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Implementation of prime decomposition of polynomial ideals over small finite fields

Masayuki Noro, Kazuhiro Yokoyama
2003 ACM SIGSAM Bulletin  
. , 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

Masayuki Noro, Kazuhiro Yokoyama
2004 Journal of symbolic computation  
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  
, 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  
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

Xiaoliang Li, Chenqi Mou, Dongming Wang
2010 Computers and Mathematics with Applications  
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

Shuhong Gao, Daqing Wan, Mingsheng Wang
2009 Mathematics of Computation  
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]

John Abbott, Anna Maria Bigatti, Elisa Palezzato, Lorenzo Robbiano
2019 arXiv   pre-print
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  
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

Daniel J. Bates, Wolfram Decker, Jonathan D. Hauenstein, Chris Peterson, Gerhard Pfister, Frank-Olaf Schreyer, Andrew J. Sommese, Charles W. Wampler
2014 Applied Mathematics and Computation  
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

Claus Fieker
2007 Experimental Mathematics  
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

Allan Steel
2005 Journal of symbolic computation  
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]

Aurore Guillevic
2015 Lecture Notes in Computer Science  
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]

Antoine Joux, Vanessa Vitse
2012 Lecture Notes in Computer Science  
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

Dan Bates, Chris Peterson, Andrew J. Sommese
2006 Journal of Complexity  
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]

Yao Sun, Dingkang Wang
2010 arXiv   pre-print
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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