### Solving the generalized Pell equation x 2 − Dy 2 = N

"... This article gives fast, simple algorithms to find integer solutions x, y to generalized Pell equations, x 2 − Dy 2 = N, for D a positive integer, not a square, and N a nonzero integer. Pell equations have fascinated for centuries. ..."

Abstract
- Add to MetaCart

This article gives fast, simple algorithms to find integer solutions x, y to generalized Pell equations, x 2 − Dy 2 = N, for D a positive integer, not a square, and N a nonzero integer. Pell equations have fascinated for centuries.

### FactInt Advanced Methods for Factoring Integers Version 1.5.2

, 2007

"... FactInt 2 This package for GAP 4 provides a general-purpose integer factorization routine, which makes use of a combination of factoring methods. In particular it contains implementations of the following algorithms: • Pollard’s p − 1 • Williams ’ p + 1 • Elliptic Curves Method (ECM) • Continued Fra ..."

Abstract
- Add to MetaCart

FactInt 2 This package for GAP 4 provides a general-purpose integer factorization routine, which makes use of a combination of factoring methods. In particular it contains implementations of the following algorithms: • Pollard’s p − 1 • Williams ’ p + 1 • Elliptic Curves Method (ECM) • Continued Fraction Algorithm (CFRAC)

### Parallel Computing in Cryptoanalysis: Experiences in a Graduate Students' Project - Workpackage WP5.1

"... This work reports on a graduate students' project on parallel computing in cryptoanalysis. Major hardware- and softwaretypes have been used to implement basic cryptoanalytic algorithms. 1 Introduction In this work we report experiences made within a graduate students' project performed at the depar ..."

Abstract
- Add to MetaCart

This work reports on a graduate students' project on parallel computing in cryptoanalysis. Major hardware- and softwaretypes have been used to implement basic cryptoanalytic algorithms. 1 Introduction In this work we report experiences made within a graduate students' project performed at the department of Computer Science and System Analysis (Univ. Salzburg). The topic of the project was "Parallel Computing in Cryptoanalysis". The security of most of the public key cryptosystems known today relies on computationally infeasible problems in computational number theory (e.g. RSA -- factoring of large integers, ElGamal -- calculating discrete logarithms in a finite field; for more examples see [10]). The goal of this project was to exploit to power of parallel and distributed computing in order to perform the necessary computations to break such cryptosystems in reasonable time. Since the projects' underlying course was not theory-focused we had to choose simple algorithms to be parallel...

### The Pseudoprimes up to 10^13

, 1995

"... . There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a back-tracking search for possible prime factorisatio ..."

Abstract
- Add to MetaCart

. There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a back-tracking search for possible prime factorisations, and the computations checked by a sieving technique. 1 Introduction A (Fermat) pseudoprime (base 2) is a composite number N with the property that 2 N \Gamma1 j 1 mod N . For background on pseudoprimes and primality tests in general we refer to Bressoud [1], Brillhart et al [2], Koblitz [4], Ribenboim [12] and [13] or Riesel [14]. Previous tables of pseudoprimes were computed by Pomerance, Selfridge and Wagstaff [11]. We have shown that there are 38975 pseudoprimes up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 ; all have at most 9 prime factors. Let P (X) denote the number of pseudoprimes less than X and let P (d; X) denote the number with exactly d prime factors. In ...

### Advantages of Parallel Processing and the

"... Many computing tasks involve heavy mathematical calculations, or analyzing large amounts of data. These operations can take a long time to complete using only one computer. Networks such as the Internet provide many computers with the ability to communicate with each other. Parallel or distributed c ..."

Abstract
- Add to MetaCart

Many computing tasks involve heavy mathematical calculations, or analyzing large amounts of data. These operations can take a long time to complete using only one computer. Networks such as the Internet provide many computers with the ability to communicate with each other. Parallel or distributed computing takes advantage of these networked computers by arranging them to work together on a problem, thereby reducing the time needed to obtain the solution. The drawback to using a network of computers to solve a problem is the time wasted in communicating between the various hosts. The application of distributed computing techniques to a space environment or to use over a satellite network would therefore be limited by the amount of time needed to send data across the network, which would typically take much longer than on a terrestrial network. This experiment shows how much faster a large job can be performed by adding more computers to the task, what role communications time plays in the total execution time, and the impact a long-delay network has on a distributed computing system.

### Factoring Integers With Large Prime Variations of the Quadratic Sieve

, 1995

"... We present the results of many factorization runs with the single and double large prime variations (PMPQS, and PPMPQS, respectively) of the quadratic sieve factorization method on SGI workstations, and on a Cray C90 vectorcomputer. Experiments with 71--, 87--, and 99--digit numbers show that for ..."

Abstract
- Add to MetaCart

We present the results of many factorization runs with the single and double large prime variations (PMPQS, and PPMPQS, respectively) of the quadratic sieve factorization method on SGI workstations, and on a Cray C90 vectorcomputer. Experiments with 71--, 87--, and 99--digit numbers show that for our Cray C90 implementations PPMPQS beats PMPQS for numbers of more than 80 digits, and this cross--over point goes down with the amount of available central memory. For PMPQS a known theoretical formula is worked out and tested that helps to predict the total running time on the basis of a short test run. The accuracy of the prediction is within 10% of the actual running time. For PPMPQS such a prediction formula is not known and the determination of an optimal choice of the parameters for a given number would require many full runs with that given number, and the use of an inadmissible amount of CPU--time. In order yet to provide measurements that can help to determine a good choic...

### 1 A Beginner’s Guide To The General Number Field Sieve

"... RSA is a very popular public key cryptosystem. This algorithm is known to be secure, but this fact relies on the difficulty of factoring large numbers. Because of the popularity of the algorithm, much research has gone into this problem of factoring a large number. The size of the number that we are ..."

Abstract
- Add to MetaCart

RSA is a very popular public key cryptosystem. This algorithm is known to be secure, but this fact relies on the difficulty of factoring large numbers. Because of the popularity of the algorithm, much research has gone into this problem of factoring a large number. The size of the number that we are able to factor increases exponentially year by year. This fact is partly due to advancements in computing hardware, but it is largely due to advancements in factoring algorithms. The General Number Field Sieve is an example of just such an advanced factoring algorithm. This is currently the best known method for factoring large numbers. This paper is a presentation of the General Number Field Sieve. It begins with a discussion of the algorithm in general and covers the theory that is responsible for its success. Because often the best way to learn an algorithm is by applying it, an extensive numerical example is included as well. I.

### Computing Science The Magic Words Are Squeamish Ossifrage

"... . bu are given two integers, a and b, and ' asked to compute their product, ab = c. An algorithm for this task is taught in the early primary grades. For those of us who were day dreaming in class that day, a computer implemen tation of the algorithm yields an answer in micro seconds, even if a and ..."

Abstract
- Add to MetaCart

. bu are given two integers, a and b, and ' asked to compute their product, ab = c. An algorithm for this task is taught in the early primary grades. For those of us who were day dreaming in class that day, a computer implemen tation of the algorithm yields an answer in micro seconds, even if a and b are rather large numbers, say 60 or 70 decimal digits. Now suppose you are given the number c and asked to discover the two factors a and b, which you may assume are prime numbers (that is, they have no factors of their own, apart from 1 and themselves). This is a much harder assignment. If a and b are in the 60-digit range, so that c has more than 120 digits, finding the factors is definitely not elementary-school homework. The dramatic asymmetry between multiplica tion and factorization is the basis of an important cryptographic system: the RSA public-key cryptosystem, named for the initials of its inventors,