Factoring with the quadratic sieve on large vector computers

AbstractThe results are presented of experiments with the multiple polynomial version of the quadratic sieve factorization method on a CYBER 205 and on a NEC SX-2 vector computer. Various numbers in the 50–92 decimal digits range have been factorized, as a contribution to (i) the Cunningham project, (ii) Brent's table of factors of Mersenne numbers, and (iii) a proof by Brent and G. Cohen of the non-existence of odd perfect numbers below 10200. The factorized 92-decimal digits number is a record for general purpose factorization methods

Factorization of a 512 bit RSA modulus

This paper reports on the factorization of the 512 bit number RSA-155 by the number field Sieve factoring method (NFS) and discusses the implications for RS

General purpose integer factoring

This chapter describes the developments since 1970 in general purpose integer factoring and highlights the contributions of Peter L. Montgomery. This article appeared as Chapter 5 of the book Topics in Computational Number Theory inspired by Peter L. Montgomery , edited by Joppe W. Bos and Arjen K. Lenstra and published by Cambridge University Press. See www.cambridge.org/9781107109353

A Distributed Security Architecture for Large Scale Systems

This thesis describes the research leading from the conception, through development, to the practical implementation of a comprehensive security architecture for use within, and as a value-added enhancement to, the ISO Open Systems Interconnection (OSI) model. The Comprehensive Security System (CSS) is arranged basically as an Application Layer service but can allow any of the ISO recommended security facilities to be provided at any layer of the model. It is suitable as an 'add-on' service to existing arrangements or can be fully integrated into new applications. For large scale, distributed processing operations, a network of security management centres (SMCs) is suggested, that can help to ensure that system misuse is minimised, and that flexible operation is provided in an efficient manner. The background to the OSI standards are covered in detail, followed by an introduction to security in open systems. A survey of existing techniques in formal analysis and verification is then presented. The architecture of the CSS is described in terms of a conceptual model using agents and protocols, followed by an extension of the CSS concept to a large scale network controlled by SMCs. A new approach to formal security analysis is described which is based on two main methodologies. Firstly, every function within the system is built from layers of provably secure sequences of finite state machines, using a recursive function to monitor and constrain the system to the desired state at all times. Secondly, the correctness of the protocols generated by the sequences to exchange security information and control data between agents in a distributed environment, is analysed in terms of a modified temporal Hoare logic. This is based on ideas concerning the validity of beliefs about the global state of a system as a result of actions performed by entities within the system, including the notion of timeliness. The two fundamental problems in number theory upon which the assumptions about the security of the finite state machine model rest are described, together with a comprehensive survey of the very latest progress in this area. Having assumed that the two problems will remain computationally intractable in the foreseeable future, the method is then applied to the formal analysis of some of the components of the Comprehensive Security System. A practical implementation of the CSS has been achieved as a demonstration system for a network of IBM Personal Computers connected via an Ethernet LAN, which fully meets the aims and objectives set out in Chapter 1. This implementation is described, and finally some comments are made on the possible future of research into security aspects of distributed systems.IBM (United Kingdom) Laboratories Hursley Park, Winchester, U

Faktorisierungsverfahren für ganze Zahlen

Die vorliegende Diplomarbeit behandelt das Faktorisierungsproblem, also von der Berechnung eines Teilers einer zusammengesetzten ganzen Zahl, und primär den Methoden, die sowohl historisch als auch aktuell dafür zur Anwendung kommen. Zuvor führen wir alle mathematischen Grundlagen, die wir zur Vorstellung der Faktorisierungsalgorithmen benötigen, ein. Die im Hauptteil beinhalteten Methoden zur Teilerberechnung, deren Funktionsweisen wir detailliert vorstellen, sind die Probedivision, Pollards - und (p-1)-Methode, die Elliptic Curve Method, die Methode nach Fermat, nach Legendre, der Kettenbruchalgorithmus, das quadratische Sieb und das Zahlkörpersieb. Für ein besseres Verständnis geben wir einige der Algorithmen als funktionstüchtigen Code des Algebra- Programms PARI/GP an und testen diese auf ihre Geschwindigkeit und Eigenschaften. Zwei Fragen schenken wir bei der Beschreibung dieser Methoden besondere Beachtung: 1) Wie schnell arbeitet der jeweilige Algorithmus? 2) Spielen Charakteristika der zu faktorisierenden Zahlen eine Rolle? Diese Fragestellungen sind im Besonderen im Anwendungsbereich des Faktorisierungsproblems, dem RSA-Verfahren in der Verschlüsselungstheorie, von zentraler Bedeutung, da die Sicherheit des Verfahrens auf deren Komplexität beruht. Danach nehmen wir aktuelle Faktorisierungsprogramme praktisch unter die Lupe, testen also mittels ausgewählter Zahlen ihre Faktorisierungsgeschwindigkeit. Abschließend erfolgt wir eine Zeitreise durch die Faktorisierungsgeschichte, eine Analyse des gegenwärtigen Forschungsstandes und ein Blick in die Zukunft der Zahlenfaktorisierung.This diploma thesis is about the factorization problem, i.e. to compute a factor of a composite integer, and primary the historical and current methods to do this. First of all we discuss the mathematical background we need to present the factorization algorithms. The main part of this thesis, the detailed introductions of the factorization algorithms, will cover the trial division algorithm, Pollard's Rho- and (p-1)-method, the elliptic curve method, Fermat's factorization method, Legendre's factorization method, the continued fraction method, the quadratic sieve and the number field sieve. For a better understanding we specify for some of the methods a working source code for the algebra software PARI/GP, which we will check for speed and properties. The two following questions will cause our special interest: 1) How fast terminates the algorithm? 2) Do the characteristics of the factoring numbers play a role? These questions are in particular important for the RSA-algorithm in the cryptography, where the factorization problem plays the central part for the security. Afterwards we test some of the current factorization softwares for their speed and finally we go on a time travel of the factorization history, analyze the state of the art and look into the future of the integer factorization