Fast Library for Number Theory: An Introduction

Hart, William B.

doi:10.1007/978-3-642-15582-6_18

Cited by 123 publications

(137 citation statements)

References 8 publications

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“…LLL and its variants are implemented in many software packages, notably in NTL [Sho], FLINT [HJP14] and fplll [CPS13]. The latter also implements L2.…”

Section: Lattice Reduction Algorithmsmentioning

confidence: 99%

On the concrete hardness of Learning with Errors

Albrecht

Player

Scott

2015

Journal of Mathematical Cryptology

512

302

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Abstract. The Learning with Errors (LWE) problem has become a central building block of modern cryptographic constructions. This work collects and presents hardness results for concrete instances of LWE. In particular, we discuss algorithms proposed in the literature and give the expected resources required to run them. We consider both generic instances of LWE as well as small secret variants. Since for several methods of solving LWE we require a lattice reduction step, we also review lattice reduction algorithms and use a refined model for estimating their running times. We also give concrete estimates for various families of LWE instances, provide a Sage module for computing these estimates and highlight gaps in the knowledge about algorithms for solving the Learning with Errors problem.

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“…LLL and its variants are implemented in many software packages, notably in NTL [Sho], FLINT [HJP14] and fplll [CPS13]. The latter also implements L2.…”

Section: Lattice Reduction Algorithmsmentioning

confidence: 99%

On the concrete hardness of Learning with Errors

Albrecht

Player

Scott

2015

Journal of Mathematical Cryptology

512

302

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“…The main reason for this choice was the existence of well-tested, high performance packages for doing such computations, such as FLINT [17] and zn poly [21]. In fact, Victor Shoup's well-tested NTL package [34] was the only library we were aware of with asymptotically fast NTTs.…”

Section: The First Polynomial Methodsmentioning

confidence: 99%

Congruent Number Theta Coefficients to 1012

Hart

Tornaría

Watkins

2010

Lecture Notes in Computer Science

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We report on a computation of congruent numbers, which subject to the Birch and Swinnerton-Dyer conjecture is an accurate list up to 10 12 . The computation involves multiplying large theta series as per Tunnell (1983). The first method, which we describe in some detail, uses a multimodular disk based technique for multiplying polynomials out-of-core which minimises expensive disk access by keeping data truncated. The second technique uses "Bailey's four-step" Fast Fourier method in combination with compression of the data to disk in intermediate stages.

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“…Assumptions: In our analysis we assume the use of fast arithmetic (see [1]), which is available in FLINT [9]. Also, only for the simplicity of bit-complexity analysis, we will assume throughout that coefficients of f and g are of O(m) bits, where m is the degree of g, the inner polynomial in the composition f (g).…”

Section: Problem Statementmentioning

confidence: 99%

“…In the original application the bit complexity was not considered, however we show that the algorithm is asymptotically fast for polynomial composition in Z[x]. The algorithm was rediscovered while implementing the number theory library FLINT [9], and we are grateful to Joris van der Hoeven for pointing out its first occurrence in the literature.…”

Section: Problem Statementmentioning

confidence: 99%

Practical Divide-and-Conquer Algorithms for Polynomial Arithmetic

Hart

Novocin

2011

Computer Algebra in Scientific Computing

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Abstract. We investigate two practical divide-and-conquer style algorithms for univariate polynomial arithmetic. First we revisit an algorithm originally described by Brent and Kung for composition of power series, showing that it can be applied practically to composition of polynomials in Z [x] given in the standard monomial basis. We offer a complexity analysis, showing that it is asymptotically fast, avoiding coefficient explosion in Z[x]. Secondly we provide an improvement to Mulders' polynomial division algorithm. We show that it is particularly efficient compared with the multimodular algorithm. The algorithms are straightforward to implement and available in the open source FLINT C library. We offer a practical comparison of our implementations with various computer algebra systems.

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Fast Library for Number Theory: An Introduction

Abstract: Abstract. We discuss FLINT (Fast Library for Number Theory), a library to support computations in number theory, including highly optimised routines for polynomial arithmetic and linear algebra in exact rings.

Cited by 123 publications

References 8 publications

On the concrete hardness of Learning with Errors

On the concrete hardness of Learning with Errors

Congruent Number Theta Coefficients to 1012

Practical Divide-and-Conquer Algorithms for Polynomial Arithmetic

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