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

Assia Mahboubi
2006 Lecture Notes in Computer Science  
We describe here a 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.  ... 
doi:10.1007/11814771_37 fatcat:xfqy6n7cozgj7nn23gn2dy2ubq

A Refinement-Based Approach to Computational Algebra in Coq [chapter]

Maxime Dénès, Anders Mörtberg, Vincent Siles
2012 Lecture Notes in Computer Science  
We describe a step-by-step approach to 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  ... 
doi:10.1007/978-3-642-32347-8_7 fatcat:s44cutj22rh7xihrdjljt4qgia

A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm

Jose Divasón, Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada
2019 Journal of automated reasoning  
We further adapt 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  ... 
doi:10.1007/s10817-019-09526-y pmid:32269396 pmcid:PMC7115093 fatcat:63gvtrcskjgsbkugj7vp2syfjy

A formalization of the Berlekamp-Zassenhaus factorization algorithm

Jose Divasón, Sebastiaan Joosten, René Thiemann, Akihisa Yamada
2017 Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2017  
We further adapt 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  ... 
doi:10.1145/3018610.3018617 dblp:conf/cpp/DivasonJT017 fatcat:sgrcj3xlzna35oyp4dk5pnpsgq

Formal Analysis Of A Public-Key Algorithm

Markus Kaiser, Johannes Buchmann
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.  ... 
doi:10.5281/zenodo.1062626 fatcat:jceju2rdofd7bcygnqmjtrne6q

A Verified Implementation of Algebraic Numbers in Isabelle/HOL

Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada
2018 Journal of automated reasoning  
We moreover provide 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  ... 
doi:10.1007/s10817-018-09504-w pmid:32226180 pmcid:PMC7089722 fatcat:pvzb5tg36jdb5bfrvnlgmajq7y

Computer algebra implemented in Isabelle's function package under Lucas-interpretation - a case study

Walther Neuper
2013 International Conference on Intelligent Computer Mathematics  
This case study was motivated by a master thesis at RISC Linz, which 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  ... 
dblp:conf/mkm/Neuper13 fatcat:meyc7ts7n5cu3csvl3735e3oyi

A Formalization of the LLL Basis Reduction Algorithm [chapter]

Jose Divasón, Sebastiaan Joosten, René Thiemann, Akihisa Yamada
2018 Lecture Notes in Computer Science  
We additionally integrate one application 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.  ... 
doi:10.1007/978-3-319-94821-8_10 fatcat:fpazlnxfyrhvzltnl23t6u36e4

Provably correct derivation of algorithms using FermaT

Martin Ward, Hussein Zedan
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!  ... 
doi:10.1007/s00165-013-0287-2 fatcat:rxkg6phmmvhwxb4z6fganvzylu

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

René Thiemann, Ralph Bottesch, Jose Divasón, Max W. Haslbeck, Sebastiaan J. C. Joosten, Akihisa Yamada
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.  ... 
doi:10.1007/s10817-020-09552-1 pmid:32831440 pmcid:PMC7413592 fatcat:fhfgozhs5zhvzg7zfooxpdiina

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]

Petr Glotov
2006 arXiv   pre-print
In this paper we present 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.  ... 
arXiv:math/0609831v3 fatcat:f6nqtmlqwzbmrogy3wyvl32tla

A Decision Procedure for Univariate Real Polynomials in Isabelle/HOL

Manuel Eberl
2015 Proceedings of the 2015 Conference on Certified Programs and Proofs - CPP '15  
Sturm sequences are a method 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.  ... 
doi:10.1145/2676724.2693166 dblp:conf/cpp/Eberl15 fatcat:ofuh7d52wfbr5bnoyj2steyuwa

Univariate polynomial factorization over finite fields

Patrice Naudin, Claude Quitté
1998 Theoretical Computer Science  
We give a detailed description 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  ... 
doi:10.1016/s0304-3975(97)80001-1 fatcat:pzgms4qpqrcqvplafkulrj7oh4

Algorithms for Computing Triangular Decompositions of Polynomial Systems [article]

Changbo Chen, Marc Moreno Maza
2011 arXiv   pre-print
We propose new 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.  ... 
arXiv:1104.0689v1 fatcat:ez5kkazvrrgpdjpstb5iosqorm
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