Lecture Notes : "Topics in Algebraic Computation"

1997

We establish that the problem of computing a gcd-free basis for a set of polynomials is in AfC~for any arbitrary field F. This leads to a proof that arithmetic for a simple algebraic extension is in N@. This result is applied to improve the complexity of the parallel deterministic algorithm to compute the Jordan normal form of a n dimensional matrix in time 0(log2 n). Basis Computations Let F be an arbitrary commutative field. The computational model used in this section is the arithmetic PRAM model. We say that a problem lies in NC! [4, 11] if there exists a parallel algorithm which solves it in time is bounded by O(log~n) using nQ1) processors for all inputs of size n.

ArXiv, 2019

We introduce a framework generalizing lattice reduction algorithms to module lattices in order to practically and efficiently solve the $\gamma$-Hermite Module-SVP problem over arbitrary cyclotomic fields. The core idea is to exploit the structure of the subfields for designing a doubly-recursive strategy of reduction: both recursive in the rank of the module and in the field we are working in. Besides, we demonstrate how to leverage the inherent symplectic geometry existing in the tower of fields to provide a significant speed-up of the reduction for rank two modules. The recursive strategy over the rank can also be applied to the reduction of Euclidean lattices, and we can perform a reduction in asymptotically almost the same time as matrix multiplication. As a byproduct of the design of these fast reductions, we also generalize to all cyclotomic fields and provide speedups for many previous number theoretical algorithms. Quantitatively, we show that a module of rank 2 over a cycl...

Programming and Computer Software, 2011

2020

Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents of finite fields. That is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. Most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. This means how integer and polynomial arithmetic operations are done for Z p such as addition, subtraction, division and multiplication in Z p followed by reduction of p (modulo p). Since addition is the same as subtraction and division is treated as the inverse of the multiplication, thus in this paper, only addition and multiplication arithmetic operations are applied for the considered small finite fields (Z 2 − Z 17 ). With polynomials, one can learn from this paper how arithmetic computational throug...

Linear Algebra and Its Applications, 2006

Barnett's method through Bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way. Two different uses of this method in computer algebra are introduced here. Firstly, we describe an algorithm for parameterizing the greatest common divisor of several polynomials in K[x, y], being x a parameter taking values in an real field K. Secondly, we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following Corless et al. [R.M. Corless, P.M. Gianni, B.M. Trager, S. Watt, The singular value decomposition for polynomial systems, in: ACM International Symposium on Symbolic and Algebraic Computation, 1995, pp. 195-207]. Given a family of polynomials whose coefficients are imperfectly known, we describe an algorithm for computing their approximate greatest common divisor by using, as main tools, Barnett's method and singular value decomposition computations. Furthermore, we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients.

Mathematics of Computation, 1990

This volume has been compiled in honour of the well known mathematician Hans Zasscnhaus on the occasion of his 75th birthday. As colleagues, collaborators and friends, we dedicate this work to him in the hope that it might inspire present and future researchers, in a similar fashion to the way in which his brilliant ideas filled us with the creati>e urge. Hans Zassenhaus was born in Koblenz (West Germany) on 28 May 1912 and brought up in Hamburg. There he studied mathematics under the supervision of E. Artin, E. Heeke, and E. Sperner. and was also a student of physics and biology. He was awarded his PhD at the early age of 22 with a thesis on "Kennzeichnung linearer Gruppen als Permutationsgruppen••. In the subsequent two years, which he spent at Rostock as a teaching assistant, he wrote his famous monograph on group theory which is still among th..: standard textbooks on that subject. In 1938-back in Hamburg-he qualified for a full teaching appointment with a paper on Lie rings of prime characteristic. In 1946 he was appointed associate professor and director of the Institute for Applied \1athcmatics which he had founded at Hamburg University in the same year. Accepting the challenge of an offer to help McGill University in the building up of Canadian graduate education in mathematics he left his country in 1949 for Montreal, Canada, where he was later joined by his wife and children. As Peter Redpath Professor at McGill University he supervised the PhD studies of many Canadian students. In 1957 he became a Canadian citizen. In 1959 he moved to the USA where he taught at Notre Dame (1959-1963) and at Ohio State University in Columbus. At OSU he hdd the position of research professor until his retirement five years ago. During these years he frequently visited other universities as a guest professor. We briefly mention the academic year 1955-56 at Princeton, two years at the California Institute of Technology as a Fairchild Distinguished Scholar, a Gauss professorship at Gottingen in 1967 and the US Senior Scientist Award of the Humboldt-Stiftung. In 1956 he became a Fellow of the Royal Canadian Society and in 1969 Editor-in-Chief of the Journal of Number Theory. Among the mathematicians of our time he is one of the few still active in different areas-we have already mentioned his contributions to group theory. Most graduate students learn his famous "butterfly lemma" which nowadays forms a substantial part of the proof of the Jordan-Holder-Schreier theorem. In 1978 (jointly with R. Bulow, J. Neubiiser and H. Wondraschek) he wrote a book on crystallographic groups which seems to be better known to physicists than to the mathematical community. Orders (and their ideal theory) are also among the central objectives of his research. The constructive approach clearly dominates. In a joint paper with E. C. Dade and 0. Taussky it was proven that the (n-1)st power of a fractional ideal of an order of rank n over the integers is always invertible. The authors obtained the idea of that theorem by numerical calculations on a computer and then each of them gave a different proof. This was an early and powerful demonstration of the usefulness of mathematical experimenting by computer. Later on he developed several algorithms for the embedding of an order into its maximal order. Each algorithm improved the preceding one and the numerical results obtained by each algorithm led to further theoretical improvements-0747-7171"87.040001

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well.

2010

Let F be a number field. There are many interesting things we can compute about F: Invariants: maximal order OF, class group Cl(F), units U(F), higher algebraic K-groups, Dedekind ζF... Subfields: Galois group, lattice of subfields. Extensions: build L/F, e.g given explicitly by primitive elements or implicitly via Kummer or class field theory. Invariants thereof (e.g in class field towers). Basic operations: elementary operations on elements and ideals of OF, mostly multiplications (at least in class field theory). XIV e Rencontres Arithmétiques de Caen (20/06/2003) – p. 2/19Setup (2/4) For most of these problems, there exist efficient algorithms, deterministic or randomized, possibly assuming some deep conjecture (GRH, density of friable elements in appropriate sets...), possibly giving a wrong result with small probability in an appropriate model, possibly not an algorithm at all but usually giving sensible results... But there are a number of pitfalls, especially when the degree...

Chapman & Hall/CRC Applied Algorithms and Data Structures series, 1998