Monotone Boolean functions

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Blocks of Monotone Boolean Functions

Blocks of Monotone Boolean Functions

Abstract This paper first proposed the method of constructing blocks of monotone Boolean functions (MBFs) is developed for classification and the analysis of these functions. Use of only nonisomorphic blocks considerably simplifies enumeration MBFs. Application of the method of construction blocks on the example of classification and analysis of MBFs from 0 to 4 variables is considered. This method can be used at counting of all MBFs of a given rank n.

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Blocks of Monotone Boolean Functions of Rank 5

Blocks of Monotone Boolean Functions of Rank 5

Abstract Based on the classification of monotone Boolean functions (MBFs) on the types and the method of building MBFs blocks, an analysis is conducted of the MBFs rank 5. Four matrixes for such MBFs are adduced. It is shown that there are 7581 MBFs rank 5, 276 of them are MBFs of maximal types. These 7581 MBFs are contained in 522 blocks or 23 groups of isomorphic blocks or 6 groups of similar blocks. The offered methods can be used to analyze large MBFs ranks. In previous articles were shown how MBFs used in telecommunications for analyzing networks and building codes for cryptosystems.
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Monotone Boolean functions as combinatorially piecewise linear maps

Monotone Boolean functions as combinatorially piecewise linear maps

This paper describes how a monotone boolean function in n variables whose prime implicants and prime clauses are non-rivial defines a partition of the symmetric group on n symbols into a[r]

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Exponential lower bounds and separation for query rewriting

Exponential lower bounds and separation for query rewriting

Abstract. We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of first- order and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove similar results for the size of rewritings that do not use non-signature constants. For example, we show that, in the worst case, positive existential and nonrecursive Datalog rewritings are exponentially longer than the original queries; nonrecursive Datalog rewritings are in general exponentially more succinct than positive exis- tential rewritings; while first-order rewritings can be superpolynomially more succinct than positive existential rewritings.
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Optimal Cryptographic Hardness of Learning Monotone Functions

Optimal Cryptographic Hardness of Learning Monotone Functions

In this section we first present a simple variant of the lower bound construction in [6], obtaining an information-theoretic lower bound on the learnability of the general class of all monotone Boolean functions. The quantitative bound our variant achieves is weaker than that of [6], but has the advantage that it can be easily derandomized. Indeed, as mentioned in Section 5 (and further discussed below), our construction uses a certain probability distribution over monotone DNFs, such that a typical random input x satisfies only poly(n) many “candidate terms” (which are terms that may be present in a random DNF drawn from our distribution). By selecting terms for inclusion in the DNF in a pseudorandom rather than truly random way, we obtain a class of poly(n)-size monotone circuits that is hard to learn to accuracy 1/2 + 1/polylog(n) (assuming one-way functions exist).
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Recent  Results  on  Balanced  Symmetric  Boolean  Functions

Recent Results on Balanced Symmetric Boolean Functions

Balancedness is a primary requirement to resist the attacks on each cryptosystem. In [8], Gathen and Rouche found all the balanced symmetric boolean functions up to 128 variables. Canteaut et al. proved that balanced symmetric functions of degree less than or equal to 7(excluding the trivial cases) only exist for eight variables [1]. Since the number of nontrivial balanced functions seems to be very small, they conjectured that balanced symmetric functions

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On  the  Classification  of  Finite  Boolean  Functions  up  to  Fairness

On the Classification of Finite Boolean Functions up to Fairness

are computable with fairness. These functions roughly correspond to the Mil- lionaire’s Problem (i.e. f (x, y) = 0 ⇔ x ≥ y). The authors also show that even certain XOR-embedded functions are fair. Using the real/ideal world paradigm, they design a protocol (that we henceforth refer to as the GHKL-protocol) and show that for certain functions, any real world adversary can be simulated in the ideal model. Thus proving that in an execution of the GHKL-protocol, the adversary cannot gain any advantage in learning the output over the honest party. On the other hand, in [5], Asharov et al give a complete characterization of finite Boolean functions that are not fair due to the coin-tossing criterion. In other words, there exists a class of functions (strictly balanced functions) that cannot be computed fairly, since the realisation of any such function yields a completely fair coin-toss. What’s more, they show that functions that are not in that class cannot be reduced to the coin-tossing problem, meaning that any new negative results with respect to fairness must rely on different criteria.
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The  Number  of  Boolean  Functions  with  Multiplicative  Complexity 2

The Number of Boolean Functions with Multiplicative Complexity 2

Abstract. Multiplicative complexity is a complexity measure defined as the minimum number of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementa- tions of ciphers with a small number of AND gates are preferred in pro- tocols for fully homomorphic encryption, multi-party computation and zero-knowledge proofs. In 2002, Fischer and Peralta [12] showed that the number of n-variable Boolean functions with multiplicative complexity one equals 2 2 3 n . In this paper, we study Boolean functions with multi- plicative complexity 2. By characterizing the structure of these functions in terms of affine equivalence relations, we provide a closed form formula for the number of Boolean functions with multiplicative complexity 2. Keywords:Affine equivalence; Boolean functions; Cryptography; Mul- tiplicative complexity; Self-mappings
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A  note  on  generalized  bent  criteria  for  Boolean  functions

A note on generalized bent criteria for Boolean functions

Abstract—In this paper, we consider the spectra of Boolean functions with respect to the action of uni- tary transforms obtained by taking tensor products of the Hadamard kernel, denoted by H , and the nega– Hadamard kernel, denoted by N. The set of all such transforms is denoted by {H, N} n

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Minimization of Boolean Functions Using Genetic Algorithm

Minimization of Boolean Functions Using Genetic Algorithm

Each adjacency may include some of statements and refuse the others. For calculation of total minimized form using supposed adjacency, we must consider whole Boolean function except the statements that are included in supposed adjacency. Instead of them, we add the minimized form of adjacency statements. Fore sure, one of the chromosomes in the last generation contains the best adjacencies. So, after finishing genetic algorithm process, we must calculate the minimized form based on the chromosomes of last generation. After reducing the repeated ones, the best minimized form that is the shortest one will be appeared.
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State of the ART in Boolean Functions Cryptographic Assessment

State of the ART in Boolean Functions Cryptographic Assessment

constant. Bent Functions exist only for even dimensions (n-even), and only if m ≤ n/2. Obviously Bent functions satisfy the nonlinearity upper bound, due to their Walsh transform coefficients, but unfortunately Bent functions are surjective unbalanced functions, so they should be carefully used cipher design. But fortunately any n- variable Bent function satisfies PC(n).

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Computational techniques for nonlinear codes and Boolean functions

Computational techniques for nonlinear codes and Boolean functions

If C GB ( A ) is the omplexity of solving a Gröbner basis using algorithm A , then the omplexity of the hybrid approah is learly.. O ( q r C GB( A )).[r]

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The Complexity of Boolean Functions   Ingo Wegener pdf

The Complexity of Boolean Functions Ingo Wegener pdf

It is easy to verify the correctness of the replacement rules, but it is difficult and more important to get a feeling why such replacement rules work. Let g be computed in a monotone circuit for f . If t ∈ PI(g) but t t ∈ PI(f) for all monoms t (including the empty monom), t is of no use for the computation of f . At ∧ -gates t can only be lengthened. At ∨ -gates either t is saved or t is eliminated by the law of simplification. Because of the conditions on t we have to eliminate t and all its lengthenings. Hence it is reasonable to conjecture that g can be replaced by h where PI(h) = PI(g) − { t } . If all prime implicants of f have a length of at most k and if all prime implicants of g have a length of at least k + 1 , we can apply the same replacement rule several times and can replace g by the constant 0 .
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On  Hardware  Implementation  of  Tang-Maitra  Boolean  Functions

On Hardware Implementation of Tang-Maitra Boolean Functions

For ASIC, we have used Synopsys Design Compiler for both the Open Source Nangate 45nm and TSMC 65nm Standard Cell Libraries, given in Tables 3 and 4. The implementations from [PCZ17] have also been synthesized for the same technology. The ASIC results also follow the theoretical estimations. However, compared to [PCZ17], the cost of the Tang-Maitra functions is higher in terms of performance, due to the depth of linear order. With respect to area, the Tang- Maitra functions show a quadratic growth rate, compared to the sub-exponential rate in [PCZ17]. In addition, while the Tang-Maitra ASIC implementations are slower than the GMM ASIC implementations, the clock frequency for 22 vari- ables is still more than 100 MHz, which is faster than the speed required by many practical applications. The hardware results in this paper combined with the cryptographic properties of the Tang-Maitra functions show that they can be a building block for promising cryptographic primitives. Besides, we have not studied the effect of advanced circuit optimization techniques on the ASIC implementations, which can further improve both the area and performance.
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On  the (Fast)  Algebraic  Immunity  of  Boolean  Power  Functions

On the (Fast) Algebraic Immunity of Boolean Power Functions

Abstract 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 function f can be represented as a monomial trace function over finite field F 2 n , f (x) = T r n

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Linear read once and related Boolean functions

Linear read once and related Boolean functions

Our result about the minimal threshold non-linear read-once functions seems to support this conjecture, since all these functions have specication number 2n . One more result proved in this paper, which can be viewed as a supporting argument for the conjecture, states that a positive function f depending on n variables has exactly n + 1 extremal points if and only if it is linear read-once. Nevertheless, rather surprisingly, we show that the conjecture is not true by exhibiting a positive threshold non-linear read-once function depending on n variables whose specication number is n + 1 .
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Monotone and convex H* algebra valued functions

Monotone and convex H* algebra valued functions

H*-algebra, trace-class, Hilbert Schmidt operators, monotone functions, saltus function, convex function, Stieltjes integral, Borel measure.. 1991 AMS SUBJECT CLASSIFICATION CODES.[r]

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Monotone local linear estimation of transducer functions

Monotone local linear estimation of transducer functions

Local polynomial regression has received a great deal of attention in the past. It is a highly adaptable regression method when the true response model is not known. However, estimates obtained in this way are not guaranteed to be monotone. In some situations the response is known to depend monotonically upon some variables. Various methods have been suggested for constraining nonparametric local polynomial regression to be monotone. The earliest of these is known as the Pool Adjacent Violators algorithm (PAVA) and was first suggested by Brunk [4]. Kappenman [29] suggested that a non-parametric estimate could be made monotone by simply increasing the bandwidth used until the estimate was monotone. Dette et al. [9] have suggested a monotonicity constraint which they call the DNP method. Their method involves calculating a density estimate of the unconstrained regression estimate, and using this to calculate an estimate of the inverse of the regression function.
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Existence Result for Nonlinear Initial Value Problems Involving the Difference of Two Monotone Functions

Existence Result for Nonlinear Initial Value Problems Involving the Difference of Two Monotone Functions

uniformly and monotonically on J and the functions v(t) and w(t) are the coupled minimal and maximal solutions of nonlinear IVP (2.2) − (2.3) respectively. Proof : Consider the following coupled linear system of fractional differential e- quations with initial conditions (LIVP)

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Implementing Groundness Analysis with Definite Boolean Functions

Implementing Groundness Analysis with Definite Boolean Functions

The eciency of groundness analysis depends critically on the way dependen- cies are represented. C and Prolog based Def analysers have been constructed around two representations: (1) Armstrong et al [1] argue that Dual Blake Canon- ical Form (DBCF) is suitable for representing Def . This represents functions as conjunctions of denite (propositional) clauses [12] maintained in a normal (orthogonal) form that makes explicit transitive variable dependencies. For ex- ample, the function ( x y ) ^ ( y z ) is represented as ( x ( y _ z )) ^ ( y z ).

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