Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.Comment: 25 page

An analysis of key generation efficiency of RSA cryptosystem in distributed environments

Thesis (Master)--Izmir Institute of Technology, Computer Engineering, Izmir, 2005Includes bibliographical references (leaves: 68)Text in English Abstract: Turkish and Englishix, 74 leavesAs the size of the communication through networks and especially through Internet grew, there became a huge need for securing these connections. The symmetric and asymmetric cryptosystems formed a good complementary approach for providing this security. While the asymmetric cryptosystems were a perfect solution for the distribution of the keys used by the communicating parties, they were very slow for the actual encryption and decryption of the data flowing between them. Therefore, the symmetric cryptosystems perfectly filled this space and were used for the encryption and decryption process once the session keys had been exchanged securely. Parallelism is a hot research topic area in many different fields and being used to deal with problems whose solutions take a considerable amount of time. Cryptography is no exception and, computer scientists have discovered that parallelism could certainly be used for making the algorithms for asymmetric cryptosystems go faster and the experimental results have shown a good promise so far. This thesis is based on the parallelization of a famous public-key algorithm, namely RSA

A Performant, Misuse-Resistant API for Primality Testing

Primality testing is a basic cryptographic task. But developers today are faced with complex APIs for primality testing, along with documentation that fails to clearly state the reliability of the tests being performed. This leads to the APIs being incorrectly used in practice, with potentially disastrous consequences. In an effort to overcome this, we present a primality test having a simplest-possible API: the test accepts a number to be tested and returns a Boolean indicating whether the input was composite or probably prime. For all inputs, the output is guaranteed to be correct with probability at least 1 - 2-128. The test is performant: on random, odd, 1024-bit inputs, it is faster than the default test used in OpenSSL by 17%. We investigate the impact of our new test on the cost of random prime generation, a key use case for primality testing. The OpenSSL developers have adopted our suggestions in full; our new API and primality test are scheduled for release in OpenSSL 3.0

Association of Christians in the Mathematical Sciences Proceedings 2017

The conference proceedings of the Association of Christians in the Mathematical Sciences biannual conference, May 31-June 2, 2017 at Charleson Southern University

The Elliptic Curve Method and Other Integer Factorization Algorithms

The goal of this paper is to describe the Elliptic Curve Method (ECM), an integer factorization algorithm first proposed by Lenstra. Before describing this algorithm, we first discuss some preliminaries. In Chapter 2, we prove some elementary results from number theory and explicitly describe the RSA algorithm. In Chapter 3, we describe several factorization algorithms, including Pollard's Rho algorithm and the Quadratic Sieve. In Chapter 4, we discuss the algebraic properties of elliptic curves and the ECM. In Chapter 5, we discuss how performing the ECM with a curve in Montgomery form reduces computations. We conclude Chapter 5 by discussing some recent developments of the ECM

Investigation of the efficiency of the Elliptic Curve Cryptosystem for multi-application smart card

Thesis (M.E.Sc.) -- University of Adelaide, Dept. of Engineering, 199

Abstract Algebra: Theory and Applications

Tom Judson\u27s Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. Rob Beezer has contributed complementary material using the open source system, Sage.An HTML version on the PreText platform is available here. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.https://scholarworks.sfasu.edu/ebooks/1022/thumbnail.jp