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We analyze different ways of constructing binary extended formulations of mixed-integer problems with bounded integer variables and compare their relative strength with respect to split cuts. We show that among all binary extended…
Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This…
This paper depicts an algorithm for solving the Decision Boolean Satisfiability Problem using the binary numerical properties of a Special Decision Satisfiability Problem, parallel execution, object oriented, and short termination. The two…
Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation…
Practically relevant problems of quadratic optimization often contain multidimensional arrays of variables interconnected by linear constraints, such as equalities and inequalities. The values of each variable depend on its specific meaning…
Quadratic Unconstrained Binary Optimization (QUBO) is recognized as a unifying framework for modeling a wide range of problems. Problems can be solved with commercial solvers customized for solving QUBO and since QUBO have degree two, it is…
In this thesis, a new class of algorithms based on Sums of Squares Programming is developed. These allow to reduce a degree-$d$ homogeneous polynomial $T = \sum_{i = 1}^m \langle a_i, X \rangle^d $ to a quadratic form being close to a…
A linear quaternionic equation in one quaternionic variable q is of the form $a_1 q b_1+a_2 q b_2+ ... +a_m q b_m = c$, where the $a_i, b_j, c$ are given quaternionic coefficients. If introducing basis elements $\bf i, j, k$ of pure…
We study the precise computational complexity of deciding satisfiability of first-order quantified formulas over the theory of fixed-size bit-vectors with binary-encoded bit-widths and constants. This problem is known to be in EXPSPACE and…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
We study a Grover-type method for Quadratic Unconstrained Binary Optimization (QUBO) problems. For an $n$-dimensional QUBO problem with $m$ nonzero terms, we construct a marker oracle for such problems with a tuneable parameter, $\Lambda…
We introduce a new class of irreducible pentanomials over $\mathbb{F}_2$ of the form $f(x) = x^{2b+c} + x^{b+c} + x^b + x^c + 1$. Let $m=2b+c$ and use $f$ to define the finite field extension of degree $m$. We give the exact number of…
In this paper, we develop a way to encode several NP-Complete problems in Abstract Argumentation to Quadratic Unconstrained Binary Optimization (QUBO) problems. In this form, a solution for a QUBO problem involves minimizing a quadratic…
We study a bi-objective optimization problem, which for a given positive real number $n$ aims to find a vector $X = \{x_0,\cdots,x_{k-1}\} \in \mathbb{R}^{k}_{\ge 0}$ such that $\sum_{i=0}^{k-1} x_i = n$, minimizing the maximum of $k$…
A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the…
For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector…
We consider a proximal operator given by a quadratic function subject to bound constraints and give an optimization algorithm using the alternating direction method of multipliers (ADMM). The algorithm is particularly efficient to solve a…
Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m} $ for certain choices…
A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in…
In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…