Fast Polynomial Multiplication over F<sub>2</sub><sub>60</sub>

Harvey, David; Hoeven, Joris van der; Lecerf, Grégoire

doi:10.1145/2930889.2930920

Cited by 11 publications

(25 citation statements)

References 30 publications

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“…In a finite field like F 2 l , there are roots of unity only for specific orders, so we would have S = {n|n divides 2 l −1}. A remarkable example is F 2 60 because there are efficient ways of computing in this field, and many roots of unity (with large, highly-composite order) are known [7].…”

Section: Goal Of This Papermentioning

confidence: 99%

“…In an in-place implementation of the Cooley-Tukey FFT, it is convenient to order the output differently (see for example [7,Section 2.1]). The purpose of this different order is to ensure that the result of the full FFT is simply the concatenation of the outputs from the outer DFTs.…”

Section: Generalized Bitwise Mirrormentioning

confidence: 99%

“…As for the in-place Cooley-Tukey FFT presented in [7], the result depends on the vector v, but not on the choice of h.…”

Section: Remark 32mentioning

confidence: 99%

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The Truncated Fourier Transform for Mixed Radices

Larrieu

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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The standard version of the Fast Fourier Transform (FFT) is applied to problems of size n = 2 k. For this reason, FFT-based evaluation/interpolation schemes often reduce a problem of size l to a problem of size n, where n is the smallest power of two with l n. However, this method presents "jumps" in the complexity at powers of two; and on the other hand, n − l values are computed that are actually unnecessary for the interpolation. To mitigate this problem, a truncated variant of the FFT was designed to avoid the computation of these unnecessary values. In the initial formulation [14], it is assumed that n is a power of two, but some use cases (for example in finite fields) may require more general values of n. This paper presents a generalization of the Truncated Fourier Transform (TFT) for arbitrary orders. This allows to benefit from the advantages of the TFT in the general case.

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Section: Goal Of This Papermentioning

confidence: 99%

Section: Generalized Bitwise Mirrormentioning

confidence: 99%

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The Truncated Fourier Transform for Mixed Radices

Larrieu

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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“…In [15], they also propose a similar algorithm for the multiplication over finite fields, achieving a Fürer-like complexity. This work led to an efficient implementation in [17], using multiplication of polynomials over the special field F 2 60 . In [7], Covanov and Thomé proposed an algorithm based on generalized Fermat primes and the same scheme as Fürer's algorithm, to multiply integers with a Fürer-like complexity.…”

Section: Introductionmentioning

confidence: 99%

Big Prime Field FFT on Multi-core Processors

Covanov

Mohajerani

Maza

et al. 2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We report on a multi-threaded implementation of Fast Fourier Transforms over generalized Fermat prime fields. This work extends a previous study realized on graphics processing units to multi-core processors. In this new context, we overcome the less fine control of hardware resources by successively using FFT in support of the multiplication in those fields. We obtain favorable speedup factors (up to 6.9x on a 6-core, 12 threads node, and 4.3x on a 4-core, 8 threads node) of our parallel implementation compared to the serial implementation for the overall application thanks to the low memory footprint and the sharp control of arithmetic instructions of our implementation of generalized Fermat prime fields.

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“…In particular, the case of q = 2 has received a lot of attention from the research communities due to its wide-ranging application, e.g., in coding theory and cryptography. Here we obviously need to go to an appropriate extension field F 2 d in order to obtain a primitive n-th root of unity for any meaningful n, and in this case, it is well known that one can use the Kronecker method to efficiently compute binary polynomial multiplication [Can89,HvdHL16]. Such FFT-based techniques have better asymptotic complexity compared with school-book and Karatsuba algorithms.…”

Section: Introductionmentioning

confidence: 99%

Frobenius Additive Fast Fourier Transform

Chen

Kuo

et al. 2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ∈ F q [x] of degree < n at all n-th roots of unity in F q d can essentially be computed d-time faster than evaluating Q ∈ F q d [x] at all these roots, assuming F q d contains a primitive n-th root of unity [vdHL17a]. Termed the Frobenius FFT, this discovery has a profound impact on polynomial multiplication, especially for multiplying binary polynomials, which finds ample application in coding theory and cryptography. In this paper, we show that the theory of Frobenius FFT beautifully generalizes to a class of additive FFT developed by Cantor and Gao-Mateer [Can89, GM10]. Furthermore, we demonstrate the power of Frobenius additive FFT for q = 2: to multiply two binary polynomials whose product is of degree < 256, the new technique requires only 29,005 bit operations, while the best result previously reported was 33,397. To the best of our knowledge, this is the first time that FFT-based multiplication outperforms Karatsuba and the like at such a low degree in terms of bit-operation count. CCS CONCEPTS• Mathematics of computing → Computations in finite fields; KEYWORDS addtitive FFT, Frobenius FFT, polynomial multiplication

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Fast Polynomial Multiplication over F₂₆₀

Cited by 11 publications

References 30 publications

The Truncated Fourier Transform for Mixed Radices

The Truncated Fourier Transform for Mixed Radices

Big Prime Field FFT on Multi-core Processors

Frobenius Additive Fast Fourier Transform

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Fast Polynomial Multiplication over F260

Cited by 11 publications

References 30 publications

The Truncated Fourier Transform for Mixed Radices

The Truncated Fourier Transform for Mixed Radices

Big Prime Field FFT on Multi-core Processors

Frobenius Additive Fast Fourier Transform

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Product

Resources

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Fast Polynomial Multiplication over F₂₆₀