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

Gaudry, Pierrick; Kruppa, Alexander I.; Zimmermann, Paul

doi:10.1145/1277548.1277572

Cited by 32 publications

(30 citation statements)

References 16 publications

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“…We modified the GMP library with an improved FFT multiplication code [8]. We compare our results with the two programs mentioned in Section 2, which compute digits of π using Formula (3).…”

Section: Resultsmentioning

confidence: 99%

Time-and space-efficient evaluation of some hypergeometric constants

Cheng

Hanrot

Thomé

et al. 2007

Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation

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Abstract:The currently best known algorithms for the numerical evaluation of hypergeometric constants such as ζ(3) to d decimal digits have time complexityFollowing work from Cheng, Gergel, Kim and Zima, we present a new algorithm with the same asymptotic complexity, but more efficient in practice. Our implementation of this algorithm improves slightly over existing programs for the computation of π, and we announce a new record of 2 billion digits for ζ(3).

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Section: Resultsmentioning

confidence: 99%

Time-and space-efficient evaluation of some hypergeometric constants

Cheng

Hanrot

Thomé

et al. 2007

Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation

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“…In this connection we mention the improvements in multiplication speed reported by [GKZ07]; their paper also gives comparisons with other implementations of large-integer arithmetic.…”

Section: Performance Datamentioning

confidence: 96%

A multimodular algorithm for computing Bernoulli numbers

Harvey¹

2010

Math. Comp.

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Abstract. We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed B k for k = 10 8 , a new record. Our method is to compute B k modulo p for many small primes p and then reconstruct B k via the Chinese Remainder Theorem. The asymptotic time complexity is O(k 2 log 2+ε k), matching that of existing algorithms that exploit the relationship between B k and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method.

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“…In our experiments we used the scaled remainder tree [9], which replaces most division steps in the remainder tree by multiplications. This algorithm is fastest when fft multiplications are done modulo numbers of the form 2 α − 1: we used this Mersenne fft multiplication as well, as implemented in Gaudry, Kruppa and Zimmermann's gmp patch [24]. Other optimizations included computing the product z only once, and treating the prime 2 separately.…”

Section: Bernstein's Smoothness Detection Algorithmmentioning

confidence: 99%

Practical Cryptanalysis of iso/iec 9796-2 and emv Signatures

Coron

Naccache

Tibouchi

et al. 2009

Advances in Cryptology - CRYPTO 2009

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Abstract. In 1999, Coron, Naccache and Stern discovered an existential signature forgery for two popular rsa signature standards, iso/iec 9796-1 and 2. Following this attack iso/iec 9796-1 was withdrawn. iso/iec 9796-2 was amended by increasing the message digest to at least 160 bits. Attacking this amended version required at least 2 61 operations.In this paper, we exhibit algorithmic refinements allowing to attack the amended (currently valid) version of iso/iec 9796-2 for all modulus sizes. A practical forgery was computed in only two days using 19 servers on the Amazon ec2 grid for a total cost of ≃ us$800. The forgery was implemented for e = 2 but attacking odd exponents will not take longer. The forgery was computed for the rsa-2048 challenge modulus, whose factorization is still unknown. The new attack blends several theoretical tools. These do not change the asymptotic complexity of Coron et al.'s technique but significantly accelerate it for parameter values previously considered beyond reach. While less efficient (us$45,000), the acceleration also extends to emv signatures. emv is an iso/iec 9796-2-compliant format with extra redundancy. Luckily, this attack does not threaten any of the 730 million emv payment cards in circulation for operational reasons. Costs are per modulus: after a first forgery for a given modulus, obtaining more forgeries is virtually immediate.

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

Cited by 32 publications

References 16 publications

Time-and space-efficient evaluation of some hypergeometric constants

Time-and space-efficient evaluation of some hypergeometric constants

A multimodular algorithm for computing Bernoulli numbers

Practical Cryptanalysis of iso/iec 9796-2 and emv Signatures

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