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.

A function F:F2n F2n is almost perfect nonlinear (APN) if, for every a 0, b in F2n, the equation F(x)+F(x+a)=b has at most two solutions in F2n. When used as an S-box in a block cipher, it contributes optimally to the resistance to differential cryptanalysis. The function F is almost bent (AB) if the minimum Hamming distance between all its component functions v F, v∈F2n {0} (where "" denotes any inner product in F2n ) and all affine Boolean functions on F2n takes the maximal value 2n-1-2. AB functions exist for n odd only and contribute optimally to the resistance to the linear cryptanalysis. Every AB function is APN, and in the n odd case, any quadratic APN function is AB. The APN and AB properties are preserved by affine equivalence: F F' if F'=A1 F A2, where A1,A2 are affine permutations. More generally, they are preserved by CCZ-equivalence, that is, affine equivalence of the graphs of F: {(x,F(xv)) | x∈ F2n} and of F'. Until recently, the only known constructions of APN and AB functions were CCZ-equivalent to power functions F(x)=xd over finite fields (F2n being identified with F2n and an inner product being x y=tr(xy) where tr is the trace function). Several recent infinite classes of APN functions have been proved CCZ-inequivalent to power functions. In this paper, we describe the state of the art in the domain and we also present original results. We indicate what are the most important open problems and make some new observations about them. Many results presented are from joint works with Lilya Budaghyan, Gregor Leander and Alexander Pott.

A tree-shellable function is a positive Boolean function which can be represented by a binary decision tree whose number of paths from the root to a leaf labeled 1 equals the number of prime implicants. In this paper, we consider the tree-shellability of DNFs with restrictions. We show that, for read-k DNFs, the number of terms in a tree-shellable function is at most k2. We also show that, for k-DNFs, recognition of ordered tree-shellable functions is NP-complete for k=4 and tree-shellable functions can be recognized in polynomial time for constant k.

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.

Horn functions are Boolean functions where each of the prime implicants contains at most one negative literal. A class of Boolean functions is considered in this letter where a single term containing two negative literals is added by logical-or operation to a Horn function. We show that the function does not have any prime implicant containing three negative literals. We also show that if two terms containing two negative literals are added to a Horn function, then it may have many prime implicants all of which contain three negative literals. We show that it is P-complete to determine whether a given Boolean function in disjunctive normal form of the considered class is a tautology.

In this study, we construct balanced Boolean functions with a high nonlinearity and an optimum algebraic degree for both odd and even dimensions. Our approach is based on modifying functions from the Maiorana-McFarland's superclass, which has been introduced by Carlet. A drawback of Maiorana-McFarland's function is that their restrictions obtained by fixing some variables in their input are affine. Affine functions are cryptographically weak functions, so there is a risk that this property will be exploited in attacks. Due to the contribution of Carlet, our constructions do not have the potential weakness that is shared by the Maiorana-McFarland construction or its modifications.

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.

In cryptography, we want a Boolean function which satisfies PC(l) of order k for many (l,k). Let PCn(l,k) be a set of Boolean functions with n input bits satisfying PC(l) of order k. From a view point of construction, it is desirable that there exists (l0,k0) such that PCn(l0, k0) PCn(li,ki) for many i 1. In this paper, we show a negative result for this problem. We prove that PCn(l1,k1) PCn(l2,k2) for a large class of l1, k1, l2 and k2.

In this paper, we consider the complexity of recognizing ordered tree-shellable Boolean functions when Boolean functions are given as OBDDs. An ordered tree-shellable function is a positive Boolean function such that the number of prime implicants equals the number of paths from the root node to a 1-node in its ordered binary decision tree representation. We show that given an OBDD, it is possible to check within polynomial time if the function is ordered tree-shellable with respect to the variable ordering of the OBDD.

In several design methods for Pass-transistor Logic (PTL) circuits, Boolean functions are expressed as OBDDs in decomposed form and then the component OBDDs are directly mapped to PTL cells. The total size of OBDDs (number of nodes) corresponds to the circuit size. In this paper, we investigate a method for PTL synthesis based on exact minimization of Free BDDs (FBDDs). FBDDs are well-studied extension of OBDDs with free variable ordering on each path. We present statistics showing that more than 56% of 616126 NPN-equivalence classes of 5-variable Boolean functions have minimum FBDDs with less size than their OBDDs. This result can be used for PTL synthesis as libraries. We also applied the exact minimization algorithm of FBDDs to the minimization of subcircuits in the synthesis for MCNC benchmarks and found up to 5% size reduction.

Some novel learning strategies based on set covering in Hamming geometrical space are presented and proved, which are related to the three-layer Boolean neural network (BNN) for implementing an arbitrary Boolean function with low-complexity. Each hidden neuron memorizes a set of learning patterns, then the output layer combines these hidden neurons for explicit output as a Boolean function. The network structure is simple, reliable and can be easily implemented by hardware.

In this paper we study n-input m-output Boolean functions (abbr. (n,m)-functions) with high nonlinearity. First, we present a basic construction method for a balanced (n,m)-function based on a primitive element in GF(2m). With an iterative procedure, we improve some lower bounds of the maximum nonlinearity of balanced (n,m)-functions. The resulting bounds are larger than the maximum nonlinearity achieved by any previous construction method for (n,m)-functions. Finally, our basic method is developed to construct an (n,m)-bent function and discuss its maximum algebraic degree.

This paper proposes a simple learning algorithm that can realize any boolean function using the three-layer binary neural networks. The algorithm has flexible learning functions. 1) moving "core" for the inputs separations,2) "don't care" settings of the separated inputs. The "don't care" inputs do not affect the successive separations. Performing numerical simulations on some typical examples, we have verified that our algorithm can give less number of hidden layer neurons than those by conventional ones.

An Ordered Binary Decision Diagram (OBDD) is a directed acyclic graph representing a Boolean function. The size of OBDDs largely depends on the variable ordering. In this paper, we show the size of the OBDD representing the i-th bit of the output of n-bit/n-bit integer division is Ω ( 2(n-i)/8 ) for any variable ordering. We also show that -OBDDs, -OBDDs and -OBDDs representing integer division has the same lower bounds on the size. We develop new methods for proving lower bounds on the size of -OBDDs, -OBDDs and -OBDDs.

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.

Ordered Binary Decision Diagrams (OBDDs) are graph-based representations of Boolean functions which are widely used because of their good properties. In this paper, we introduce nondeterministic OBDDs (NOBDDs) and their restricted forms, and evaluate their expressive power. In some applications of OBDDs, canonicity, which is one of the good properties of OBDDs, is not necessary. In such cases, we can reduce the required amount of storage by using OBDDs in some non-canonical form. A class of NOBDDs can be used as a non-canonical form of OBDDs. In this paper, we focus on two particular methods which can be regarded as using restricted forms of NOBDDs. Our aim is to show how the size of OBDDs can be reduced in such forms from theoretical point of view. Firstly, we consider a method to solve satisfiability problem of combinational circuits using the structure of circuits as a key to reduce the NOBDD size. We show that the NOBDD size is related to the cutwidth of circuits. Secondly, we analyze methods that use OBDDs to represent Boolean functions as sets of product terms. We show that the class of functions treated feasibly in this representation strictly contains that in OBDDs and contained by that in NOBDDs.

An ordered binary decision diagram (OBDD) is a directed acyclic graph for representing a Boolean function. OBDDs are widely used in various areas which require Boolean function manipulation, since they can represent efficiently many practical Boolean functions and have other desirable properties. However, there is very little theoretical research on the complexity of constructing an OBDD. In this paper, we prove that the optimal variable ordering problem of a shared BDD is NP-complete, and briefly discuss the approximation hardness of this problem and related OBDD problems.

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.

The present paper is concerned with an algorithm for the minimization of multiple-valued input, binary-valued output functions. The algorithm is an extension to muitiple-valued logic of an algorithm for the minimization of ordinary single-output Boolean functions. It is based on a local covering approach. Basically, it uses a "divide and conquer" technique, consisting of two steps called expansion and selection. The present algorithm preserves two important features of the original one. First, a lower bound on the number of prime implicants in the minimum cover of the given function is furnished as a by-product of the minimization. Second, all the essential primes of the function are identified and selected during the expansion process. That usually improves efficiency when handling functions with many essential primes. Results of a comparison of the proposed algorithm with the program ESPRESSO-IIC developed at Berkeley are presented.