A sparse modular GCD algorithm for polynomials over algebraic function fields

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

2009

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We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1, . . . , xv] where L is an algebraic function field with k ≥ 0 parameters t1, . . . , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. 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.

Section: Benchmarksmentioning

confidence: 99%

“…p 8 > 2||fi||∞. The last column in Table 1 is the time for computing gcd(f1f2, f1(f2 + 1)) using our SparseModGcd algorithm in [3]. One can see that our factorization algorithm is often as fast as the GCD algorithm on a problem of comparable size, except for problem 6.…”

Section: Benchmarksmentioning

confidence: 99%

On factorization of multivariate polynomials over algebraic number and function fields

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

2009

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“…Sparse interpolation is a key part of many algorithms in computer algebra such as polynomial GCD computation [17,7,13] over Z. Here one computes the GCD modulo p where p is chosen to be a machine size prime.…”

Section: Introductionmentioning

confidence: 99%

“…A motivation for our new algorithm is to use the Ben-Or/Tiwari approach in modular algorithms (e.g. GCD computations in characteristic 0 -see [7]) where the prime p is chosen to be a machine prime so that arithmetic in Zp is efficient.…”

Section: Introductionmentioning

confidence: 99%

Parallel sparse polynomial interpolation over finite fields

Proceedings of the 4th International Workshop on Parallel and Symbolic Computation

2010

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We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of BenOr and Tiwari from 1988 for interpolating polynomials over rings with characteristic zero to characteristic p by doing additional probes.To interpolate a polynomial in n variables with t non-zero terms, Zippel's (1990) algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. It interpolates each variable independently using O(t) probes which allows us to parallelize the main loop giving an advantage over Zippel's algorithm.We have implemented both Zippel's algorithm and the new algorithm in C and we have done a parallel implementation of our algorithm using Cilk [2]. In the paper we provide benchmarks comparing the number of probes our algorithm does with both Zippel's algorithm and Kaltofen and Lee's hybrid of the Zippel and Ben-Or/Tiwari algorithms.

“…Rayes, Wang and Weber attempted a parallel implementation in [8]. In [4] we extended it to compute GCDs over algebraic function fields.The second such algorithm was developed by Ben-Or and Tiwari in 1988 for interpolation in characteristic 0. …”

mentioning

confidence: 99%

On sparse interpolation over finite fields

ACM Commun. Comput. Algebra

2011

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Extended Abstract of Poster Presented at ISSAC 2010Let f be a polynomial in n variables x 1 , x 2 , ..., x n of degree d with t non-zero terms over a finite field with characteristic p. It is well known that f can be interpolated from (1 + d) n points using Newton interpolation. But this is exponential in n. If t(1 + d) n , that is, f is sparse, we seek algorithms for interpolating f whose complexity is polynomial in d, t and n.The first such algorithm was engineered by Richard Zippel in 1979 (see [9]). Zippel's algorithm interpolates one variable at a time using O(ndt) points. Zippel's algorithm is probabilistic; it needs p ndt to succeed with high probability. Zippel developed his algorithm to interpolate a sparse polynomial GCD with integer coefficients. Zippel's algorithm is now used in Maple, Magma and Mathematica as the default GCD algorithm for multivariate GCD computation over the integers. Rayes, Wang and Weber attempted a parallel implementation in [8]. In [4] we extended it to compute GCDs over algebraic function fields.The second such algorithm was developed by Ben-Or and Tiwari in 1988 for interpolation in characteristic 0.