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Implementation of prime decomposition of polynomial ideals over small finite fields
2003
ACM SIGSAM Bulletin
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. , P u } : 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 ( ...
doi:10.1145/990353.990366
fatcat:bp2fk55advcg3bxbcnl4vllscm
Implementation of prime decomposition of polynomial ideals over small finite fields
2004
Journal of symbolic computation
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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. ...
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 ...
doi:10.1016/j.jsc.2003.08.004
fatcat:xskdukgarrdpjeuqz3eksrfhwq
Index to Volumes 37 and 38
2004
Journal of symbolic computation
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, S., Elementary 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, ...
doi:10.1016/s0747-7171(04)00109-9
fatcat:q3cckydpknhjhinygacsvlj52y
Page 419 of IEEE Transactions on Computers Vol. 52, Issue 4
[page]
2003
IEEE Transactions on Computers
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The Internet Archive has digitized a microfilm copy of this work. It may be possible to borrow a copy for reading.
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 ...
Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case
2010
Computers and Mathematics with Applications
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This paper presents algorithms for decomposing any zero-dimensional 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. ...
doi:10.1016/j.camwa.2010.09.059
fatcat:e7hzggp2hnafhjsajwjub455pe
Primary decomposition of zero-dimensional ideals over finite fields
2009
Mathematics of Computation
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A new algorithm is presented for computing primary decomposition of zero-dimensional ideals over finite fields. ...
directly to root finding of univariate polynomials over the ground field. ...
over finite fields. ...
doi:10.1090/s0025-5718-08-02115-7
fatcat:37utjutzxnfjnatkq26prz64je
Computing and Using Minimal Polynomials
[article]
2019
arXiv
pre-print
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2019 pre-print arXiv:1702.07262v2 fatcat:rox5azrd5jagtfp6tt4t4mkrgqOther Versions
We also present some applications 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. ...
arXiv:1702.07262v3
fatcat:y72jdiv4vndk5iuhyx3bkero3q
Algorithmic Properties of Polynomial Rings
1998
Journal of symbolic computation
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In this paper we investigate how algorithms for computing heights, radicals, unmixed and primary 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. ...
doi:10.1006/jsco.1998.0227
fatcat:lmw5uzaf6zddxfcsfb7mxfl2ai
Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety
2014
Applied Mathematics and Computation
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In particular, we compare running times for homotopy-based numerical 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. ...
doi:10.1016/j.amc.2013.12.165
fatcat:3loemn6l6ba6phdqv3s64nu6uu
Sparse Representation for Cyclotomic Fields
2007
Experimental Mathematics
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We 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. ...
doi:10.1080/10586458.2007.10129012
fatcat:eyxxcnes25f23nswdoty3eyvqe
Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic
2005
Journal of symbolic computation
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As a corollary, the problem 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. ...
doi:10.1016/j.jsc.2005.03.002
fatcat:l7ni76nazfbw5h5gtuqpqw3fti
Computing Individual Discrete Logarithms Faster in $${{\mathrm{GF}}}(p^n)$$ with the NFS-DL Algorithm
[chapter]
2015
Lecture Notes in Computer Science
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The first step outputs two 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. ...
doi:10.1007/978-3-662-48797-6_7
fatcat:guobsnva6jdv3ah75ss5wta2b4
Cover and Decomposition Index Calculus on Elliptic Curves Made Practical
[chapter]
2012
Lecture Notes in Computer Science
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We give a real-size example 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 ...
doi:10.1007/978-3-642-29011-4_3
fatcat:23ppmsbcyrgf7atw55jgrbgztq
A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set
2006
Journal of Complexity
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This algebraic set decomposes into a union 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. ...
doi:10.1016/j.jco.2006.04.003
fatcat:43kwrs4brrajhljts5cqz2erby
An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field
[article]
2010
arXiv
pre-print
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From these factors, the factorization 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. ...
arXiv:0907.2300v2
fatcat:2wmvxidei5bd3mkldrvu3jotv4
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