Factoring Polynomials Over Finite Fields: A Survey

Gathen, Joachim von zur; Panario, Daniel

doi:10.1006/jsco.1999.1002

Cited by 81 publications

(47 citation statements)

References 116 publications

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“…Milestones in the development of polynomial-time algorithms for factoring in F q [X] are the algorithms of Berlekamp [Ber70], Cantor & Zassenhaus [CZ81], von zur Gathen & Shoup [vzGS92] and Kaltofen & Shoup [KS98]. See the surveys [vzGP01,Kal03,vzG06]. Presently, there are practical algorithms that factor degree n polynomials over F q using a quadratic number of operations (ignoring for a moment the dependence on q), and subquadratic algorithms that rely on fast matrix multiplication [KS98].…”

Section: Introductionmentioning

confidence: 99%

Fast Polynomial Factorization and Modular Composition

Kedlaya¹,

Umans²

2011

SIAM J. Comput.

181

208

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We obtain randomized algorithms for factoring degree n univariate polynomials over F q requiring O(n 1.5+o(1) log 1+o(1) q + n 1+o(1) log 2+o(1) q) bit operations. When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q ≥ n, it matches the asymptotic running time of the best known algorithms.The improvements come from new algorithms for modular composition of degree n univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use O(n (ω+1)/2 ) field operations, where ω is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)).We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent α implies an algorithm for the other with exponent α + o(1), and vice versa. We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most n o(1) , and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese Remainder Theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest.Our algorithms use techniques that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication. * The material in this paper appeared in conferences as [Uma08] and [KU08].

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Section: Introductionmentioning

confidence: 99%

Fast Polynomial Factorization and Modular Composition

Kedlaya¹,

Umans²

2011

SIAM J. Comput.

181

208

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“…This problem can nevertheless be solved efficiently by randomized algorithms in average polynomial time. Concerning related results and historical surveys on this topic, the reader might consult [17,18,25,33].…”

Section: Related Workmentioning

confidence: 99%

Deterministic root finding over finite fields using Graeffe transforms

Grenet

Hoeven

Lecerf

2015

AAECC

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We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field F q . Our algorithms were designed to be particularly efficient in the case when the cardinality q − 1 of the multiplicative group of F q is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders.

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“…Milestones in the development of polynomial-time algorithms for factoring in F p [X] are the algorithms of Berlekamp [8], Cantor & Zassenhaus [12], von zur Gathen & Shoup [19] and Kaltofen & Shoup [38]. See the surveys [18,37,17]. A straightforward implementation of Berlekamp's algorithm [8] uses O(n 3 + n 1+o(1) log p) operations in F p .…”

Section: Computation Efficiencymentioning

confidence: 99%

Compress Multiple Ciphertexts Using Elgamal Encryption Schemes

Kim¹,

Kim²,

Cheon³

2013

Journal of the Korean Mathematical Society

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Abstract. In this work we deal with the problem of how to squeeze multiple ciphertexts without losing original message information. To do so, we formalize the notion of decomposability for public-key encryption and investigate why adding decomposability is challenging. We construct an ElGamal encryption scheme over extension fields, and show that it supports the efficient decomposition. We then analyze security of our scheme under the standard DDH assumption, and evaluate the performance of our construction.

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Factoring Polynomials Over Finite Fields: A Survey

Abstract: This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an up-to-date bibliography of the problem.

Cited by 81 publications

References 116 publications

Fast Polynomial Factorization and Modular Composition

Fast Polynomial Factorization and Modular Composition

Deterministic root finding over finite fields using Graeffe transforms

Compress Multiple Ciphertexts Using Elgamal Encryption Schemes

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