26,554 research outputs found
Equivalence of operations with respect to discriminator clones
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
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
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 -Variable Functions with Cards
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 -variable Boolean function requires at least cards. We are
interested in optimal protocols that use exactly 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 -card
protocols to compute -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 , , , and
Heuristic search of (semi-)bent functions based on cellular automata
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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