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Implementation of prime decomposition of polynomial ideals over small finite fields

2003
*
ACM SIGSAM Bulletin
*

. , P u } :

doi:10.1145/990353.990366
fatcat:bp2fk55advcg3bxbcnl4vllscm
*prime*divisors*of*sc(H) ⇒ { √ Q 1 , . . . , Q u )}, Q k = φ(P k ) are*prime*divisors*of*H. ------------------------------------------------------- 3*Implementation**Polynomial*factorization ... (E)*Decomposition**of*0-dimensional*ideals*• Frobenius map φ • Separable closure sc(H) = φ −1 (H) 1.2 Pre-*decomposition*√ I = ∩ s i=1 √ I i , where I i is generated by irreducible*polynomials*by using ( ...##
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Implementation of prime decomposition of polynomial ideals over small finite fields

2004
*
Journal of symbolic computation
*

An algorithm for the

doi:10.1016/j.jsc.2003.08.004
fatcat:xskdukgarrdpjeuqz3eksrfhwq
*prime**decomposition**of**polynomial**ideals**over**small**finite**fields*is proposed and*implemented*on the basis*of*previous work*of*the second author. ... The practicality*of*the algorithm is examined by testing the*implementation*experimentally, which also reveals information about the quality*of*the*implementation*. ... Concluding remarks We have*implemented*an algorithm for*prime**decomposition**of**polynomial**ideals**over**small**finite**fields*on a computer, and have evaluated the practicality and quality*of*our*implementation*...##
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Index to Volumes 37 and 38

2004
*
Journal of symbolic computation
*

, S., Elementary

doi:10.1016/s0747-7171(04)00109-9
fatcat:q3cckydpknhjhinygacsvlj52y
*decompositions**of*arbitrary maps*over**finite*sets, 305 CANNON, J. and HOLT, D.F., Computing maximal subgroups*of**finite*groups, 589 Certified dense linear system solving, 485 Characterization ...*ideal**of*a general projective curve, 295 BELABAS, K., A relative van Hoeij algorithm*over*number*fields*, 641 BERNSTEIN, D., The computational complexity*of*rules for the character table*of*S n , 727 BURCKEL ... solving, 1343 New algorithms for generating Conway*polynomials**over**finite**fields*, 1003 NORO, M. and YOKOYAMA, K.,*Implementation**of**prime**decomposition**of**polynomial**ideals**over**small**finite**fields*, ...##
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Page 419 of IEEE Transactions on Computers Vol. 52, Issue 4
[page]

2003
*
IEEE Transactions on Computers
*

However, irreducible all-1

*polynomials**of*degree m exist if and only if (m 1) is*prime*and 2 is a generator*of*the*field*GF(m + 1). ... Wedderburn’s Theorem also gives a is the*finite*extension*field**over*procedure to map the ring elements to*field*elements. This ...##
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Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case

2010
*
Computers and Mathematics with Applications
*

This paper presents algorithms for decomposing any zero-dimensional

doi:10.1016/j.camwa.2010.09.059
fatcat:e7hzggp2hnafhjsajwjub455pe
*polynomial*set into simple sets*over*an arbitrary*finite**field*, with an associated*ideal*or zero*decomposition*. ... As a key ingredient*of*these algorithms, we generalize the squarefree*decomposition*approach for univariate*polynomials**over*a*finite**field*to that*over*the*field*product determined by a simple set. ... Acknowledgements The authors wish to thank Evelyne Hubert for beneficial discussions on some*of*the problems treated in the paper and the referees for their helpful suggestions. ...##
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Primary decomposition of zero-dimensional ideals over finite fields

2009
*
Mathematics of Computation
*

A new algorithm is presented for computing primary

doi:10.1090/s0025-5718-08-02115-7
fatcat:37utjutzxnfjnatkq26prz64je
*decomposition**of*zero-dimensional*ideals**over**finite**fields*. ... directly to root finding*of*univariate*polynomials**over*the ground*field*. ...*over**finite**fields*. ...##
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Computing and Using Minimal Polynomials
[article]

2019
*
arXiv
*
pre-print

We also present some applications

arXiv:1702.07262v3
fatcat:y72jdiv4vndk5iuhyx3bkero3q
*of*minimal*polynomials*, namely algorithms for computing radicals and primary*decompositions**of*zero-dimensional*ideals*, and also for testing radicality and maximality. ... Given a zero-dimensional*ideal*I in a*polynomial*ring, many computations start by finding univariate*polynomials*in I. ... Moreover, it is easy to*implement*this class in an efficient way*over*a (*small**prime*)*finite**field*; in CoCoALib its core is the class LinDepFp which uses machine integers. ...##
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Algorithmic Properties of Polynomial Rings

1998
*
Journal of symbolic computation
*

In this paper we investigate how algorithms for computing heights, radicals, unmixed and primary

doi:10.1006/jsco.1998.0227
fatcat:lmw5uzaf6zddxfcsfb7mxfl2ai
*decompositions**of**ideals*can be lifted from a Noetherian commutative ring R to*polynomial*rings*over*R. ... Efficient algorithms for computing unmixed*decomposition**of**ideals**over**fields*resp. ...*ideals*in multivariate*polynomial*rings*over**fields*are based on these results. ...##
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Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

2014
*
Applied Mathematics and Computation
*

In particular, we compare running times for homotopy-based numerical

doi:10.1016/j.amc.2013.12.165
fatcat:3loemn6l6ba6phdqv3s64nu6uu
*decomposition*algorithms*over*C with running times for*finite**field*-based symbolic*decomposition*algorithms*over*Q for systems*of**polynomial*... Thus, these calculations are carried through*over**fields*such as the*field**of*rational numbers,*finite**prime**fields*, or algebraic extensions thereof. ...##
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Sparse Representation for Cyclotomic Fields

2007
*
Experimental Mathematics
*

We

doi:10.1080/10586458.2007.10129012
fatcat:eyxxcnes25f23nswdoty3eyvqe
*implemented*our ideas in Magma and used it for*fields**of*degree > 10 6*over*Q. ... Currently, all major*implementations**of*cyclotomic*fields*as well as number*fields*, are based on a dense model where elements are represented either as dense*polynomials*in the generator*of*the*field*or ... In particular, we need to evaluate the sum*of*eigenvalues*of*matrices*over*a*finite**field*lifted into a cyclotomic*field*. ...##
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Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic

2005
*
Journal of symbolic computation
*

As a corollary, the problem

doi:10.1016/j.jsc.2005.03.002
fatcat:l7ni76nazfbw5h5gtuqpqw3fti
*of*computing the primary*decomposition**of*a positive-dimensional*ideal**over*a*finite**field*is also solved. ... This paper presents practical algorithms for the first time for (1) computing the primary*decomposition**of**ideals**of**polynomial*rings defined*over*such*fields*and (2) factoring arbitrary multivariate*polynomials*... Introduction Let K be a*field**of*positive characteristic p which is*finitely*generated*over*its*prime**field*. ...##
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Computing Individual Discrete Logarithms Faster in $${{\mathrm{GF}}}(p^n)$$ with the NFS-DL Algorithm
[chapter]

2015
*
Lecture Notes in Computer Science
*

The first step outputs two

doi:10.1007/978-3-662-48797-6_7
fatcat:guobsnva6jdv3ah75ss5wta2b4
*polynomials*defining two number*fields*, and a map from the*polynomial*ring*over*the integers modulo each*of*these*polynomials*to F_p^n. ... The Number*Field*Sieve (NFS) algorithm is the best known method to compute discrete logarithms (DL) in*finite**fields*F_p^n, with p medium to large and n ≥ 1*small*. ... The author thanks the anonymous reviewers for their constructive comments and the generalization*of*Lemma 3. The author is grateful to Pierrick Gaudry, François Morain and Ben Smith. ...##
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Cover and Decomposition Index Calculus on Elliptic Curves Made Practical
[chapter]

2012
*
Lecture Notes in Computer Science
*

We give a real-size example

doi:10.1007/978-3-642-29011-4_3
fatcat:23ppmsbcyrgf7atw55jgrbgztq
*of*discrete logarithm computations on a curve*over*a 151-bit degree 6 extension*field*, which would not have been practically attackable using previously known algorithms. ... This attack applies, at least theoretically, to all composite degree extension*fields*, and is particularly well-suited for curves defined*over*F p 6 . ... Cover and*Decomposition*attack Let F q d /F p be an extension*of**finite**fields*, where q is a power*of*p (in most applications p denotes a large*prime*but in general, it can be any*prime*power), and let ...##
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A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set

2006
*
Journal of Complexity
*

This algebraic set decomposes into a union

doi:10.1016/j.jco.2006.04.003
fatcat:43kwrs4brrajhljts5cqz2erby
*of*simpler, irreducible components. The set*of**polynomials*imposes on each component a positive integer known as the multiplicity*of*the component. ... The common zero locus*of*these*polynomials*, V (F 1 , F 2 , . . . , F t ) = {p ∈ C n |F i (p) = 0 for 1 i t}, determines an algebraic set. ... Frequently, a symbolic version*of*roundoff is done to prevent coefficient blowup. The computations are carried out with exact arithmetic but*over*a*field*with*finite*characteristic. ...##
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An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field
[article]

2010
*
arXiv
*
pre-print

From these factors, the factorization

arXiv:0907.2300v2
fatcat:2wmvxidei5bd3mkldrvu3jotv4
*of*the*polynomial**over*the extension*field*is obtained. ... A new efficient algorithm is proposed for factoring*polynomials**over*an algebraic extension*field*. The extension*field*is defined by a*polynomial*ring modulo a maximal*ideal*. ... Let now suppose that Q is a zero dimensional radical*ideal**of*R and Q has a minimal*prime**decomposition*: Q = Q 1 ∩ · · · ∩ Q t , 3 where each Q i is a*prime**ideal**of*R. ...
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