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Efficient SIMD arithmetic modulo a Mersenne number

Arjen Lenstra, Joppe Willem Bos, Thorsten Kleinjung

This paper describes carry-less arithmetic operations modulo an integer 2^M − 1 in the thousand-bit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implemen ...

IEEE Computer Soc Press2011

Efficient SIMD Arithmetic Modulo a Mersenne Number

Arjen Lenstra, Joppe Willem Bos, Thorsten Kleinjung

This paper describes carry-less arithmetic operations modulo an integer 2^M-1 in the thousand-bit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementa ...

IEEE2011

A kilobit special number field sieve factorization

Arjen Lenstra, Thorsten Kleinjung, Dag Arne Osvik

We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 21039 -1. Although this factorization is orders of magnitude ...

2007

Global mixed-integer dynamic optimization

Benoît Chachuat

Recent advances in process synthesis, design, operations, and control have created an increasing demand for efficient numerical algorithms for optimizing a dynamic system coupled with discrete decisions; these problems are termed mixed-integer dynamic opti ...

2005

The Overhead of Consensus Failure Recovery

Rachid Guerraoui, Partha Dutta

Many reliable distributed systems are consensus-based and typically operate under two modes: a fast normal mode in failure-free periods, and a slower recovery mode following failures. A lot of work has been devoted to optimizing the normal mode, but little ...

2004

Irregular primes and Cyclotomic Invariants to Twelve Million

Mohammad Amin Shokrollahi

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to twelve million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes (Shokrollahi (1996)). ...

2001

Factorization of RSA-140 using the number field sieve

Arjen Lenstra, Pénélope Leyland, Patric Zimmermann

On February 2, 1999, we completed the factorization of the 140-digit number RSA-140 with the help of the Number Field Sieve factoring method (NFS). This is a new general factoring record. The previous record was established on April 10, 1996 by the factori ...

Springer-Verlag New York, Ms Ingrid Cunningham, 175 Fifth Ave, New York, Ny 10010 Usa1999

Soft real-time analysis of asynchronous agreement algorithms using Petri nets

Nicoleta Sergent

A distributed algorithm comprises a collection of sequential-code entities, which are spread over different computers connected to a network. The process of designing a distributed algorithm is influenced by the assumptions we make on the computational env ...

EPFL1998

Deciding properties of number fields without factoring

Mohammad Amin Shokrollahi

The polynomial time algorithm of Lenstra, Lenstra, and Lovász [15] for factoring integer polynomials and variants thereof have been widely used to show that various computational problems in number theory have polynomial time solutions. Among them is the p ...

1997

The magic words are squeamish ossifrage [RSA public key cryptography]

Arjen Lenstra, Pénélope Leyland

We describe the computation which resulted in the title of this paper. Furthermore, we give an analysis of the data collected during this computation. From this data, we derive the important observation that in the final stages, the progress of the double ...

1995