. Firstly, we discuss Boolean functions with n variables that satisfy the PC with respect to all but three elements in {0,1}n-{(0,...,0)}. For even n4, a necessary and sufficient condition is presented for Boolean functions with n variables to satisfy the PC with respect to all but three elements in {0,1}n-{(0,...,0)}. From this condition, it is proved that all of these Boolean functions are constructed from all perfectly nonlinear Boolean functions with n-2 variables. For odd n3, it is shown that Boolean functions with n variables satisfying the PC with respect to all but three elements in {0,1}n-{(0,...,0)} satisfy the PC with respect to all but one elements in it. Secondly, Boolean functions satisfying the PC of degree n-2 and their balancedness are considered. For even n4, it is proved that an upper bound on the degree of the PC is n-3 for balanced Boolean functions with n variables. This bound is optimal for n=4,6. It is also proved that, for odd n3, balanced Boolean functions with n variables satisfying the PC of degree n-2 satisfy the PC with respect to all but one elements in {0,1}n-{(0,...,0)}.

Manipulation of Boolean functions is one of the most important techniques for implementing of VLSI logic design systems. This paper presents a fast method for generating prime-irredundant covers from Binary Decision Diagrams (BDDs), which are efficient representation of Boolean functions. Prime-irredundant covers are forms in which each cube is a prime implicant and no cube can be eliminated. This new method generates compact cube sets from BDDs directly, in contrast to the conventional cube set reduction algorithms, which commonly manipulate redundant cube sets or truth tables. Our method is based on the idea of a recursive operator, proposed by Morreale. Morreale's algorithm is also based on cube set manipulation. We found that the algorithm can be improved and rearranged to fit BDD operations efficiently. The experimental results demonstrate that our method is efficient in terms of time and space. In practical time, we can generate cube sets consisting of more than 1,000,000 literals from multi-level logic circuits which have never previously been flattened into two-level logics. Our method is more than 10 times faster than ESPRESSO in large-scale examples. It gives quasi-minimum numbers of cubes and literals. This method should find many useful applications in logic design systems.