boolean functions are often represented by ordered binary-decision diagrams (OBDD's) introduced by Bryant. Liaw and Lin have proved upper and lower bounds on the minimal OBDD size of almost all boolean functions. ...
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boolean functions are often represented by ordered binary-decision diagrams (OBDD's) introduced by Bryant. Liaw and Lin have proved upper and lower bounds on the minimal OBDD size of almost all boolean functions. Now tight bounds are proved for the minimal OBDD size for arbitrary or optimal variable orderings and for the minimal read-once branching program size of almost all functions. Almost all boolean functions have a sensitivity of almost 1, i.e., the minimal OBDD size for an optimal variable ordering differs from the minimal OBDD size for a worst variable ordering by a factor of at most 1 + epsilon(n) where epsilon(n) converges exponentially fast to 0.
Unnormalized Haar spectra and Ordered Binary Decision Diagrams (OBDDs) are two standard representations of boolean functions used in logic design. In this article, mutual relationships between those two representation...
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Unnormalized Haar spectra and Ordered Binary Decision Diagrams (OBDDs) are two standard representations of boolean functions used in logic design. In this article, mutual relationships between those two representations have been derived. The method of calculating the Haar spectrum from OBDD has been presented. The decomposition of the Haar spectrum, in terms of the cofactors of boolean functions, has been introduced. Based on the above decomposition, another method to synthesize OBDD directly from the Haar spectrum has been presented.
A quantum algorithm to determine approximations of linear structures of boolean functions is presented and analysed. Similar results have already been published (see Simon's algorithm) but only for some promise ve...
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A quantum algorithm to determine approximations of linear structures of boolean functions is presented and analysed. Similar results have already been published (see Simon's algorithm) but only for some promise versions of the problem, and it has been shown that no exponential quantum speedup can be obtained for the general (no promise) version of the problem. In this paper, no additional promise assumptions are made. The approach presented is based on the method used in the Bernstein-Vazirani algorithm to identify linear boolean functions and on ideas from Simon's period finding algorithm. A proper combination of these two approaches results here to a polynomial-time approximation to the linear structures set. Specifically, we show how the accuracy of the approximation with high probability changes according to the running time of the algorithm. Moreover, we show that the time required for the linear structure determine problem with high success probability is related to so called relative differential uniformity delta(f) of a boolean function f. Smaller differential uniformity is, shorter time is needed.
We characterise the aperiodic autocorrelation for a boolean function, f, and define the Aperiodic Propagation Criteria (APC) of degree l and order q. We establish the strong similarity between APC and the Extended Pro...
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We characterise the aperiodic autocorrelation for a boolean function, f, and define the Aperiodic Propagation Criteria (APC) of degree l and order q. We establish the strong similarity between APC and the Extended Propagation Criteria as defined by Preneel et al. in 1991, although the criteria are not identical. We also show how aperiodic autocorrelation can be related to the first derivative off. We further propose the metric APC distance and show that quantum error correcting codes are natural candidates for boolean functions with favourable APC distance. (C) 2006 Elsevier Inc. All rights reserved.
We study a procedure for estimating an upper bound of an unknown noise factor in the frequency domain. A learning algorithm using a Fourier transformation method was originally given by Linial, Mansour and Nisan. Whil...
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We study a procedure for estimating an upper bound of an unknown noise factor in the frequency domain. A learning algorithm using a Fourier transformation method was originally given by Linial, Mansour and Nisan. While Linial, Mansour and Nisan assumed that the learning algorithm estimates Fourier coefficients from noiseless data, Bshouty, Jackson, and Tamon, and also Ohtsuki and Tomita extended the algorithm to ones that are robust for noisy data. The noise process that we consider is as follows: for an example < x, f (x)>, where x is an element of {0, 1}(n), f(x) is an element of {-1, 1}, each bit of x and f (x) gets flipped independently with probability eta during a learning process. The previous learning algorithms for noisy data all assume that the noise factor eta or an upper bound of eta is known in advance. The learning algorithm proposed in this paper works without this assumption. We estimate an upper bound of the noise factor by evaluating a noisy power spectrum in the frequency domain and by using a sampling trick. Combining this procedure with Ohtsuki and Tomita's algorithm, we obtain a quasi-polynomial-time learning algorithm that can cope with noise without knowing any information about the noise in advance. (C) 2011 Elsevier B.V. All rights reserved.
Identification of disjunctive decomposition in a balanced boolean function through Walsh spectra is performed. Balanced boolean functions occur frequently iii common logic circuits such as adders, parity checkers and ...
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Identification of disjunctive decomposition in a balanced boolean function through Walsh spectra is performed. Balanced boolean functions occur frequently iii common logic circuits such as adders, parity checkers and multiplexers. Various types of decomposition of the balanced functions are considered and the corresponding Walsh spectral conditions that have to be satisfied for their existence are listed.
A necessary condition for the security of cryptographic functions is to be "sufficiently distant" from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that...
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A necessary condition for the security of cryptographic functions is to be "sufficiently distant" from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that six common measures, nonlinearity, algebraic degree, annihilator immunity, algebraic thickness, normality, and multiplicative complexity, are incomparable in the sense that for each pair of measures, mu(1), mu(2), there exist functions f(1), f(2) with f(1) being more nonlinear than f(2) according to mu(1), but less nonlinear according to mu(2). We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative complexity of collision-free functions.
We give an explicit formula for the number of n-variable clique functions 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 ...
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We give an explicit formula for the number of n-variable clique functions 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.
One well-known method of generating key stream sequences for stream ciphers is to combine the outputs of several linear-feedback shift registers (LFSR) using a combining boolean function. Here we concentrate on the de...
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One well-known method of generating key stream sequences for stream ciphers is to combine the outputs of several linear-feedback shift registers (LFSR) using a combining boolean function. Here we concentrate on the design of good combining boolean functions. We provide resilient boolean functions with currently best known nonlinearity. These functions were not known earlier and the issues related to their existence were posed as open questions in the literature. Some of the functions we construct here achieve the provable upper bound on nonlinearity for resilient boolean functions. Our technique interlinks mathematical results with classical computer search.
A new algorithm is given that converts a reduced representation of boolean functions in the form of disjoint cubes to sign Walsh spectra. Since the known algorithms that generate sign Walsh spectra always start from t...
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A new algorithm is given that converts a reduced representation of boolean functions in the form of disjoint cubes to sign Walsh spectra. Since the known algorithms that generate sign Walsh spectra always start from the truth table of boolean functions, the method presented computes faster with a smaller computer memory. The method is especially efficient for such boolean functions that are described by only few disjoint cubes. (C) 2002 Elsevier Science Ltd. All rights reserved.
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