In recent years many papers have derived polyhedral and non-polyhedral convex envelopes for different classes of functions. Except for the univariate cases, all these classes of functions share the property that the g...
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In recent years many papers have derived polyhedral and non-polyhedral convex envelopes for different classes of functions. Except for the univariate cases, all these classes of functions share the property that the generating set of their convex envelope is a subset of the border of the region over which the envelope is computed. In this paper we derive the convex envelope over a rectangular region for a class of functions which does not have this property, namely the class of bivariate cubic functions without univariate third-degree terms.
This paper estimates the integral I-f((p)) = integral(infinity)(-infinity)(-p) dx, where p > 1, f(x) is a polynomial and := 1+ vertical bar center dot vertical bar. When f(x) = x(2) + a(1)x + a(0) is a quadratic f...
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This paper estimates the integral I-f((p)) = integral(infinity)(-infinity)< f (x)>(-p) dx, where p > 1, f(x) is a polynomial and
:= 1+ vertical bar center dot vertical bar. When f(x) = x(2) + a(1)x + a(0) is a quadratic function, it is well known that I-f((p)) approximate to p{1/(1/2) if Delta 2 >= 0, 1/(p-1/2) if Delta(2) < 0, where Delta(2) := a(1)(2) - 4a(0) is the discriminant of f(x). But when f(x) = x(3) + b(1)x + b(0) is a cubic function, it is much more challenging to give a precise estimate of I-f((p)). In fact, the discriminant alone is not enough to quantify I-f((p)) accurately. Denote Delta(3) to be the discriminant of f and define lambda = vertical bar b(1)vertical bar(3) + b(0)(2). By delicately analyzing how Delta(3) and how the roots of f affect I-f((p)), we conclude I-f((p)) approximate to p 1/, where phi is some function in Delta(3) and lambda with an explicit formula. (C) 2021 Elsevier Ltd. All rights reserved.
We consider the problem of identifying the classes of Boolean functions having high second-order non-linearities. In this paper, we demonstrate that the cubic bent functions obtained by Leander and McGuire (J. Combin....
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We consider the problem of identifying the classes of Boolean functions having high second-order non-linearities. In this paper, we demonstrate that the cubic bent functions obtained by Leander and McGuire (J. Combin. Theory Ser. A, 116 (2009), pp. 960-970), which are concatenations of the quadratic Gold functions, possess high second-order nonlinearities.
Plateaued functions play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Comparing to their importance, those functions have not been studied in detail in a...
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Plateaued functions play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Comparing to their importance, those functions have not been studied in detail in a general framework. Our motivation is to bring further results on the characterizations of bent and plateaued functions, and to introduce new tools which allow us firstly a better understanding of their structure and secondly to get methods for handling and designing such functions. We first characterize bent functions in terms of all even moments of the Walsh transform, and then plateaued (vectorial) functions in terms of the value distribution of the second-order derivatives. Moreover, we devote to cubic functions the characterization of plateaued functions in terms of the value distribution of the second-order derivatives, and hence this reveals non-existence of homogeneous cubic bent (and also (homogeneous) cubic plateaued for some cases) functions in odd characteristic. We use a rank notion which generalizes the rank notion of quadratic functions. This rank notion reveals new results about (homogeneous) cubic plateaued functions. Furthermore, we observe non-existence of a function whose absolute Walsh transform takes exactly 3 distinct values (one being zero). We finally provide a new class of functions whose absolute Walsh transform takes exactly 4 distinct values (one being zero).
Natural numbers that defy decomposition into factors lower than themselves provide the basis for entrenching security in the public key encryption world. A variant algorithm of trial division of sieve of Eratosthenes,...
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ISBN:
(纸本)9781509025497
Natural numbers that defy decomposition into factors lower than themselves provide the basis for entrenching security in the public key encryption world. A variant algorithm of trial division of sieve of Eratosthenes, with its digits split into two unequal components and whose growth centered on displacement from its immediate neighbors (i.e., Polignac and Goldbach theorems) was fashioned out. As challenged by Euler's verdict on non-existence of a function that completely generates only primes, our formulation was based both upon multiple linear regression analysis hoping to find a multi-variate function, of degree at most 3, which can predict primes and comparison to Akaike Information Criterion (AIC) for model selection. About 20 billion lower primes of digits less than or equal 12 were subjected to various validation techniques (e.g., Sloane's A006988) and other heuristics for benchmarking purposes. Our results reveal certain desirable features illuminating prime patterns regardless of their chaotic camouflage.
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