A quantified Boolean formula is true, if for any existentially quantified variable there exists a Boolean function depending on the preceding universal variables, such that substituting the existential variables by the Boolean functions results in a true formula. We call a satisfying set of Boolean functions a model. In this paper, we investigate for various classes of quantified Boolean formul...

We de ne the notions of reducibility and completeness in (two party and multi-party) private computations. Let g be an n-argument function. We say that a function f is reducible to a function g if n honest-but-curious players can compute the function f n-privately, given a black-box for g (for which they secretly give inputs and get the result of operating g on these inputs). We say that g is c...

Linear algebra is introduced by Faugère in F4 to speed up the reduction procedure during Gröbner basis computations. Linear algebra has also been used in fast implementations of F5 and other signature-based Gröbner basis algorithms. To use linear algebra for reductions, an important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function...

This paper develops theoretical foundations and presents a practical algorithm for optimizing multi-output Boolean function M(x) whose outputs are combined using Boolean OR operator into a single-output Boolean function S(x). The proposed algorithm simplifies the logic structure of function M(x) and may change or remove some of its outputs, while preserving the functionality of function S(x). A...

– This paper gives a new and efficient method of determining the equivalence of two given Boolean functions. We define a characteristic polynomial directly from the sum-of-product form of the logic function. The polynomial contains a real variable corresponding to each Boolean variable. Logical operations on the Boolean function correspond to arithmetic operations on the polynomial. We show tha...

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...

The (fast) algebraic immunity, including (standard) algebraic immunity and the resistance against fast algebraic attacks, has been considered as an important cryptographic property for Boolean functions used in stream ciphers. This paper is on the determination of the (fast) algebraic immunity of a special class of Boolean functions, called Boolean power functions. An n-variable Boolean power f...

The decision tree complexity of a boolean function F of n arguments is the depth of a minimum-depth decision tree that computes F correctly on every input. A boolean function F of n arguments is cn-evasive if its decision tree complexity is cn, (c < 1). F is completely evasive, (n-evasive), if all of its arguments need to be probed in the worst case. When we restrict the properties and form of ...

K. Y. Sill Information Systems Laboratory Stanford University Stanford, CA, 94305 T. Kailath Informat.ion Systems Laboratory Stanford U ni versity Stanford, CA, 94305 '~le introduce a geometric approach for investigating the power of threshold circuits. Viewing n-variable boolean functions as vectors in 'R'2", we invoke tools from linear algebra and linear programming to derive new results on t...

The minimum number of NOT gates in a logic circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that ⌈log 2 (d(f)+ 1)⌉ NOT gates are necessary and sufficient to compute any Boolean function f (where d(f) is the maximum number of value changes from greater to smaller...