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Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials
[chapter]

2006
*
Lecture Notes in Computer Science
*

We describe here a

doi:10.1007/11814771_37
fatcat:xfqy6n7cozgj7nn23gn2dy2ubq
*formal*proof in*the*Coq system*of**the*structure theorem*for*subresultants, which allows to*prove**formally**the*correction*of*our*implementation**of**the*subresultants*algorithm*. ... Up to our knowledge it is*the*first mechanized proof*of*this result. ... in*the*pedestrian proofs*of**the*ring axioms*for**polynomials*and Marie-Françoise Roy*for*her detailed explanations on subresultants and elimination theory. ...##
###
A Refinement-Based Approach to Computational Algebra in Coq
[chapter]

2012
*
Lecture Notes in Computer Science
*

We describe a step-by-step approach to

doi:10.1007/978-3-642-32347-8_7
fatcat:s44cutj22rh7xihrdjljt4qgia
*the**implementation*and*formal*verification*of**efficient*algebraic*algorithms*. ... We illustrate this methodology on key applications: matrix rank computation, Winograd's fast matrix product, Karatsuba's*polynomial*multiplication, and*the**gcd**of*multivariate*polynomials*. **The*research ... Using that any*polynomial*can be split in a primitive part and a non-primitive part by dividing by*the**gcd**of*its coefficients (this is called*the*content*of**the**polynomial*) we get*an**algorithm**for*computing ...##
###
A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm

2019
*
Journal of automated reasoning
*

We further adapt

doi:10.1007/s10817-019-09526-y
pmid:32269396
pmcid:PMC7115093
fatcat:63gvtrcskjgsbkugj7vp2syfjy
*an*existing*formalization**of*Yun's square-free factorization*algorithm*to integer*polynomials*, and thus provide*an**efficient*and certified factorization*algorithm**for*arbitrary univariate ... We*formally*verify*the*Berlekamp-Zassenhaus*algorithm**for*factoring square-free integer*polynomials*in Isabelle/HOL. ... We thank Florian Haftmann*for*integrating our changes in*the**polynomial*library into*the*Isabelle distribution; we thank Manuel Eberl*for*discussions on factorial rings in Isabelle; and we thank*the*anonymous ...##
###
A formalization of the Berlekamp-Zassenhaus factorization algorithm

2017
*
Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2017
*

We further adapt

doi:10.1145/3018610.3018617
dblp:conf/cpp/DivasonJT017
fatcat:sgrcj3xlzna35oyp4dk5pnpsgq
*an*existing*formalization**of*Yun's square-free factorization*algorithm*to integer*polynomials*, and thus provide*an**efficient*and certified factorization*algorithm**for*arbitrary univariate ... We*formalize**the*Berlekamp-Zassenhaus*algorithm**for*factoring square-free integer*polynomials*in Isabelle/HOL. ... We thank Florian Haftmann*for*integrating our changes in*the**polynomial*library into*the*Isabelle distribution; we thank Manuel Eberl*for*discussions on factorial rings in Isabelle; and we thank*the*anonymous ...##
###
Formal Analysis Of A Public-Key Algorithm

2007
*
Zenodo
*

*The*analysis uses a given database to

*prove*

*formal*properties

*of*our

*implemented*functions with computer support. ... A major objective

*of*this article is

*the*presentation

*of*

*the*first computer-

*proved*

*implementation*

*of*

*the*Rabin public-key scheme in Isabelle/HOL. ... A verification

*of*

*an*

*algorithm*can

*prove*crucial properties or in

*the*optimal case all relevant facts needed

*for*its application. ...

##
###
A Verified Implementation of Algebraic Numbers in Isabelle/HOL

2018
*
Journal of automated reasoning
*

We moreover provide

doi:10.1007/s10817-018-09504-w
pmid:32226180
pmcid:PMC7089722
fatcat:pvzb5tg36jdb5bfrvnlgmajq7y
*algorithms*that can identify all*the*real or complex roots*of*rational*polynomials*, and two*implementations*to display algebraic numbers,*an*approximative version and*an*injective precise ...*The*development combines various existing*formalizations*such as matrices, Sturm's theorem, and*polynomial*factorization, and it includes new*formalizations*about bivariate*polynomials*, unique factorization ... Open Access This article is distributed under*the*terms*of**the*Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution ...##
###
Computer algebra implemented in Isabelle's function package under Lucas-interpretation - a case study

2013
*
International Conference on Intelligent Computer Mathematics
*

This case study was motivated by a master thesis at RISC Linz, which

dblp:conf/mkm/Neuper13
fatcat:meyc7ts7n5cu3csvl3735e3oyi
*implemented*a CA*algorithm**for**the*greatest common divisor*of*multivariate*polynomials*[10] in SML. ...*For*sake*of**efficiency*in development and*of*coherence*of*Isabelle's knowledge*the*first step*of*this task will be to adopt Isabelle's multivariate*Polynomials*4 -from*the*side*of*Isabelle development ...##
###
A Formalization of the LLL Basis Reduction Algorithm
[chapter]

2018
*
Lecture Notes in Computer Science
*

We additionally integrate one application

doi:10.1007/978-3-319-94821-8_10
fatcat:fpazlnxfyrhvzltnl23t6u36e4
*of*LLL, namely a verified factorization*algorithm**for*univariate integer*polynomials*which runs in*polynomial*time. ... It thereby approximates*an*NP-hard problem where*the*approximation quality solely depends on*the*dimension*of**the*lattice, but not*the*lattice itself. ... Some*of**the*research was conducted while Sebastiaan Joosten and Akihisa Yamada were working in*the*University*of*Innsbruck. ...##
###
Provably correct derivation of algorithms using FermaT

2013
*
Formal Aspects of Computing
*

*The*transformational programming method

*of*

*algorithm*derivation starts with a

*formal*specification

*of*

*the*result to be achieved, plus some informal ideas as to what techniques will be used in

*the*

*implementation*...

*The*

*formal*specification is then transformed into

*an*

*implementation*, by means

*of*correctness-preserving refinement and transformation steps, guided by

*the*informal ideas. ... Extensive testing with various sizes

*of*random

*polynomials*demonstrated that

*the*abstract

*algorithm*was in reality more

*efficient*than Knuth's

*algorithm*

*for*every case tested! ...

##
###
Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL

2020
*
Journal of automated reasoning
*

*The*

*algorithm*has applications in number theory, computer algebra and cryptography. In this paper, we provide

*an*

*implementation*

*of*

*the*LLL

*algorithm*. ... We additionally integrate one application

*of*LLL, namely a verified factorization

*algorithm*

*for*univariate integer

*polynomials*which runs in

*polynomial*time. ... Sebastiaan is now working at University

*of*Twente,

*the*Netherlands, and supported by

*the*NWO VICI 639.023.710 Mercedes project. ...

##
###
Page 1661 of Mathematical Reviews Vol. , Issue 91C
[page]

1991
*
Mathematical Reviews
*

Summary: “

*An**efficient*parallel*algorithm*(RNC?)*for**the*two- processor scheduling problem is presented. ... This paper deals with lower and upper bounds*for**the*topological complexity*of**algorithms**for**the*approximate solution*of*systems*of**polynomial*equations. ...##
###
Generalized subresultants and generalized subresultant algorithm
[article]

2006
*
arXiv
*
pre-print

In this paper we present

arXiv:math/0609831v3
fatcat:f6nqtmlqwzbmrogy3wyvl32tla
*the*notions*of*trail (pseudo-)division, generalized subresultants and generalized subresultant*algorithm*. ...*Implementation**The**algorithms**for**gcd*and resultant computing above was*implemented*with*the*Axiomxl computer algebra system, which allows to get*an**efficient*executing code. ... From*the*considerations above we can derive*the**algorithms**for*computing*the**gcd*and resultants. Bellow we present*the**algorithm**for**gcd*computation. ...##
###
A Decision Procedure for Univariate Real Polynomials in Isabelle/HOL

2015
*
Proceedings of the 2015 Conference on Certified Programs and Proofs - CPP '15
*

Sturm sequences are a method

doi:10.1145/2676724.2693166
dblp:conf/cpp/Eberl15
fatcat:ofuh7d52wfbr5bnoyj2steyuwa
*for*computing*the*number*of*real roots*of*a univariate real*polynomial*inside a given interval*efficiently*. ... Building upon this,*an*Isabelle/HOL proof method was then*implemented*to*prove*interesting statements about*the*number*of*real roots*of*a univariate real*polynomial*and related properties such as non-negativity ... Larry Paulson contributed interesting insights into*the*performance*of*our proof method in practice and encouraged us to*implement*non-strict inequalities. ...##
###
Univariate polynomial factorization over finite fields

1998
*
Theoretical Computer Science
*

We give a detailed description

doi:10.1016/s0304-3975(97)80001-1
fatcat:pzgms4qpqrcqvplafkulrj7oh4
*of**an**efficient**implementation**of**the*Cantor-Zassenhaus*algorithm*used in*the*release 2*of**the*Axiom computer algebra system. ... Explicit*algorithms*are presented in a form suitable*for*almost immediate*implementation*. ... This*implementation*makes use*of**the*trace function to eliminate most*of**the*exponentiations needed by*the**algorithm*, and this is achieved by using tables allowing*an**efficient*computation*of*Frobenius ...##
###
Algorithms for Computing Triangular Decompositions of Polynomial Systems
[article]

2011
*
arXiv
*
pre-print

We propose new

arXiv:1104.0689v1
fatcat:ez5kkazvrrgpdjpstb5iosqorm
*algorithms**for*computing triangular decompositions*of**polynomial*systems incrementally. ... With respect to previous works, our improvements are based on a weakened notion*of*a*polynomial**GCD*modulo a regular chain, which permits to greatly simplify and optimize*the*sub-*algorithms*. ...*The*authors would like to thank*the*support*of*Maplesoft, Mitacs and Nserc*of*Canada. ...
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