Fast Fourier Transform and Convolution Algorithms

Nussbaumer, Henri J.

doi:10.1007/978-3-642-81897-4

Cited by 447 publications

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“…Polynomial multiplication can be performed with O(n log n) operations in Z q using the convolution property of the number theoretic transform (NTT) [Pol71,Nus82,Win96,GG03,Bla10]. The NTT is basically a fast Fourier transformation (FFT) defined over a finite field or ring so that no inaccurate floating point or complex arithmetic is needed.…”

Section: The Number Theoretic Transformmentioning

confidence: 99%

“…Additionally, in this work we assume for simplicity that the modulus q is prime. For more details on the NTT with composite moduli we refer to [Nus82] and first introduce the definition of primitive roots of unity.…”

Section: The Number Theoretic Transformmentioning

confidence: 99%

“…Definition 17 (NTT Supporting Circular Convolutions, see [Nus82]). For a given primitive nth root of unity ω in Z q , where q is a prime, the generic forward NTT that is supporting circular convolutions and transformation of a length-n vector a = (a 0 , .…”

Section: The Number Theoretic Transformmentioning

confidence: 99%

“…Theorem 4 (Existence of the NTT Supporting Circular Convolutions, see [Nus82]). An NTT of length n that supports circular convolutions and that is defined modulo a prime q exists if and only if n|(q − 1).…”

Section: The Number Theoretic Transformmentioning

confidence: 99%

“…We also evaluated RMult realized with Nussbaumer's method [Nus82,BCNS15] (denoted RMult N b ) and obtained better performance than with the NTT using q . But as a fast software implementation is not in the scope of this work we did not investigate the root cause for the better performance but consider a detailed comparison of both algorithms as valuable future work.…”

Section: Software Performancementioning

confidence: 99%

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Efficient implementation of ideal lattice-based cryptography

Pöppelmann¹

2017

It - Information Technology

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Digital signatures and public-key encryption are used to protect almost any secure communication channel on the Internet or between embedded devices. Currently, protocol designers and engineers usually rely on schemes that are either based on the factoring assumption (RSA) or on the hardness of the discrete logarithm problem (DSA/ECDSA). But in case of advances in classical cryptanalysis or progress on the development of quantum computers the hardness of these closely related problems might be seriously weakened. In order to prepare for such an event, research on alternatives is required to provide long-term security.In this thesis, we focus on the efficient implementation of such alternative public-key cryptosystems whose security is based on the intractability of certain computational problems on ideal lattices. While an extensive theoretical background exists for lattice-based and ideal latticebased cryptography, not much is known about the efficiency of practical instantiations, especially on constrained and cost-sensitive platforms. We thus investigate novel algorithms and implementation techniques for fast and flexible polynomial multiplication and Gaussian sampling and then use these building blocks to implement public-key encryption and signature schemes. The results provided in this thesis show that lattice-based schemes can be optimized for high performance or resource efficiency on embedded microcontrollers and reconfigurable hardware. Our implementations of a public-key encryption scheme based on the ring learning with errors problems (RLWE) or of the bimodal lattice signature scheme (BLISS) can even outperform classical ECC-and RSA-based implementations.Lattice-based cryptography can also be used to realize homomorphic cryptography that allows computation on encrypted data. However, due to the large parameter sets and complex operations required, even for simple homomorphic evaluation operations, the performance of these schemes is a major issue preventing practical usage. In this thesis we investigate options for acceleration of homomorphic cryptography in a cloud environment using reconfigurable hardware. We implement all evaluation operations of the YASHE homomorphic encryption scheme and propose methods to deal with large ciphertext and key sizes as well as limited memory bandwidth. KeywordsPost-quantum cryptography, public-key cryptosystem, embedded system, microcontroller, FPGA KurzfassungDigitale Signaturen und Public-Key-Verschlüsselung werden für den Schutz nahezu jeder sicheren Kommunikation über das Internet oder zwischen eingebetteten Systemen genutzt. Die Sicherheit basiert dabei entweder auf der Faktorisierungsannahme (RSA) oder der Annahme, dass es schwer ist, das diskrete Logarithmus-Problem (DSA/ECDSA) zu lösen. Durch Fortschritte in der klassischen Kryptoanalyse oder bei der Entwicklung von Quantencomputern könnten diese Probleme allerdings in Zukunft ernsthaft geschwächt oder gelöst werden. Daher ist Forschung zu alternativen Public-Key-Kryptosystemen erforderlich, die in ...

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Section: The Number Theoretic Transformmentioning

confidence: 99%

Section: The Number Theoretic Transformmentioning

confidence: 99%

Section: The Number Theoretic Transformmentioning

confidence: 99%

Section: The Number Theoretic Transformmentioning

confidence: 99%

Section: Software Performancementioning

confidence: 99%

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Efficient implementation of ideal lattice-based cryptography

Pöppelmann¹

2017

It - Information Technology

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High resolution fast quantitative docking using fourier domain correlation techniques

Blom

Sygusch

1997

Proteins

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A 'docking' method based on finite grid forcefield sampling is proposed for fast evaluation of interaction energies between macromolecules and ligands. Forcefield used to calculate interaction energies utilizes a potential energy function composed of a 1/r-dependent electrostatic term and a (6-12) Lennard-Jones term for van der Waals interactions. Fast evaluation makes use of the convolu-tion theorem allowing a point-by-point N-dimensional correlation in direct space to be replaced by a simple multiplication in spatial frequency space. Predictive accuracy was assessed by using seven protein-ligand complexes available from the Brookhaven Data Bank and determined crystallographically to high resolution. Successful prediction of ligand position and determination of ligand-protein interaction en-thalpy was dependent on forcefield sampling grid size. Minimum interaction enthalpy calculated for four protein-ligand complexes coincided with crystallographic structures that used sampling grid sizes of 0.25 Å resolution and was independent of ligand starting position and orientation. Successful docking was obtained for the remaining complexes at same grid resolution provided ligand starting positions were not randomized. Sensitivity of the docking algorithm to starting orientation was a consequence of tight fit of respective ligand structures with their protein target sites for these three cases and can be circumvented by using finer rotational sampling grids for the ligand. Boltzmann statistics derived from calculated interaction energies successfully extracted the observed ribonuclease A cytidylic acid complex from a manifold of similar interaction energies. The proposed method was able to reproduce the observed crystallographic complex by using a dynamical description of ligand.

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Fast Fourier Transforms in Electromagnetics

Guo

2014

Wiley Encyclopedia of Electrical and Electronics Engineering

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This Chapter review the fast Fourier transform (FFT) technique and its application to computational electromagnetics, especially to the fast solver algorithms including the Conjugate Gradient Fast Fourier Transform (CG‐FFT) method, Precorrected Fast Fourier Transform (pFFT) method, Adaptive Integral Method (AIM), Greens Function Interpolation with FFT (GI‐FFT) method and Integral Equations with FFT (IE‐FFT) method. The basic ideas used in the FFT applications are addressed while the brief introduction to integral equation method is conducted. The general formulation and procedure in the integral equation method, surface integral equations, volume integral equations, solutions to integral equations, and their implementations of fast Fourier transform algorithm are also briefed together with fast convolution using fast Fourier transform. Fast integral equation method developed based on fast Fourier transform are reviewed where conjugate gradient fast Fourier transform method, and precorrected fast Fourier transform method (where projection operators and interpolation operators are also highlighted), adaptive integral method, Greens function interpolation with FFT approach and integral equations with FFT method are also described. While the matching schemes for gradients of Green's functions are addressed, accuracy and complexity, memory requirement and computational cost, and error controls and estimations are also discussed.

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Fast Fourier Transform and Convolution Algorithms

Cited by 447 publications

References 0 publications

Efficient implementation of ideal lattice-based cryptography

Efficient implementation of ideal lattice-based cryptography

High resolution fast quantitative docking using fourier domain correlation techniques

Fast Fourier Transforms in Electromagnetics

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