26,554 research outputs found

    Equivalence of operations with respect to discriminator clones

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    For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.Comment: 17 page

    Бульові алгебри на базі неперервних і неперервно-диференційовних функций

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    В даній роботі на множинах неперервних і неперервно-диференційовних функцій встановлюється відношення еквівалентності, яке розбиває ці множини на класи еквівалентності H₀, Hm. В доповнення до алгебро-логічних побудов, започаткованих раніше, показано, як на множинах класів еквівалентності H₀, Hm можна побудувати бульові алгебри, ізоморфні бульовій алгебрі множин простору Rⁿ.In this work the relation of equivalence is determine on sets of continuous and continuously differentiable functions, which breaks up these sets on the classes of equivalence H₀, Hm. In addition to algebro-logical constructions, founded before, it is shown, how on the sets of classes of equivalence of H₀, Hm can be built Boolean algebra isomorphic Boolean algebra of sets of Rⁿ space

    Бульові алгебри на базі неперервних і неперервно-диференційовних функций

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    В даній роботі на множинах неперервних і неперервно-диференційовних функцій встановлюється відношення еквівалентності, яке розбиває ці множини на класи еквівалентності H₀, Hm. В доповнення до алгебро-логічних побудов, започаткованих раніше, показано, як на множинах класів еквівалентності H₀, Hm можна побудувати бульові алгебри, ізоморфні бульовій алгебрі множин простору Rⁿ.In this work the relation of equivalence is determine on sets of continuous and continuously differentiable functions, which breaks up these sets on the classes of equivalence H₀, Hm. In addition to algebro-logical constructions, founded before, it is shown, how on the sets of classes of equivalence of H₀, Hm can be built Boolean algebra isomorphic Boolean algebra of sets of Rⁿ space

    The Landscape of Computing Symmetric nn-Variable Functions with 2n2n Cards

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    Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Many card-based protocols to securely compute various Boolean functions have been developed. As each input bit is typically encoded by two cards, computing an nn-variable Boolean function requires at least 2n2n cards. We are interested in optimal protocols that use exactly 2n2n cards. In particular, we focus on symmetric functions, where the output only depends on the number of 1s in the inputs. In this paper, we formulate the problem of developing 2n2n-card protocols to compute nn-variable symmetric Boolean functions by classifying all such functions into several NPN-equivalence classes. We then summarize existing protocols that can compute some representative functions from these classes, and also solve some of the open problems by developing protocols to compute particular functions in the cases n=4n=4, 55, 66, and 77

    Heuristic search of (semi-)bent functions based on cellular automata

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    An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata (CA), focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter
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