In many applications that involve identification of an unknown monotone boolean function (MBF), the cost of inferring the value of a vector using monotonicity is negligible compared to the cost of querying its value. ...
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The algebraic immunity of boolean functions is studied in this paper. More precisely, having the prominent Carlet-Feng construction as a starting point, we propose a new method to construct a large number of functions...
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The algebraic immunity of boolean functions is studied in this paper. More precisely, having the prominent Carlet-Feng construction as a starting point, we propose a new method to construct a large number of functions with maximum algebraic immunity. The new method is based on deriving new properties of minimal codewords of the punctured Reed-Muller code for any n, allowing-if n is odd-for efficiently generating large classes of new functions that cannot be obtained by other known constructions. It is shown that high nonlinearity, as well as good behavior against fast algebraic attacks, is also attainable.
In this paper we study relationships between CNF representations of a given boolean function f and essential sets of implicates off. It is known that every CNF representation and every essential set must intersect. Th...
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In this paper we study relationships between CNF representations of a given boolean function f and essential sets of implicates off. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets off provides a lower bound on the size of any CNF representation off. We are interested in functions, for which this lower bound is tight, and call such functions covetable. We prove that for every covetable function there exists a polynomially verifiable certificate (witness) for its minimum CNF size. On the other hand, we show that not all functions are covetable, and construct examples of non-covetable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is NP-hard under various restrictions on the function and on its input representation. (c) 2011 Elsevier B.V. All rights reserved.
Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). boolean functions with MC 1 and 2 have been characterized in...
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Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). boolean functions with MC 1 and 2 have been characterized in Fisher and Peralta (2002), and Find et al. (IJICoT4(4), 222-236,2017), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order to achieve this, we utilize the notion of the dimensiondim(f) of a boolean function in relation to its linearity dimension, and provide a new lower bound suggesting that the multiplicative complexity offis at least left ceiling dim(f)/2 right ceiling . For MC 3, this implies that there are no equivalence classes other than those 24 identified in calik et al. (2018). Using the techniques from calik et al. and the new relation between the dimension and MC, we identify all 1277 equivalence classes having MC 4. We also provide a closed formula for the number ofn-variable functions with MC 3 and 4. These results allow us to construct AND-optimal circuits for boolean functions that have MC 4 or less, independent of the number of variables they are defined on.
This paper describes a fundamental correspondence between boolean functions and projection operators in Hilbert space. The correspondence is widely applicable, and it is used in this paper to provide a common mathemat...
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This paper describes a fundamental correspondence between boolean functions and projection operators in Hilbert space. The correspondence is widely applicable, and it is used in this paper to provide a common mathematical framework for the design of both additive and nonadditive quantum error correcting codes. The new framework leads to the construction of a variety of codes including an infinite class of codes that extend the original ((5, 6, 2)) code found by Rains et al It also extends to operator quantum error correcting codes.
Let f : F-2(n) ->{0, 1} be a boolean function, and suppose that the spectral norm parallel to f parallel to(A) := Sigma(r) vertical bar(f) over cap (r)vertical bar of f is at most M. Then [GRAPHICS] where L <= 2...
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Let f : F-2(n) ->{0, 1} be a boolean function, and suppose that the spectral norm parallel to f parallel to(A) := Sigma(r) vertical bar(f) over cap (r)vertical bar of f is at most M. Then [GRAPHICS] where L <= 2(2CM4) and each H-j is a subgroup of F-2(n). This result may be regarded as a quantitative analogue of the Cohen Helson - Rudin structure theorem for idempotent measures in locally compact abelian groups.
We study boolean functions derived from Fermat quotients modulo p using the Legendre symbol. We prove bounds on several complexity measures for these boolean functions: the nonlinearity, sparsity, average sensitivity,...
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We study boolean functions derived from Fermat quotients modulo p using the Legendre symbol. We prove bounds on several complexity measures for these boolean functions: the nonlinearity, sparsity, average sensitivity, and combinatorial complexity. Our main tools are bounds on character sums of Fermat quotients modulo p.
We study the behaviour of the algebraic degree of vectorial boolean functions when their inputs are restricted to an affine subspace of their domain. functions which maintain their degree on all subspaces of as high a...
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Extending Sparks’s theorem, we determine the cardinality of the lattice of (C1, C2)-clonoids of boolean functions for certain pairs (C1, C2) of clones of boolean functions. Namely, when C1 is a subclone (a proper sub...
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For each boolean function in n variables, from the expression of the product of all its Walsh spectrum values derived in a precedent paper, we deduce a new characterization of the parity of its distance from the set o...
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For each boolean function in n variables, from the expression of the product of all its Walsh spectrum values derived in a precedent paper, we deduce a new characterization of the parity of its distance from the set of all the affine functions. This characterization uses a subset of permutations on F-2(n), and some new properties on this subset are deduced.
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