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Evaluation of the heuristic polynomial GCD

Hsin-Chao Liao, Richard J. Fateman
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.  ... 
doi:10.1145/220346.220376 dblp:conf/issac/LiaoF95 fatcat:aox7tsovordg7ky6v3l5bt4zcm

GCDHEU: Heuristic polynomial GCD algorithm based on integer GCD computation

Bruce W. Char, Keith O. Geddes, Gaston H. Gonnet
1989 Journal of symbolic computation  
A 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.  ... 
doi:10.1016/s0747-7171(89)80004-5 fatcat:sqjk4aow3ncixa4kbn2p2rtrom

A correct proof of the heuristic GCD algorithm [article]

Bernard Parisse
2002 arXiv   pre-print
In this note, we fill a gap in 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  ... 
arXiv:cs/0206032v1 fatcat:lickgk5zbnf5dkosppmapjk5fa

A heuristic irreducibility test for univariate polynomials

Michael B. Monagan
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 .  ... 
doi:10.1016/0747-7171(92)90005-o fatcat:3kbkbfh4d5ectbvi7xswlfxpr4

Efficient multivariate factorization over finite fields [chapter]

Laurent Bernardin, Michael B. Monagan
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  ... 
doi:10.1007/3-540-63163-1_2 fatcat:uwugewy5nraprhouopnomjtbvu

Prime Values of Quadratic Polynomials [article]

N.A. Carella
2021 arXiv   pre-print
This note investigates 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).  ... 
arXiv:2009.00417v4 fatcat:b7pi2egwrzb4ncn6tgy345dzje

A parallel algorithm to compute the greatest common divisor of sparse multivariate polynomials

Jiaxiong Hu, Michael Monagan
2014 ACM Communications in Computer Algebra  
Wang's algorithm 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  ... 
doi:10.1145/2576802.2576817 fatcat:szl5ww5j5je65chmfuig6l6uy4

On factorization of multivariate polynomials over algebraic number and function fields

Seyed Mohammad Mahdi Javadi, Michael B. Monagan
2009 Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09  
This enables us to avoid using poor bounds on 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.  ... 
doi:10.1145/1576702.1576731 dblp:conf/issac/JavadiM09 fatcat:mnwjc2nsfzgqrhyb7uuupg6equ

Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm

Mark Giesbrecht, Erich Kaltofen, Wen-shin Lee
2002 Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02  
Note: many 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.  ... 
doi:10.1145/780506.780519 dblp:conf/issac/GiesbrechtKL02 fatcat:stbmtl4tcbewllqivxe5mfpuj4

High-performance polynomial GCD computations on graphics processors

Pavel Emeliyanenko
2011 2011 International Conference on High Performance Computing & Simulation  
Our approach exhibits block structure to distribute 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  ... 
doi:10.1109/hpcsim.2011.5999827 dblp:conf/ieeehpcs/Emeliyanenko11 fatcat:saz6fddhqjbgtdbslwpmvojdyi

In-place arithmetic for polynomials over Zn [chapter]

Michael Monagan
1993 Lecture Notes in Computer Science  
These algorithms provide 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.  ... 
doi:10.1007/3-540-57272-4_21 fatcat:cnum46gw4bbkhjy7tvuae6uenq

A Period Assignment Algorithm for Real-Time System Design [chapter]

Minsoo Ryu, Seongsoo Hong
1999 Lecture Notes in Computer Science  
This paper proposes a 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.  ... 
doi:10.1007/3-540-49059-0_3 fatcat:nvcdyhyqsbd3rdlktbq2vh7474

Lattice Reduction Algorithms: Theory and Practice [chapter]

Phong Q. Nguyen
2011 Lecture Notes in Computer Science  
On 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  ... 
doi:10.1007/978-3-642-20465-4_2 fatcat:i7bjoolptfcanjnl7knuo4cl6a

Faster Algorithms for Approximate Common Divisors: Breaking Fully-Homomorphic-Encryption Challenges over the Integers [chapter]

Yuanmi Chen, Phong Q. Nguyen
2012 Lecture Notes in Computer Science  
We present a new PACD algorithm whose running time is essentially 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.  ... 
doi:10.1007/978-3-642-29011-4_30 fatcat:njp4w45wibdlddi6uza73hay6u

The supersingular isogeny path and endomorphism ring problems are equivalent

Benjamin Wesolowski
2022 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)  
As an essential tool, we develop a rigorous algorithm for 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.  ... 
doi:10.1109/focs52979.2021.00109 fatcat:uxtllin6djfizn3qsmgydudhm4
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