The twenty-fourth Fermat number is composite

Mayer, Ernst W.; Papadopoulos, Jason S.

doi:10.1090/s0025-5718-02-01479-5

Cited by 19 publications

(10 citation statements)

References 27 publications

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“…This is especially true in light of the heuristic error estimates given in [3] which, while giving an expected error far below our proved bound (by a factor of roughly √ N ), appear to agree with numerical experiment.…”

Section: Resultssupporting

confidence: 85%

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Rapid multiplication modulo the sum and difference of highly composite numbers

Percival

2002

Math. Comp.

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Abstract. We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form a ± b where p|ab p is small. In particular this allows rapid computation modulo numbers of the form k ·2 n ±1.In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.

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Section: Resultssupporting

confidence: 85%

“…As we will see later, these estimates are pessimistic by a factor of √ N . It is suggested in a later paper that the maximum errors should be roughly O( √ N log N log log N ) [3]. This is argued based on the hypothesis that the errors in the FFT resemble a random walk of length N log N .…”

Section: Previous Fft Error Estimatesmentioning

confidence: 99%

Rapid multiplication modulo the sum and difference of highly composite numbers

Percival

2002

Math. Comp.

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“…The optimal choice of ℓ for 170 bits is about 2 21 . Since the Amazon instances are memory-constrained (less than 1gb of ram per core), we preferred to use ℓ = 2 20 . This choice had the additional advantage of making the final linear algebra step faster, which is convenient since this step was run on a single off-line pc.…”

Section: The Experiment: Outline Details and Resultsmentioning

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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“…The only known FermatPrimes are 2 2 k + 1 for k = 0, 1, 2, 3, 4. It is currently known that 2 2 k + 1 is composite for 5 k 32 [3]. Moreover, it is conjectured that the number of Fermat-Primes is finite with 2 2 4 + 1 = 65537 being the largest Fermat-Prime (see, for example, [5]).…”

Section: Claimmentioning

confidence: 99%

The complexity of membership problems for circuits over sets of integers

Travers

2006

Theoretical Computer Science

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We investigate the complexity of membership problems for {∪, ∩, − , +, ×}-circuits computing sets of integers. These problems are a natural modification of the membership problems for circuits computing sets of natural numbers studied by McKenzie and Wagner [The complexity of membership problems for circuits over sets of natural numbers, We show that there are several membership problems for which the complexity in the case of integers differs significantly from the case of the natural numbers: testing membership in the subset of integers produced at the output of a {∪, +, ×}-circuit is NEXPTIME-complete, whereas it is PSPACE-complete for the natural numbers. As another result, evaluating { − , +}-circuits is shown to be P-complete for the integers and PSPACE-complete for the natural numbers. The latter result extends McKenzie and Wagner's work in nontrivial ways. Furthermore, evaluating {×}-circuits is shown to be NL ∧ L-complete, and several other cases are resolved.

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The twenty-fourth Fermat number is composite

Cited by 19 publications

References 27 publications

Rapid multiplication modulo the sum and difference of highly composite numbers

Rapid multiplication modulo the sum and difference of highly composite numbers

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

The complexity of membership problems for circuits over sets of integers

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