Random primes in arithmetic progressions

Giorgi, Pascal; Grenet, Bruno; Cray, Armelle Perret du; Roche, Daniel S.

doi:10.48550/arxiv.2202.05955

Cited by 1 publication

(2 citation statements)

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“…The next fact presents a variant with better probability bounds and a larger range of validity. We provide the complete proof in a short note [17]. Fact 2.2.…”

Section: Prime Number Generationmentioning

confidence: 99%

“…While the rigorous proof of this fact implies to have large values for 𝜆, it is not too difficult to see by running few experiments that such triples exist with good probability even for smaller values. One can find some preliminary experiments in our short note [17]. In this paper, we rely on Fact 2.2 to provide rigorously proven algorithm, thus implying limitations on its practicability.…”

Section: Prime Number Generationmentioning

confidence: 99%

See 1 more Smart Citation

Sparse Polynomial Interpolation and Division in Soft-linear Time

Giorgi

Grenet

Cray

et al. 2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of computing the exact quotient of two given polynomials. These methods are efficient in terms of the bit-length of the sparse representation, that is, the number of nonzero terms, the size of coefficients, the number of variables, and the logarithm of the degree. At the core of our results is a new Monte Carlo randomized algorithm to recover a polynomial 𝑓 (𝑥) with integer coefficients given a way to evaluate 𝑓 (𝜃 ) mod 𝑚 for any chosen integers 𝜃 and 𝑚. This algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is softly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with soft-linear bit complexity in the total output size. For polynomials with integer coefficients, the best previously known results have at least a cubic dependency on the bit-length of the exponents. CCS CONCEPTS• Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms; • Mathematics of computing → Probabilistic algorithms.

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“…The next fact presents a variant with better probability bounds and a larger range of validity. We provide the complete proof in a short note [17]. Fact 2.2.…”

Section: Prime Number Generationmentioning

confidence: 99%

Section: Prime Number Generationmentioning

confidence: 99%