Integer Linear Programming applied to determining monic hyperbolic irreducible polynomials with integer coefficients and span less than 4

Irreducible polynomials play an important role, till now in construction of 8-bit S-Boxes in ciphers. The 8-bit S-Box of Advanced Encryption Standard is a list of decimal equivalents of Multiplicative Inverses (MI) of all the elemental polynomials of a monic irreducible polynomial over Galois Field GF(28) [1]. In this paper a new method to search for monic Irreducible Polynomials (IPs) over Galois fields GF(pq) has been introduced. Here the decimal equivalents of each monic elemental polynomial (ep), two at a time, are split into the p-nary coefficients of each term, of those two monic elemental polynomials. From those coefficients, the p-nary coefficients of the resultant monic basic polynomials (BP) have been obtained. The decimal equivalents of resultant basic polynomials with p-nary coefficients are treated as decimal equivalents of the monic reducible polynomials, since monic reducible polynomials must have two monic elemental polynomials as its factor. The decimal equivalents of polynomials belonging to the list of reducible polynomials are cancelled leaving behind the monic irreducible polynomials. A non-monic irreducible polynomial is computed by multiplying a monic irreducible polynomial by α where α GF(pq) and assumes values from 2 to (p-1).

Ranking in quadratic integer programming problems

We investigate some monic integer irreducible polynomials which have two close roots. If P(x) is a separable polynomial in Z[x] of degree d \geq 2 with the Remak height R(P) and the minimal distance between the quotient of two distinct roots and unity Sep(P), then the inequality 1/Sep(P) \ll R(P)^{d-1} is true with the implied constant depending on d only. Using a recent construction of Bugeaud and Dujella we show that for each d \geq 3 there exists an irreducible monic polynomial P \in Z[x] of degree d for which R(P)^{(2d-3)(d-1)/(3d-5)} \ll 1/Sep(P). For d=3 the exponent 3/2 is improved to 5/3 and it is shown that the exponent 2 is optimal in the class of cubic (not necessarily monic) irreducible polynomials in Z[x].

Some relations between the irreducible polynomials over a finite field and its quadratic extension

This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses infinite-precision arithmetic and Sturm's Theorem to compute intervals containing the real roots of the polynomial and whose lengths are less than the given error bound, The algorithms also provide a simple means of determining the number of real roots in any interval.The first set of algorithms uses infinite-precision integer arithmetic to compute the sequence of polynomials required by Sturm's Theorem and also to do all the required polynomial evaluations. The second set uses congruence arithmetic to compute the images of the sequence of polynomials over the finite fields GF(p i ) for several primes p i and then uses the natural homomorphisms from the integers onto the finite fields GF(p i ) and the Chinese Remainder Theorem to do the evaluations. The primary advantage of the use of congruence arithmetic in the second set of algorithms is that the computing time to multiply any two elements of GF(p i ) is a constant whereas the time to multiply any two integers is proportional to the square of their lengths.The computing times of both sets of algorithms are analyzed in detail using the techniques of computing time analysis developed in the past few years by G. E. Collins and D. Knuth. It is shown that if P is a primitive, positive, univariate polynomial over the integers of positive degree, m, whose coefficients are bounded in magnitude by d, and which has L real roots, and ε is some positive rational number, 0 < ε ≤ 1, then the computing time to approximate the roots of P to within is:[EQUATION]using either set of algorithms.The empirical results of applying the two sets of algorithms to random polynomials and the first ten Legendre and Chebyshev polynomials are reported. Neither set of algorithms proved to be faster for all the polynomials tested.

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

A Survey of Lagrangean Techniques for Discrete Optimization

Irreducible polynomials with several prescribed coefficients

Diagnosability of faults in discrete event systems modeled with Petri nets can be assessed either via graph-based techniques (also called diagnoser, verifier/twin-plant based techniques), or via the solution of optimization problems. The approaches that belong to the former class are based on the analysis of the net reachability or coverability graphs (or of a more compact version of them). The latter approach exploits the mathematical representation of the net itself to specify and solve optimization problems, which are usually expressed as integer linear programming (ILP) problems. In this paper we exploit the railway Petri net model originally proposed in [16], and extended in [14] to be used as a benchmark for diagnosability analysis, to assess the efficiency of the approach based on the solution of ILP problems proposed in [3]. In order to show the effectiveness of the proposed technique, a comparison with a graph-based approach for analyzing diagnosability is also presented.

This paper presents a distributed server allocation model with preventive start-time optimization against a single server failure. The presented model preventively determines the assignment of servers to users under each failure pattern to minimize the largest maximum delay among all failure patterns. We formulate the proposed model as an integer linear programming (ILP) problem. We prove the NP-completeness of the considered problem. As the number of users and that of servers increase, the size of ILP problem increases; the computation time to solve the ILP problem becomes excessively large. We develop a heuristic approach that applies simulated annealing and the ILP approach in a hybrid manner to obtain the solution. Numerical results reveal that the developed heuristic approach reduces the computation time by 26% compared to the ILP approach while increasing the largest maximum delay by just 3.4% in average. It reduces the largest maximum delay compared with the start-time optimization model; it avoids the instability caused by the unnecessary disconnection permitted by the run-time optimization model.

A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of irreducible polynomials arising through an arbitrary quadratic transformation. In the present paper we provide an explanation for this extension, and a simpler proof of Ahmadi's result, by a reduction to the known special case of self-reciprocal polynomials and a minor variation. We also prove further results on polynomials arising through a quadratic transformation, and through some special transformations of higher degree.

Congruence arithmetic algorithms for polynomial real zero determination

This paper discusses a set of algorithms which given a univariate polynomial with integer coefficients (with possible multiple zeros) and a positive rational error bound, uses infinite-precision integer arithmetic and Sturm's Theorem to compute intervals containing the real zeros of the polynomial and whose lengths are less than the given error bound. The algorithms also provide a simple means of determining the number of real zeros in any interval. Theoretical computing time bounds are developed for the algorithms and some empirical results are reported.

An accelerated Benders decomposition approach for virtual base station formation in stochastic Cloud-RANs

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L=Qα is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of OB(N10) for isolating the real roots of Bα, where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of Descartes' algorithm introduced by Sagraloff. For the former we prove a complexity bound of OB(N8) and for the latter a bound of OB(N7). We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.