Mark S. Manasse

Competitive snoopy caching

In a snoopy cache multiprocessor system, each processor has a cache in which it stores blocks of data. Each cache is connected to a bus used to communicate with the other caches and with main memory. Each cache monitors the activity on the bus and in its own processor and decides which blocks of data to keep and which to discard. For several of the proposed architectures for snoopy caching systems, we present new on-line algorithms to be used by the caches to decide which blocks to retain and which to drop in order to minimize communication over the bus. We prove that, for any sequence of operations, our algorithms communication costs are within a constant factor of the minimum required for that sequence; for some of our algorithms we prove that no on-line algorithm has this property with a smaller constant.

Competitive algorithms for server problems

Abstract The k-server problem is that of planning the motion of k mobile servers on the vertices of a graph under a sequence of requests for service. Each request consists of the name of a vertex, and is satisfied by placing a server at the requested vertex. The requests must be satisfied in their order of occurrence. The cost of satisfying a sequence of requests is the distance moved by the servers. In this paper we study on-line algorithms for this problem from the competitive point of view. That is, we seek to develop on-line algorithms whose performance on any sequence of requests is as close as possible to the performance of the optimum off-line algorithm. We obtain optimally competitive algorithms for several important cases. Because of the flexibility in choosing the distances in the graph and the number of servers, the k-server problem can be used to model a number of important paging and caching problems. It can also be used as a building block for solving more general problems. We show how server algorithms can be used to solve a seemingly more general class of problems known as task systems.

Competitive algorithms for on-line problems

An on-line problem is one in which an algorithm must handle a sequence of requests, satisfying each request without knowledge of the future requests. Examples of on-line problems include scheduling the motion of elevators, finding routes in networks, allocating cache memory, and maintaining dynamic data structures. A competitive algorithm for an on-line problem has the property that its performance on any sequence of requests is within a constant factor of the performance of any other algorithm on the same sequence. This paper presents several general results concerning competitive algorithms, as well as results on specific on-line problems.

The number field sieve

The number field sieve is an algorithm to factor integers of the form re − s for small positive r and |s|. The algorithm depends on arithmetic in an algebraic number field. We describe the algorithm, discuss several aspects of its implementation, and present some of the factorizations obtained. A heuristic run time analysis indicates that the number field sieve is asymptotically substantially faster than any other known factoring method, for the integers that it applies to. The number field sieve can be modified to handle arbitrary integers. This variant is slower, but asymptotically it is still expected to beat all older factoring methods.

Factoring by electronic mail

In this paper we describe our distributed implementation of two factoring algorithms. the elliptic curve method (ecm) and the multiple polynomial quadratic sieve algorithm (mpqs). Since the summer of 1987. our erm-implementation on a network of MicroVAX processors at DECs Systems Research Center has factored several most and more wanted numbers from the Cun- ningham project. In the summer of 1988. we implemented the multiple polynomial quadratic sieve algorithm on rhe same network On this network alone. we are now able to factor any !@I digit integer, or to find 35 digit factors of numbers up to 150 digits long within one month. To allow an even wider distribution of our programs we made use of electronic mail networks For the distribution of the programs and for inter-processor communicatton. Even during the mitial stage of this experiment machines all over the United States and at various places in Europe and Ausnalia conhibuted 15 percent of the total factorization effort. At all the sites where our program is running we only use cycles that would otherwise have been idle. This shows that the enormous computational task of factoring 100 digit integers with the current algoritluns can be completed almost for free. Since we use a negligible fraction of the idle cycles of alI the machines on the worldwide elecnonic mail networks. we could factor 100 digit integers within a few days with a little more help.

The factorization of the ninth Fermat number

In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 2(512) + 1. It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, which is a factoring algorithm that depends on arithmetic in an algebraic number field. In the present case, the number field used was Q(fifth-root 2) . The calculations were done on approximately 700 workstations scattered around the world, and in one of the final stages a supercomputer was used. The entire factorization took four months.

Factoring with two large primes

The study of integer factoring algorithms and the design of faster factoring algorithms is a subject of great importance in cryptology (cf. [1]), and a constant concern for cryptographers. In this paper we present a new technique that proved to be extremely useful, not only to achieve a considerable speed-up of an older and widely studied factoring algorithm, but also, and more importantly, to make practical application of a new factoring algorithm feasible. While this first application does not pose serious threats to factorization-based cryptosystems, the consequences of the second application could be very encouraging (from the cryptanalysts point of view).

On the Factorization of RSA-120

We present data concerning the factorization of the 120-digit number RSA-120, which we factored on July 9, 1993, using the quadratic sieve method. The factorization took approximately 825 MIPS years and was completed within three months real time. At the time of writing RSA-120 is the largest integer ever factored by a general purpose factoring algorithm. We also present some conservative extrapolations to estimate the difficulty of factoring even larger numbers, using either the quadratic sieve method or the number field sieve, and discuss the issue of the crossover point between these two methods.

Addendum: The Factorization of the Ninth Fermat Number

In this paper we exhibit the full prime factorization of the ninth Fermat number Fg = 2512 + 1 . It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, which is a factoring algorithm that depends on arithmetic in an algebraic number field. In the present case, the number field used was Q(. The calculations were done on approximately 700 workstations scattered around the world, and in one of the final stages a supercomputer was used. The entire factorization took four months.