Inside the FFT Black Box

Chu, Eleanor; George, Alan

doi:10.1201/9780367802332

Cited by 18 publications

(14 citation statements)

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“…Gather access to twiddle factors pictured on Figure 2 1: _vr indexes = _vel_vseq_vl(VL); ⊳ 0,1,2,3,… We apply another optimization to the loading of twiddle factors, present in both 8-Pease and 8-Pease-gt. This one follows the observation that for each group of three phases, half of the vector registers have the same twiddle factors (e.g., in Figure 1, the dark blue register and the light red register of the second phase both hold W[0] and W [2] as their first and second element). In the third phase, the elements are identical for all registers.…”

Section: Optimizationssupporting

confidence: 74%

“…Therefore, a large body of research about FFT optimization on many architectures has been published in the last decades. The key reference publications used as background for our implementations are the book of Chu et al 2 the article of Pease, 3 and the article of Swarztrauber. 4 More recently, the research community is focusing on developing efficient FFT implementations targeting emerging architectures with different degrees of parallelism, for example, high number of cores and long SIMD or vector units.…”

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Acceleration with long vector architectures: Implementation and evaluation of the FFT kernel on NEC SX‐Aurora and RISC‐V vector extension

Vizcaino

Mantovani

Ferrer

et al. 2022

Concurrency and Computation

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SummaryNovel architectures leveraging long and variable vector lengths like the NEC SX‐Aurora or the vector extension of RISCV are appearing as promising solutions on the supercomputing market. These architectures often require re‐coding of scientific kernels. For example, traditional implementations of algorithms for computing the fast Fourier transform (FFT) cannot take full advantage of vector architectures. In this article, we present the implementation of FFT algorithms able to leverage these novel architectures. We evaluate these codes on NEC SX‐Aurora , comparing them with the optimized NEC libraries; and in a prototype of a RISC‐V core with a vector processing unit. We present the benefits and limitations of two approaches of RADIX‐2 FFT vector implementations. We show that our approach makes better use of the vector unit of the NEC SX‐Aurora , reaching higher or equal performance than the optimized NEC library. More generally, we prove the importance of maximizing the vector length usage of the algorithm, taking advantage of the FFT properties to reduce long‐latency vector operations, and reordering the instructions according to the specific hardware features to boost the performance of FFT‐like computational kernels.

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Section: Optimizationssupporting

confidence: 74%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Acceleration with long vector architectures: Implementation and evaluation of the FFT kernel on NEC SX‐Aurora and RISC‐V vector extension

Vizcaino

Mantovani

Ferrer

et al. 2022

Concurrency and Computation

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Section: Introductionmentioning

confidence: 99%

Accelerating FFT Using NEC SX-Aurora Vector Engine

Vizcaino

Mantovani

Labarta

2022

Euro-Par 2021: Parallel Processing Workshops

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Novel architectures leveraging long and variable vector lengths like the NEC SX-Aurora or the vector extension of RISCV are appearing as promising solutions on the supercomputing market. These architectures often require re-coding of scientic kernels. For example, traditional implementations of algorithms for computing the fast Fourier transform (FFT) cannot take full advantage of vector architectures. In this paper, we present the implementation of FFT algorithms able to leverage these novel architectures. We evaluate these codes on NEC SX-Aurora, comparing them with the optimized NEC libraries. We present the benets and limitations of two approaches of RADIX-2 FFT vector implementations. We show that our approach makes better use of the vector unit, reaching higher performance than the optimized NEC library for FFT sizes under 64k elements. More generally, we prove the importance of maximizing the vector length usage of the algorithm and that adapting the algorithm to replace memory instructions with register shuing operations can boost the performance of FFT-like computational kernels.1 This version of the contribution has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections.

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“…The multiplication

modulo

is done by computing

and

, for

, computing the component-wise multiplication

and, finally, using the inverse operation to obtain the polynomial c such that

, as discussed by Lyubashevsky and Seiler [ 27 ]. Although the asymptotic cost of an algorithm based on this technique is

, the hidden constants may be large due to the increased number of reductions modulo q in comparison with CRT-based algorithms for power-of-two cyclotomic number fields [ 27 , 46 ]. Another important aspect of the defining polynomial is captured by the expansion factor , a property introduced by Lyubashevsky and Micciancio [ 47 ].…”

Section: Applications Of the Twisted Ring-lwementioning

confidence: 99%

The Ring-LWE Problem in Lattice-Based Cryptography: The Case of Twisted Embeddings

Ortiz

Araujo

Aranha

et al. 2021

Entropy

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Several works have characterized weak instances of the Ring-LWE problem by exploring vulnerabilities arising from the use of algebraic structures. Although these weak instances are not addressed by worst-case hardness theorems, enabling other ring instantiations enlarges the scope of possible applications and favors the diversification of security assumptions. In this work, we extend the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. By doing so, we allow the Ring-LWE problem to be used over maximal real subfields of cyclotomic number fields. We prove that Twisted Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the twist factor does not affect the asymptotic approximation factors in the worst-case to average-case reductions. Thus, Twisted Ring-LWE maintains the consolidated hardness guarantee of Ring-LWE and increases the existing scope of algebraic lattices that can be considered for cryptographic applications. Additionally, we expand on the results of Ducas and Durmus (Public-Key Cryptography, 2012) on spherical Gaussian distributions to the proposed class of lattices under certain restrictions. As a result, sampling from a spherical Gaussian distribution can be done directly in the respective number field while maintaining its format and standard deviation when seen in Zn via twisted embeddings.

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Inside the FFT Black Box

Cited by 18 publications

References 0 publications

Acceleration with long vector architectures: Implementation and evaluation of the FFT kernel on NEC SX‐Aurora and RISC‐V vector extension

Acceleration with long vector architectures: Implementation and evaluation of the FFT kernel on NEC SX‐Aurora and RISC‐V vector extension

Accelerating FFT Using NEC SX-Aurora Vector Engine

The Ring-LWE Problem in Lattice-Based Cryptography: The Case of Twisted Embeddings

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