Publications related to A kilobit special number field sieve factorization

Moments of the number of points in a bounded set for number field lattices

Maryna Viazovska, Nihar Prakash Gargava, Vlad Serban

We examine the moments of the number of lattice points in a fixed ball of volume

V

for lattices in Euclidean space which are modules over the ring of integers of a number field

K

. In particular, denoting by

ω_K

the number of roots of unity in

K

, we ...

arXiv2023

The double exponential runtime is tight for 2-stage stochastic ILPs

Kim-Manuel Klein, Klaus Jansen, Alexandra Anna Lassota

We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an integer program of the form min{c(T)x vertical bar Ax = ...

SPRINGER HEIDELBERG2022

Recoloring subgraphs of K-2n for sports scheduling

Dominique de Werra

The exploration of one-factorizations of complete graphs is the foundation of some classical sports scheduling problems. One has to traverse the landscape of such one-factorizations by moving from one of those to a so-called neighbor one-factorization. Thi ...

ELSEVIER2021

A new elementary proof of the Prime Number Theorem

Florian Karl Richter

Let

\Omega(n)

denote the number of prime factors of

n

. We show that for any bounded

f\colon\mathbb{N}\to\mathbb{C}

one has [ \frac{1}{N}\sum_{n=1}^N, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1). ] This yields a ...

2021

Computation of a 768-Bit Prime Field Discrete Logarithm

Arjen Lenstra, Thorsten Kleinjung

This paper reports on the number field sieve computation of a 768-bit prime field discrete logarithm, describes the different parameter optimizations and resulting algorithmic changes compared to the factorization of a 768-bit RSA modulus, and briefly disc ...

Springer International Publishing Ag2017

On the Analysis of Public-Key Cryptologic Algorithms

Andrea Miele

The RSA cryptosystem introduced in 1977 by Ron Rivest, Adi Shamir and Len Adleman is the most commonly deployed public-key cryptosystem. Elliptic curve cryptography (ECC) introduced in the mid 80's by Neal Koblitz and Victor Miller is becoming an increasin ...

EPFL2015

Efficient matrix computation for tensor-product isogeometric analysis: the use of sum factorization

Annalisa Buffa, Pablo Antolin Sanchez, Maurizio Martinelli

In this paper we discuss the use of the sum-factorization for the calculation of the integrals arising in Galerkin isogeometric analysis. While introducing very little change in an isogeometric code based on element-by-element quadrature and assembling, th ...

2015

Cofactorization on Graphics Processing Units

Arjen Lenstra, Joppe Willem Bos, Thorsten Kleinjung, Andrea Miele

We show how the cofactorization step, a compute-intensive part of the relation collection phase of the number field sieve (NFS), can be farmed out to a graphics processing unit. Our implementation on a GTX 580 GPU, which is integrated with a state-of-the-a ...

Springer-Verlag Berlin2014

A heterogeneous computing environment to solve the 768-bit RSA challenge

Arjen Lenstra, Pénélope Leyland, Pascal Jermini, Joppe Willem Bos, Thorsten Kleinjung, Dag Arne Osvik, Michela Thiémard, Heinz Stockinger

In December 2009 the 768-bit, 232-digit number RSA-768 was factored using the number field sieve. Overall, the computational challenge would take more than 1700 years on a single, standard core. In the article we present the heterogeneous computing approac ...

Springer Verlag2012

On the Cryptanalysis of Public-Key Cryptography

Joppe Willem Bos

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the ...

EPFL2012

A kilobit special number field sieve factorization

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Moments of the number of points in a bounded set for number field lattices

The double exponential runtime is tight for 2-stage stochastic ILPs

Recoloring subgraphs of K-2n for sports scheduling

A new elementary proof of the Prime Number Theorem

Computation of a 768-Bit Prime Field Discrete Logarithm

On the Analysis of Public-Key Cryptologic Algorithms

Efficient matrix computation for tensor-product isogeometric analysis: the use of sum factorization

Cofactorization on Graphics Processing Units

A heterogeneous computing environment to solve the 768-bit RSA challenge

On the Cryptanalysis of Public-Key Cryptography

Moments of the number of points in a bounded set for number field lattices

A new elementary proof of the Prime Number Theorem

Recoloring subgraphs of K-2n for sports scheduling

The double exponential runtime is tight for 2-stage stochastic ILPs

Cofactorization on Graphics Processing Units

Efficient matrix computation for tensor-product isogeometric analysis: the use of sum factorization

On the Cryptanalysis of Public-Key Cryptography

Computation of a 768-Bit Prime Field Discrete Logarithm

On the Analysis of Public-Key Cryptologic Algorithms

A heterogeneous computing environment to solve the 768-bit RSA challenge