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Research Interests: Mathematics, Applied Mathematics, Computer Science, Iterative Methods, Convergence, and 11 moreIntegral Equations, Nonlinear Systems, Mathematical Sciences, Cauchy Problem, Computers and Mathematics with Applications 59 (2010) 35783582, Recurrence Relation, Linear Equations, Nonlinear Equations, Order of Convergence, nonlinear equation, and Iterations
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We show the use of a reconfigurable computer in computing the correlation immunity of Boolean functions of up to 6 variables. Boolean functions with high correlation immunity are desired in cryptographic systems because they are immune to... more
We show the use of a reconfigurable computer in computing the correlation immunity of Boolean functions of up to 6 variables. Boolean functions with high correlation immunity are desired in cryptographic systems because they are immune to correlation attacks. The SRC-6 reconfigurable computer was programmed in Verilog to compute the correlation immunity of functions. This computation is performed at a rate that is 190 times faster than a conventional computer. Our analysis of the correlation immunity is across all n-variable Boolean functions, for 2 ≤ n ≤ 6, thus obtaining, for the first time, a complete distribution of such functions. We also compare correlation immunity with two other cryptographic properties, nonlinearity and degree.
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This paper explores the distribution of algebraic thickness of Boolean functions (that is, the minimum number of terms in the ANF of the functions in the orbit of a Boolean function, through all affine transformations), in four and five... more
This paper explores the distribution of algebraic thickness of Boolean functions (that is, the minimum number of terms in the ANF of the functions in the orbit of a Boolean function, through all affine transformations), in four and five variables, and the complete distribution is presented. Additionally, a complete analysis of some complexity properties (e.g., nonlinearity, balancedness, etc.) of all relevant orbits of Boolean functions is presented. Some properties of our notion of rigid function (which enabled us to reduce significantly the computation) are shown and some open questions are proposed, providing some further explanation of one of these questions.
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We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and,... more
We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and, assuming the generalised Riemann hypothesis, we find all the exceptions.
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... They were quite happy together and five daughters were born: Mary Ellen (b. 1856), Margaret (b. 1858), Alicia (later Alicia Stott) (b. 1860), Lucy Everest (b. 1862), and Ethel Lilian (b. 1864). The works of Boole are contained in... more
... They were quite happy together and five daughters were born: Mary Ellen (b. 1856), Margaret (b. 1858), Alicia (later Alicia Stott) (b. 1860), Lucy Everest (b. 1862), and Ethel Lilian (b. 1864). The works of Boole are contained in about 50 articles and a few other publications. ...
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We prove various results on monotone Boolean functions. In particular, we prove a conjecture proposed recently, stating that there are no monotone bent Boolean functions. Further, we give an upper bound on the nonlinearity of monotone... more
We prove various results on monotone Boolean functions. In particular, we prove a conjecture proposed recently, stating that there are no monotone bent Boolean functions. Further, we give an upper bound on the nonlinearity of monotone functions in odd dimension, we describe the Walsh–Hadamard spectrum and investigate some other cryptographic properties of monotone Boolean functions.
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In this paper we �nd a general approach to �nd closed forms of sums of products of arbitrary sequences satisfying the same recurrence with di�erent initial conditions. We apply successfully our technique to sums of products of such... more
In this paper we �nd a general approach to �nd closed forms of sums of products of arbitrary sequences satisfying the same recurrence with di�erent initial conditions. We apply successfully our technique to sums of products of such sequences with indices in (arbitrary) arithmetic progressions. It generalizes many results from literature. We propose also an extension where the sequences satisfy di�erent recurrences.
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Here, we prove some conjectures on the monotony of combinatorial and number-theoretical sequences from a recent paper of Zhi-Wei Sun.
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Here we prove some conjectures on the monotony of combinatorial sequences from the recent preprint of Zhi--Wei Sun.
Here, we look at the Fibonacci and Lucas numbers whose Euler function is a factorial, as well as Lucas numbers whose Euler function is a product of power of two and power of three.
The Erdős-Debrunner inequality referred to in the title states that “If a triangle XYZ is inscribed in a triangle ABC – with X,Y,Z on the sides BC,CA, and AB – then σ(XYZ)≥min(σ(BXZ),σ(CXY),σ(AYZ)) with equality if and only if X,Y, and Z... more
The Erdős-Debrunner inequality referred to in the title states that “If a triangle XYZ is inscribed in a triangle ABC – with X,Y,Z on the sides BC,CA, and AB – then σ(XYZ)≥min(σ(BXZ),σ(CXY),σ(AYZ)) with equality if and only if X,Y, and Z are the midpoints of the sides BC,CA, and AB,” where σ(MNP) stands for the area of the triangle MNP. In [Elem. Math. 61, No. 1, 32–35 (2006; Zbl 1135.51017)], W. Janous has generalized this inequality in the following manner: if by M p (x,y,z) we denote the mean (x p +y p +z p 3) 1/p for p≠0 and min(x,y,z) for p=-∞, then σ(XYZ)≥M p (σ(BXZ),σ(CXY),σ(AYZ))(1) holds for certain negative values of p. He also asked for the greatest value of p for which (1) holds and formulated two conjectures aimed at establishing that maximum value of p. The authors of this paper prove both conjectures by methods of calculus – different from those of V. Mascioni [JIPAM, J. Inequal. Pure Appl. Math. 8, No. 2, Paper No. 32, 5 p. (2007; Zbl 1134.51017)], who also settled t...
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ABSTRACT Recently much progress has been made on the old problem of determining the equivalence classes of Boolean functions under permutation of the variables. In this paper we prove an asymptotic formula for the number of equivalence... more
ABSTRACT Recently much progress has been made on the old problem of determining the equivalence classes of Boolean functions under permutation of the variables. In this paper we prove an asymptotic formula for the number of equivalence classes under permutation for degree d monomial rotation symmetric (MRS) functions, in the cases where d ≥ 3 is arbitrary and the number of variables n is a prime. Our counting formula has two main terms and an error term; this is the first instance of such a detailed result for Boolean function equivalence classes which is valid for arbitrary degree and infinitely many n. We also prove an exact formula for the count of the equivalence classes when d = 5; this extends previous work for d = 3 and 4.