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Evaluation of the heuristic polynomial GCD

1995
*
Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95
*

*The*

*Heuristic*

*Polynomial*

*GCD*procedure (GCDHEU) is Richard Zippel, Effective

*Polynomial*Computation, Kluwer Academic Publishers, Boston 1993. ... Acknowledgments We thank Daniel Lichtblau

*of*Wolfram Research Inc. for his information on

*the*

*GCD*methods used in Mathematical. ... We also thank Keith Geddes for his encouragement and explanations

*of*

*the*fine details in Maple. ...

##
###
GCDHEU: Heuristic polynomial GCD algorithm based on integer GCD computation

1989
*
Journal of symbolic computation
*

A

doi:10.1016/s0747-7171(89)80004-5
fatcat:sqjk4aow3ncixa4kbn2p2rtrom
*heuristic*algorithm, GCDHEU, is described for*polynomial**GCD*computation over*the*integers. ...*The*algorithm is based on*evaluation*at a single large integer value (for each variable), integer*GCD*computation, and a single-point interpolation scheme. ... Acknowledgements*The*students and staff associated with*the*Maple project have contributed to this work in various ways. ...##
###
A correct proof of the heuristic GCD algorithm
[article]

2002
*
arXiv
*
pre-print

In this note, we fill a gap in

arXiv:cs/0206032v1
fatcat:lickgk5zbnf5dkosppmapjk5fa
*the*proof*of**the**heuristic**GCD*in*the*multivariate case made by Char, Geddes and Gonnet (JSC 1989) and give some additionnal information on this method. ... Context*The**heuristic**gcd*algorithm is used to computed*the**gcd**of*two*polynomials*P and Q with integer coefficients in one or a few variables :*the*main idea is to*evaluate*one*of**the*variable X k at ... a sufficient large integer z, compute*the**gcd**of**the**evaluations*recursively or as integers and reconstruct a candidate*gcd*from*the**gcd**of**the**evaluations*using*the*representation*of*coefficients in basis ...##
###
A heuristic irreducibility test for univariate polynomials

1992
*
Journal of symbolic computation
*

*The*irreducibility test is based on finding a prime

*evaluation*

*of*a

*polynomial*which, under appropriate conditions, is a witness to

*the*irreducibility

*of*

*the*

*polynomial*. ... This paper describes a

*heuristic*irreducibility test for univariate

*polynomials*over

*the*integers . ... It. can be seen that

*the*time taken by

*the*

*heuristic*is typically less than 10% that

*of*

*the*time taken to factor

*the*

*polynomials*by any

*of*

*the*systems . ...

##
###
Efficient multivariate factorization over finite fields
[chapter]

1997
*
Lecture Notes in Computer Science
*

*The*efficiency

*of*our implementation is illustrated by

*the*ability to factor bivariate

*polynomials*with over a million monomials over a small prime field. ... We give selected details

*of*

*the*algorithms and show several ideas that were used to improve its efficiency. Most

*of*

*the*improvements presented here are incorporated in Maple V Release 4. ... Total ...... 1136894si 100% i: Notes:

*The*square-fl'ee factorization quickly realizes that

*the*

*GCD*

*of*

*the*

*polynomial*and its derivative is one because

*the*modular

*GCD*algorithm succeeds in finding a ...

##
###
Prime Values of Quadratic Polynomials
[article]

2021
*
arXiv
*
pre-print

This note investigates

arXiv:2009.00417v4
fatcat:b7pi2egwrzb4ncn6tgy345dzje
*the*prime values*of**the**polynomial*f(t)=qt^2+a for any fixed pair*of*relatively prime integers a≥ 1 and q≥ 1*of*opposite parity. ... For a large number x≥1, an asymptotic result*of**the*form ∑_n≤ x^1/2, n oddΛ(qn^2+a)≫ qx^1/2/2φ(q) is achieved for q≪ (log x)^b, where b≥ 0 is a constant. ... Quadratic To Linear Identity*The*quadratic to linear inequality trades off*the**evaluation**of*n≤x 1/2 ,odd n Λ(qn 2 + a) for*the**evaluation**of*a product*of*some exponential sums and n≤x,odd n Λ(qn + a). ...##
###
A parallel algorithm to compute the greatest common divisor of sparse multivariate polynomials

2014
*
ACM Communications in Computer Algebra
*

Wang's algorithm

doi:10.1145/2576802.2576817
fatcat:szl5ww5j5je65chmfuig6l6uy4
*heuristically*determines that*the*leading coefficient*of**the*true*GCD*is C(z, u) = (z + u). ... . , x n ] are*the*input*polynomials*and let g =*gcd*(a, b) = l i=1 c i M i (x 1 , x 2 ) where l is*the*number*of*terms*of*g(x 1 , x 2 ), M i is*the*ith monomial*of*g(x 1 , x 2 ) and c i ∈ Z[x 3 , ..., x ...##
###
On factorization of multivariate polynomials over algebraic number and function fields

2009
*
Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09
*

This enables us to avoid using poor bounds on

doi:10.1145/1576702.1576731
dblp:conf/issac/JavadiM09
fatcat:mnwjc2nsfzgqrhyb7uuupg6equ
*the*size*of**the*integer coefficients in*the*factorization*of*f when using Hensel lifting. We have implemented our algorithm in Maple 13. ... We provide timings demonstrating*the*efficiency*of*our algorithm. ... Now Hensel lifting fails if either*the**evaluation*point is unlucky or*the**heuristic*bound T is not big enough. In this case, we will double*the**heuristic*bound, i.e. ...##
###
Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm

2002
*
Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02
*

Note: many

doi:10.1145/780506.780519
dblp:conf/issac/GiesbrechtKL02
fatcat:stbmtl4tcbewllqivxe5mfpuj4
*of**the*authors' publications cited below are accessible through links in their Internet homepages. ...*The*first simply says that*the**GCD**of*an*evaluation**of*two relatively prime integer*polynomials*is generally smooth. Lemma 2. ... We propose*the*use*of**the**GCD**of*two or three subsequent discrepancies in practice, but so far this must be considered a*heuristic*. ...##
###
High-performance polynomial GCD computations on graphics processors

2011
*
2011 International Conference on High Performance Computing & Simulation
*

Our approach exhibits block structure to distribute

doi:10.1109/hpcsim.2011.5999827
dblp:conf/ieeehpcs/Emeliyanenko11
fatcat:saz6fddhqjbgtdbslwpmvojdyi
*the*computation*of*a single modular*GCD*over several thread blocks, and thus to remove any hardware limitations on*the*maximal size*of**polynomials*that ... We propose an algorithm to compute a greatest common divisor (*GCD*)*of*univariate*polynomials*with large integer coefficients on Graphics Processing Units (GPUs). ... This is because*the*competing algorithms are*of*different nature:*the*modular approach needs to process more images when*the*bitlength increases while, for instance,*heuristic**GCD**of*Maple maps*polynomials*...##
###
In-place arithmetic for polynomials over Zn
[chapter]

1993
*
Lecture Notes in Computer Science
*

These algorithms provide

doi:10.1007/3-540-57272-4_21
fatcat:cnum46gw4bbkhjy7tvuae6uenq
*the*key tools for*the*efficient implementation*of**polynomial*resultant*gcd*and factorization computation over Z, without having to write large amounts*of*code in a systems implementation ... We present space and time efficient algorithms for univariate*polynomial*arithmetic operations over Z mod n where*the*modulus n does not necessarily fit into is not a machine word. ...*The*first method [17] is a*heuristic*method that reduces*the*problem to a single*Gcd*computation over Zn for a large possibly composite modulus n. ...##
###
A Period Assignment Algorithm for Real-Time System Design
[chapter]

1999
*
Lecture Notes in Computer Science
*

This paper proposes a

doi:10.1007/3-540-49059-0_3
fatcat:nvcdyhyqsbd3rdlktbq2vh7474
*polynomial*time approximation algorithm which produces a solution whose utilization does not exceed twice*the*optimal utilization. ... Our experimental analysis shows that*the*proposed algorithm finds solutions which are very close to*the*optimal ones in most cases*of*practical interest. ... Optimal*GCD*assignment Our first*heuristic*is*the*optimal*GCD*(greatest common divisor) assignment. ...##
###
Lattice Reduction Algorithms: Theory and Practice
[chapter]

2011
*
Lecture Notes in Computer Science
*

On

doi:10.1007/978-3-642-20465-4_2
fatcat:i7bjoolptfcanjnl7knuo4cl6a
*the*one hand, lattice reduction algorithms are widely used in publickey cryptanalysis, for instance to attack special settings*of*RSA and DSA/ECDSA. ... On*the*other hand, there are more and more cryptographic schemes whose security require that certain lattice problems are hard. ... SVP can be viewed as a geometric generalization*of**gcd*computations: Euclid's algorithm actually computes*the*smallest (in absolute value) non-zero linear combination*of*two integers, since*gcd*(a, b)Z ...##
###
Faster Algorithms for Approximate Common Divisors: Breaking Fully-Homomorphic-Encryption Challenges over the Integers
[chapter]

2012
*
Lecture Notes in Computer Science
*

We present a new PACD algorithm whose running time is essentially

doi:10.1007/978-3-642-29011-4_30
fatcat:njp4w45wibdlddi6uza73hay6u
*the*"square root"*of*that*of*exhaustive search, which was*the*best attack in practice. ...*The*seemingly easier problem PACD was recently used by Coron et al. at CRYPTO '11 to build a more efficient variant*of**the*FHE scheme by van Dijk et al.. ... Part*of*this work is supported by*the*Commission*of**the*European Communities through*the*ICT program under contract ICT-2007-216676 ECRYPT II. ...##
###
The supersingular isogeny path and endomorphism ring problems are equivalent

2022
*
2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
*

As an essential tool, we develop a rigorous algorithm for

doi:10.1109/focs52979.2021.00109
fatcat:uxtllin6djfizn3qsmgydudhm4
*the*quaternion analog*of**the*path-finding problem, building upon*the**heuristic*method*of*Kohel, Lauter, Petit and Tignol. ... We prove that*the*path-finding problem in -isogeny graphs and*the*endomorphism ring problem for supersingular elliptic curves are equivalent under reductions*of**polynomial*expected time, assuming*the*generalised ... Recall that an efficient representation means that there is an algorithm that*evaluates*α(P ) for any P ∈ E(F p k ) in time*polynomial*in*the*length*of**the*representation*of*α and in k log p. ...
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