On linear complexity of sequences over [formula omitted

The minimal polynomial over [formula omitted] of linear recurring sequence over [formula omitted

We show that the binary expansions of algebraic numbers do not form secure pseudorandom sequences; given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables us to devise a simple algorithm to factor polynomials with rational coefficients. All algorithms work in polynomial time. Introduction. Manuel Blum raised the following question: Suppose we are given an approximate root of an unknown polynomial with integral coefficients and a bound on the degree and size of the coefficients of the polynomial. Is it possible to infer the polynomial? We answer his question in the affirmative. We show that if a complex number a satisfies an irreducible polynomial h(X) of degree d with integral coefficients in absolute value at most H, then given 0(d? + d ■ logH) bits of the binary expansion of the real and complex parts of a, we can find h(X) in deterministic polynomial time (and then compute in polynomial time any further bits of a). Using the concept of secure pseudorandom sequences formulated by Shamir [23], Blum and Micali [3] and Yao [25], we then show that the binary (or m-ary for any m) expansions of algebraic numbers do not form secure sequences in a certain well-defined sense. We are able to extend our results with the same techniques to transcendental numbers of the form log(a),cos_1(a), etc., where a is algebraic. The technique is based on the lattice basis reduction algorithm from [16]. Our answer to Blum's question enables us to devise a simple polynomial-time algorithm to factor polynomials with rational coefficients: We find an approximate root of the polynomial and use our algorithm to find the irreducible polynomial satisfied by the exact root, which must then be a factor of the given polynomial. This is repeated until all the factors are found. This algorithm was found independently by Schonhage [22], and was already suggested in [16]. The technique of the paper also provides a natural, efficient method to compute with algebraic numbers. This paper is the final journal version of [13], which contains essentially the entire contents of this paper. 1. A Polynomial-Time Algorithm for Blum's Question. Throughout this paper, Z denotes the set of the integers, Q the set of the rationals, R the set of the reals, and C the set of the complex numbers. The ring of polynomials with integral Received December 23, 1985; revised October 13, 1986 and April 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 68Q15, 68Q25, 68Q40. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page

The method of representing Markov functions with minimal characteristic polynomials over a finite field is proposed. These polynomials are defined on the basis of integrated stochastic matrices. The representation accuracy of stochastic matrices is linearly dependent on the minimum degree of the polynomials. The algorithmic implementation of the method is shown to build a sequence of the Markov functions class considered, with a given linear complexity.

Three classes of binary sequences of period 4N with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period N , i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4N and optimal autocorrelation value/magnitude are discussed in the splitting field of xN-1 . As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on m-sequences instead of generalized GMW sequences.

Linear complexity over [formula omitted] and over [formula omitted] for linear recurring sequences

This thesis describes new results on computing bounds on the values of the positive roots of polynomials. Bounding the roots of polynomials is an important sub-problem in many disciplines of scientific computing. Many numerical methods for finding roots of polynomials begin with an estimate of an upper bound on the values of the positive roots. If one can obtain a more accurate estimate of the bound, one can reduce the amount of work used in searching within the range of possible values to find the root (e.g. using a bisection method). Also, the computation of the real roots of higher degree univariate polynomials with real coefficients is based on their isolation. Isolation of the real roots of a polynomial is the process of finding real disjoint intervals such that each contains one real root and every real root is contained in some interval. To isolate the real positive roots, it is necessary to compute, in the best possible way, an upper bound on the value of the largest positive root. Although, several bounds are known, the first of which were obtained by Lagrange and Cauchy, this thesis revealed that there was much room for improvement on this topic. Today, two of the algorithms presented in this thesis, are regarded as the best (one of linear computational complexity and the other of quadratic complexity) and have already been incorporated in the source code of major computer algebra systems such as Mathematica and Sage. A certain part of this thesis is also devoted to the analytical presentation of the continued fraction real root isolation method. Its algorithm and its underlying components are presented thoroughly along with a new implementation of the method using the above mentioned bounds. Intensive computational tests verify that this

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant role. We revisit a theorem of Niederreiter on (i) linear complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence, proved using partial quotients. We prove (i) and its converse from first principles and generalise (ii) to rational functions where the denominator need not have minimal degree. We prove (ii) in two parts: firstly for geometric sequences and then for sequences with a jump in linear complexity. The basic idea is to decompose the denominator as a sum of polynomial multiples of two polynomials of minimal degree; there is a similar decomposition for the numerators. The decomposition is unique when the denominator has degree at most the length of the sequence. The proof also applies to rational functions related to finite sequences, generalising a result of Massey. We give a number of applications to rational functions associated to sequences.

The Comment by Friedrich et al. does not dispute the central result of our paper [Phys. Rev. A 85, 052508 (2012)] that nonanalytic behavior is present in long-established mathematical pathologies arising in the solution of finite basis optimized effective potential (OEP) equations. In the Comment, the terms ``balancing of basis sets'' and ``basis-set convergence'' imply a particular order towards the limit of a large orbital basis sets where the large-orbital-base limit is always taken first, before the large-auxiliary-base limit, until overall convergence is achieved, at a high computational cost. The authors claim that, on physical grounds, this order of limits is not only sufficient, but also necessary in order to avoid the mathematical pathologies. In response to the Comment, we remark that it is already written in our paper that the nonanalyticity trivially disappears with large orbital basis sets. We point out that the authors of the Comment give an incorrect proof of this statement. We also show that the order of limits towards convergence of the potential is immaterial. A recent paper by the authors of the Comment proposes a partial correction for the incomplete orbital basis error in the full-potential linearized augmented-plane-wave method. Similar to the correction developed in our paper, this correction also benefits from an effectively complete orbital basis, even though only a finite orbital basis is employed in the calculation. This shows that it is unnecessary to take, in practice, the limit of an infinite orbital basis in order to avoid mathematical pathologies in the OEP. Our paper is a significant contribution in that direction with general applicability to any choice of basis sets. Finally, contrary to an allusion in the abstract and assertions in the main text of the Comment that unphysical oscillations of the OEP are supposedly attributed to the common energy denominator approximation, in fact, such anomalies arise with the full treatment of the small eigenvalues of the density response function. This characteristic of the finite basis OEP is well known in the literature but also is clearly demonstrated in our paper.

A method has been proposed to represent lumped Markov chains by minimal polynomials over a finite field. The accuracy of representing lumped stochastic matrices, the law of lumped Markov chains depends linearly on the minimum degree of polynomials over field GF(q). The method allows constructing the realizations of lumped Markov chains on linear shift registers with a pre-defined “linear complexity”.

It is proved that the product of arbitrary periodic GF(q) sequences attains maximum linear complexity if their periods are pairwise coprime. The necessary and sufficient conditions are derived for maximum linear complexity of the product of two periodic GF(q) sequences with irreducible minimal characteristic polynomials. For a linear combination of products of arbitrary periodic GF(q) sequences, it is shown that maximum linear complexity is achieved if their periods are pairwise coprime and the polynomial x-1 does not divide any of their minimal characteristic polynomials; assuming only that their periods are pairwise coprime, the author establishes a lower bound on the linear complexity which is of the same order of magnitude as maximum linear complexity. Boolean functions are derived that are optimal with respect to the maximum linear complexity. Possible applications of the results in the design of sequence generators are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We investigate the linear complexity and the minimal polynomial over the finite fields of sequences constructed on the basis of cubic residue classes. In particular, we find the linear complexity and the minimal polynomial of the characteristic sequence of cubic residue class and balanced cyclotomic sequences of order three.

Driven by the continued advance of integrated circuit (IC) technology, large-scale electromagnetic (EM) modeling and simulation problems have been encountered across the design of on-chip circuits, package, and die-package interfaces. To give a few examples, full-chip post-layout performance verification that includes transmission line and full-wave effects, system-level signal and noise integrity analysis, and the characterization of broadband die-package interaction for clean power delivery. Two major difficulties exist for the analysis of the aforementioned large-scale problems. One is the large problem size. The modeling of a die, a package, and a combined die-package system results in numerical problems of ultra-large scale, requiring billions of parameters to describe them accurately. The other is the multiscale nature of the problem. An electromagnetic simulator is required to span scale ranges of at least 10000:1 to analyze a combined die-package system. Recently, a time domain Orthogonal Finite-Element Reduction-Recovery method (OrFE-RR) was developed to handle the large problem size associated with the design of very large-scale integrated circuits [1]. In this method, a set of orthogonal prism vector bases are developed. With this set of bases, the original ultralarge scale 3-D system of order TV is rigorously reduced to a 2-D single-layered system of order Mwith negligible computational cost, where Mis much less than N. The reduced 2D system is diagonal, and hence can be solved readily. After obtaining the solution of the reduced system, the rest of the solutions can be recovered in linear complexity. This method entails no theoretical approximation. It applies to any arbitrarily-shaped multilayer structure involving inhomogeneous materials. Numerical experiments have demonstrated its accuracy and capacity in simulating very large-scale on-chip, package, and die-package interaction problems.

A theoretical investigation into the cooperativity effect involving anionic hydrogen bond, thermodynamic property and aromaticity in Cl−⋯benzonitrile⋯H2O ternary complex

Conditions are derived which guarantee that products of linear recurring sequences attain maximum linear complexity. It is shown that the product of any number of maximum-length GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> sequences has maximum linear complexity, provided only the degrees of the corresponding minimal polynomials are distinct and greater than two. It is also shown that if the roots of any number of (not necessarily irreducible) minimal polynomials are simple and lie in extension fields of pairwise relatively prime degrees, then the product of the corresponding GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> sequences attains maximum linear complexity, provided only that no two roots of any minimal polynomial are linearly dependent over the groundfield GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> (which is automatically satisfied when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q = 2)</tex> . The results obtained for products are extended to arbitrary linear combinations of product sequences.

The linear complexity of a periodic sequence over GF(pm) plays an important role in cryptography and communication (see Menezes, van Oorschort, and Vanstone, Handbook of Applied Cryptography. Boca Raton, FL: CRC, 1997). In this correspondence, we prove a result which reduces the computation of the linear complexity and minimal connection polynomial of a period un sequence over GF(pm) to the computation of the linear complexities and minimal connection polynomials of u period n sequences. The conditions u|pm-1 and gcd(n,pm-1)=1 are required for the result to hold. Some applications of this reduction in fast algorithms to determine the linear complexities and minimal connection polynomials of sequences over GF(pm) are presented