Rapid  multiplication   modulo the sum and difference   of highly composite numbers

Percival, Colin

doi:10.1090/s0025-5718-02-01419-9

Cited by 14 publications

(12 citation statements)

References 11 publications

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“…At n = 2 22 the floating point precision was observed to introduce 6 errors (this is consistent with the findings reported in the literature [13], [17]. Following the established practices in literature we used radix r = 2 16 for the FFT computations, so that each n bit integer becomes a string of N = n 16 digits, with each digit being a 16 bit number.…”

Section: Methodssupporting

confidence: 82%

Fast Modular Reduction for Large Wordlengths via One Linear and One Cyclic Convolution

Phatak¹,

Goff²

17th IEEE Symposium on Computer Arithmetic (ARITH'05)

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Abstract-Modular reduction is a fundamental operation in cryptographic systems. Most well known modular reduction methods including Barrett's and Montgomery's algorithms leverage some-pre computations to avoid divisions so that the main complexity of these methods lies in a sequence of two long multiplications. For large wordlengths a multiplication which is tantamount to a linear convolution is performed via the Fast Fourier Transform (FFT) or other transform-based techniques as in the Schonhage-Strassen multiplication algorithm.We show a fundamental property (the separation principle): in a modular reduction based on long multiplications, the linear convolution required by one of the two long multiplications can be replaced by a cyclic convolution, and the halves can be separated using other information available due to the intrinsic redundancy of the operations. This reduces the number of operations by about 25%. We demonstrate that both Barrett's and Montgomery's methods can be sped up by using the aforementioned fundamental principle. It is shown that a direct application of this algorithm to modular exponentiation (either using Barrett's or Montgomery's methods) can be expected to yield about about 17% speedup.

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Section: Methodssupporting

confidence: 82%

Fast Modular Reduction for Large Wordlengths via One Linear and One Cyclic Convolution

Phatak¹,

Goff²

17th IEEE Symposium on Computer Arithmetic (ARITH'05)

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“…It is written mainly for testing large Mersenne numbers 2 p − 1 for primality in the in the Great Internet Mersenne Prime Search [24]. It uses a DWT for multiplication mod a2 n ± c, with a and c not too large, see [17]. We compared multiplication modulo 2 2wn − 1 in Prime95 version 24.14.2 with multiplication of n-word integers using our SSA implementation on a Pentium 4 at 3.2 GHz, and on an Opteron 250 at 2.4 GHz, see Figure 4.…”

Section: Results and Conclusionmentioning

confidence: 99%

“…Some differences between Prime95 and our implementation need to be pointed out: due to the floating point nature of Prime95's FFT, rounding errors can build up for particular input data to the point where the result are incorrectly rounded to integers. The floating point FFT can be made provably correct, see again [17], but at the cost of using larger FFT lengths. For example, for a length 2 25 FFT, [17] allows 9 bits per double, whereas Prime95 uses up to 17.76.…”

Section: Results and Conclusionmentioning

confidence: 99%

A gmp-based implementation of schönhage-strassen's large integer multiplication algorithm

Gaudry

Kruppa

Zimmermann

2007

Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation

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Schönhage-Strassen's algorithm is one of the best known algorithms for multiplying large integers. Implementing it efficiently is of utmost importance, since many other algorithms rely on it as a subroutine. We present here an improved implementation, based on the one distributed within the GMP library. The following ideas and techniques were used or tried: faster arithmetic modulo 2 n + 1, improved cache locality, Mersenne transforms, Chinese Remainder Reconstruction, the √ 2 trick, Harley's and Granlund's tricks, improved tuning.

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“…Note also that the NTT is not considered as a very efficient method for most applications in signal processing as approximate solutions are usually sufficient (see [Per03] for calculations of the required accuracy for polynomial multiplication). Especially, the complexity of the butterfly structure in the PE requires a lot of resources as multiplication by the twiddle factor is just a general purpose multiplication followed by a modular reduction.…”

Section: Design Decisions For Lattice-based Cryptographymentioning

confidence: 99%

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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Rapid multiplication modulo the sum and difference of highly composite numbers

Cited by 14 publications

References 11 publications

Fast Modular Reduction for Large Wordlengths via One Linear and One Cyclic Convolution

Fast Modular Reduction for Large Wordlengths via One Linear and One Cyclic Convolution

A gmp-based implementation of schönhage-strassen's large integer multiplication algorithm

Efficient implementation of ideal lattice-based cryptography

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