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Efficient multivariate factorization over finite fields [chapter]

Laurent Bernardin, Michael B. Monagan
<span title="">1997</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We describe the Maple [23] implementation of multivariate factorization over general finite fields. Our first implementation is available in Maple V Release 3.  ...  The efficiency of our implementation is illustrated by the ability to factor bivariate polynomials with over a million monomials over a small prime field.  ...  , that in order to factor multivariate polynomials over finite fields, we need an efficient implementation of factorization of bivariate polynomials over finite fields.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-63163-1_2">doi:10.1007/3-540-63163-1_2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uwugewy5nraprhouopnomjtbvu">fatcat:uwugewy5nraprhouopnomjtbvu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810220932/http://www.cecm.sfu.ca/~mmonagan/papers/AAECC.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4d/92/4d927478a4240876a9f3b90ddd07af760fd5899d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-63163-1_2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

On square-free factorization of multivariate polynomials over a finite field

Laurent Bernardin
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field.  ...  Our algorithm is based on Yun's square-free factorization algorithm for characteristic 0.  ...  An algorithm for multivariate polynomials over finite fields based on Musser's algorithm [lo] can be found in [l] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0304-3975(97)00059-5">doi:10.1016/s0304-3975(97)00059-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3zucsxzip5dchdpeayu5sijx2m">fatcat:3zucsxzip5dchdpeayu5sijx2m</a> </span>
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Implementation of prime decomposition of polynomial ideals over small finite fields

Masayuki Noro, Kazuhiro Yokoyama
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
An algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented on the basis of previous work of the second author.  ...  If the minimal polynomials are computed over fields of rational functions, then a multivariate factorizer over finite fields is required.  ...  Implementation details Multivariate factorization and GCD over finite fields To decompose an ideal, it is necessary to factorize the minimal polynomials.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jsc.2003.08.004">doi:10.1016/j.jsc.2003.08.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xskdukgarrdpjeuqz3eksrfhwq">fatcat:xskdukgarrdpjeuqz3eksrfhwq</a> </span>
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Theoretical Properties [chapter]

<span title="2013-06-18">2013</span> <i title="Chapman and Hall/CRC"> Handbook of Finite Fields </i> &nbsp;
Here we have a situation where factoring over the rational numbers is provably easier than factoring over a sufficiently large finite field.  ...  However, as proved in [Gao and Lauder 2002] the cost of the recombination process behaves in softly linear time in average over finite fields, which explains the practical efficiency of this approach  ...  Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable factorization, 3 sparse factorization, 9 sparse polynomial representation, 9 straight-line program, 11, 12 straight-line  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1201/b15006-5">doi:10.1201/b15006-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cubpnr7y3fbfpivjinvw2dqmvy">fatcat:cubpnr7y3fbfpivjinvw2dqmvy</a> </span>
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Comparison of Bit and Word Level Algorithms for Evaluating Unstructured Functions over Finite Rings [chapter]

B. Sunar, D. Cyganski
<span title="">2005</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We study the problem of implementing multivariate functions defined over finite rings or fields as parallel circuits.  ...  We present a modification to Horner's algorithm for evaluating arbitrary n-variate functions defined over finite rings and fields.  ...  .), or public-key schemes defined over polynomial rings or finite fields.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11545262_18">doi:10.1007/11545262_18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/n2fip4iqmnh5batbdyyrw6csie">fatcat:n2fip4iqmnh5batbdyyrw6csie</a> </span>
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Modular arithmetic and finite field theory

E. Horowitz
<span title="">1971</span> <i title="ACM Press"> Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC &#39;71 </i> &nbsp;
A recent algorithm for polynomial factorization over a finite field has led to faster algorithms for factorization over the field of rationals.  ...  Hence, an effort to develop efficient methods for solution of problems over the integers has lead to a search for efficient solutions in the domain of integers modulo p.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/800204.806287">doi:10.1145/800204.806287</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dhyhlimhefcshjsgacfnbmptie">fatcat:dhyhlimhefcshjsgacfnbmptie</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20140628181125/http://f3.tiera.ru/3/M_Mathematics/MT_Number%20theory/Horowitz.%20Finite%20fields%20and%20modular%20arithmetic,%20tutorial(7s).pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/58/f1/58f1aa091a082e2a3fff445759cf64851eca2001.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/800204.806287"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Theoretical Properties [chapter]

<span title="2015-09-14">2015</span> <i title="WORLD SCIENTIFIC"> The Fence Methods </i> &nbsp;
Here we have a situation where factoring over the rational numbers is provably easier than factoring over a sufficiently large finite field.  ...  However, as proved in [Gao and Lauder 2002] the cost of the recombination process behaves in softly linear time in average over finite fields, which explains the practical efficiency of this approach  ...  Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable factorization, 3 sparse factorization, 9 sparse polynomial representation, 9 straight-line program, 11, 12 straight-line  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814596077_0009">doi:10.1142/9789814596077_0009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7rmqxc74afagha2moms4jjz35u">fatcat:7rmqxc74afagha2moms4jjz35u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20120712161040/http://www4.ncsu.edu/~kaltofen/bibliography/11/KL11.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/23/2a/232acd67bf39e2e52a93ff421b2f6483d7730993.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814596077_0009"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> worldscientific.com </button> </a>

Theoretical Properties [chapter]

<span title="2011-03-28">2011</span> <i title="Chapman and Hall/CRC"> Chapman &amp; Hall/CRC Biostatistics Series </i> &nbsp;
Here we have a situation where factoring over the rational numbers is provably easier than factoring over a sufficiently large finite field.  ...  However, as proved in [Gao and Lauder 2002] the cost of the recombination process behaves in softly linear time in average over finite fields, which explains the practical efficiency of this approach  ...  Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable factorization, 3 sparse factorization, 9 sparse polynomial representation, 9 straight-line program, 11, 12 straight-line  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1201/b10783-7">doi:10.1201/b10783-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zqyyjtuzsjf7zpekrozupsqwye">fatcat:zqyyjtuzsjf7zpekrozupsqwye</a> </span>
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Special issue computational algebraic complexity editorial

Erich Kaltofen, Bruno Buchberger
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
Bach and Shoup investigate the problem of factoring univariate polynomials over finite fields, which is one of the first problems in computer science on which the usefulness of randomization was demonstrated  ...  The article by R6nyal studies, from an algorithmic point of view, the Wedderburn-Artin structure theory of finite dimensional associative algebras over a finite field.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0747-7171(08)80010-7">doi:10.1016/s0747-7171(08)80010-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wrri7sabmjb4tpqtqyfbwuoa6e">fatcat:wrri7sabmjb4tpqtqyfbwuoa6e</a> </span>
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Page 834 of Mathematical Reviews Vol. , Issue 97B [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
As applications, we obtain efficient parallel algorithms for sparse multivariate polynomial factorization and GCD over finite fields.  ...  Summary: “We develop a randomized parallel algorithm which performs interpolation of sparse multivariate polynomials over finite fields.  ... 
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Fast Evaluation of Multivariate Quadratic Polynomials over GF(2^32) using Grahpics Processing Units

Satoshi Tanaka, Takanori Yasuda, Kouichi Sakurai
<span title="">2014</span> <i title="Innovative Information Science &amp; Technology Research Group (ISYOU)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7hlkkmw3qjhrnliyaattoye5ty" style="color: black;">Journal of Internet Services and Information Security</a> </i> &nbsp;
More specifically, the security of QUAD depends on the hardness of solving non-linear multivariate quadratic systems over a finite field, which is known as an NP-complete problem.  ...  We propose an efficient implementation for computing with multivariate polynomials in multivariate cryptography on GPU and evaluate the efficiency of the proposal.  ...  Acknowledgement This work is partly supported by "Study on Secure Cryptosystem using Multivariate Polynomials," no. 0159-0172, Strategic Information and Communications R&D Promotion Programme (SCOPE),  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.22667/jisis.2014.08.31.001">doi:10.22667/jisis.2014.08.31.001</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/journals/jisis/TanakaYS14.html">dblp:journals/jisis/TanakaYS14</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iqkfipiqlferxc6ijw4kf7hcti">fatcat:iqkfipiqlferxc6ijw4kf7hcti</a> </span>
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Page 90 of Mathematical Reviews Vol. , Issue 99a [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Math. 8 (1991), no. 3, 357-375; MR 92j:12002] regarding approxi- mate factorization of multivariate polynomials over C.  ...  {For the entire collection see MR 97j:11001.} 99a:11140 11T06 Zhou, Yong Quan Factorization of multivariate polynomials over GF(p). (Chinese.  ... 
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Computer algebra with Rings library

Stanislav Poslavsky
<span title="">2020</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wxgp7pobnrfetfizidmpebi4qy" style="color: black;">Journal of Physics, Conference Series</a> </i> &nbsp;
Polynomial arithmetic, GCDs, factorization, and Gröbner bases are implemented with the use of modern asymptotically fast algorithms.  ...  For example, the algorithms for computing GCDs or factoring univariate polynomials over finite fields F q [x] are formulated independently of the specific type of F q : they are the same for both modular  ...  polynomial ring over Galois field GF(17 3 ): Figure 2 .Figure 3 . 23 Dependence of multivariate GCD performance on the number of variables.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1742-6596/1525/1/012020">doi:10.1088/1742-6596/1525/1/012020</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zibf72pljvdjzepjop6fn2vxbu">fatcat:zibf72pljvdjzepjop6fn2vxbu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200722061750/https://iopscience.iop.org/article/10.1088/1742-6596/1525/1/012020/pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a4/48/a448f2f324310032a4a0c7dfe3343ed0cc17f849.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1742-6596/1525/1/012020"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> iop.org </button> </a>

Factoring Multivariate Polynomials over Large Finite Fields

Daqing Wan
<span title="">1990</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5bz4zmidbngqxk6yv4msbkm54u" style="color: black;">Mathematics of Computation</a> </i> &nbsp;
The algorithm can be easily generalized to factor multivariate polynomials over finite fields. We shall give two further applications of the idea involved in the algorithm.  ...  A simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field.  ...  However, an efficient algorithm for factoring univariate polynomials over finite fields was not presented until the late 1960's.  ... 
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Factoring multivariate polynomials over large finite fields

Da Qing Wan
<span title="1990-05-01">1990</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5bz4zmidbngqxk6yv4msbkm54u" style="color: black;">Mathematics of Computation</a> </i> &nbsp;
The algorithm can be easily generalized to factor multivariate polynomials over finite fields. We shall give two further applications of the idea involved in the algorithm.  ...  A simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field.  ...  However, an efficient algorithm for factoring univariate polynomials over finite fields was not presented until the late 1960's.  ... 
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