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Fermat's Primality Test
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Concept Fermat's little theorem states that if ''p'' is prime and ''a'' is not divisible by ''p'', then :a^ \equiv 1 \pmod. If one wants to test whether ''p'' is prime, then we can pick random integers ''a'' not divisible by ''p'' and see whether the congruence holds. If it does not hold for a value of ''a'', then ''p'' is composite. This congruence is unlikely to hold for a random ''a'' if ''p'' is composite. Therefore, if the equality does hold for one or more values of ''a'', then we say that ''p'' is probably prime. However, note that the above congruence holds trivially for a \equiv 1 \pmod, because the congruence relation is compatible with exponentiation. It also holds trivially for a \equiv -1 \pmod if ''p'' is odd, for the same reason. That is why one usually chooses a random ''a'' in the interval 1 < a < p - 1. Any ''a'' such that : |
Randomized Algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. There is a distinction between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result ( Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solovay–Strassen Primality Test
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic primality test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967 (see Theorem E in the paper). This test has been largely superseded by the Baillie–PSW primality test and the Miller–Rabin primality test, but has great historical importance in showing the practical feasibility of the RSA cryptosystem. Concepts Euler proved that for any odd prime number ''p'' and any integer ''a'', :a^ \equiv \left(\frac\right) \pmod p where \left(\tfrac\right) is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to \left(\tfrac\right), where ''n'' can be any odd integer. The Jacobi symbol can be computed in time O((log ''n'')²) using Jacobi's generalization of the law of quadratic reciprocity. Given an odd number ''n'' one can contemplate whether or not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Stein
Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Science. Stein is chair of the Industrial Engineering and Operations Research Department at Columbia University. Prior to joining Columbia, Stein was a professor at Dartmouth College in New Hampshire. Stein's research interests include the design and analysis of algorithms, combinatorial optimization, operations research, network algorithms, scheduling, algorithm engineering and computational biology. Stein has published many influential papers in the leading conferences and journals in his fields of research, and has occupied a variety of editorial positions including in the journals ''ACM Transactions on Algorithms'', ''Mathematical Programming'', ''Journal of Algorithms'', '' SIAM Journal on Discrete Mathematics'' and ''Operations Research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald L
Ronald is a masculine given name derived from the Old Norse ''Rögnvaldr'', Hanks; Hardcastle; Hodges (2006) p. 234; Hanks; Hodges (2003) § Ronald. or possibly from Old English '' Regenweald''. In some cases ''Ronald'' is an Anglicised form of the Gaelic '' Raghnall'', a name likewise derived from ''Rögnvaldr''. The latter name is composed of the Old Norse elements ''regin'' ("advice", "decision") and ''valdr'' ("ruler"). ''Ronald'' was originally used in England and Scotland, where Scandinavian influences were once substantial, although now the name is common throughout the English-speaking world. A short form of ''Ronald'' is ''Ron''. Pet forms of ''Ronald'' include ''Roni'' and '' Ronnie''. ''Ronalda'' and ''Rhonda'' are feminine forms of ''Ronald''. ''Rhona'', a modern name apparently only dating back to the late nineteenth century, may have originated as a feminine form of ''Ronald''. Hanks; Hardcastle; Hodges (2006) pp. 230, 408; Hanks; Hodges (2003) § Rhona. The names ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles E
Charles is a masculine given name predominantly found in English language, English and French language, French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic, Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was ''Churl, Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinisation of names, Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as ''Carolus (other), Carolus''. Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as wikt:churl, churl (< Old English ''ċeorl''), which developed its deprecating sense in the Middle English period. Some Germanic languages, for example Dutch language, Dutch and German ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas H
Thomas may refer to: People * List of people with given name Thomas * Thomas (name) * Thomas (surname) * Saint Thomas (other) * Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church * Thomas the Apostle * Thomas (bishop of the East Angles) (fl. 640s–650s), medieval Bishop of the East Angles * Thomas (Archdeacon of Barnstaple) (fl. 1203), Archdeacon of Barnstaple * Thomas, Count of Perche (1195–1217), Count of Perche * Thomas (bishop of Finland) (1248), first known Bishop of Finland * Thomas, Earl of Mar (1330–1377), 14th-century Earl, Aberdeen, Scotland Geography Places in the United States * Thomas, Idaho * Thomas, Illinois * Thomas, Oklahoma * Thomas, Oregon * Thomas, South Dakota * Thomas, Virginia * Thomas, Washington * Thomas, West Virginia * Thomas County (other) * Thomas Township (other) Elsewhere * Thomas Glacier (Greenland) Arts and entertainment * ''Thomas'' (Burton novel), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNU Privacy Guard
GNU Privacy Guard (GnuPG or GPG) is a free-software replacement for Symantec's cryptographic software suite PGP. The software is compliant with the now obsoleted , the IETF standards-track specification of OpenPGP. Modern versions of PGP are interoperable with GnuPG and other OpenPGP v4-compliant systems. November 2023 saw two drafts aiming to update the 2007 OpenPGP v4 specification (RFC4880), ultimately resulting in thRFC 9580standard in July 2024. The proposal from the GnuPG developers, which is called LibrePGP, was not taken up by the OpenPGP Working Group and future versions of GnuPG will not support the current version of OpenPGP. GnuPG is part of the GNU Project and received major funding from the German government in 1999. Overview GnuPG is a hybrid-encryption software program because it uses a combination of conventional symmetric-key cryptography for speed, and public-key cryptography for ease of secure key exchange, typically by using the recipient's publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pretty Good Privacy
Pretty Good Privacy (PGP) is an encryption software, encryption program that provides cryptographic privacy and authentication for data communication. PGP is used for digital signature, signing, encrypting, and decrypting texts, Email, e-mails, files, directories, and whole disk partitions and to increase the security of e-mail communications. Phil Zimmermann developed PGP in 1991. PGP and similar software follow the OpenPGP standard (RFC 4880), an open standard for encryption, encrypting and decrypting data. Modern versions of PGP are interoperability, interoperable with GnuPG and other OpenPGP-compliant systems. The OpenPGP standard has received criticism for its long-lived keys and the difficulty in learning it, as well as the EFAIL, Efail security vulnerability that previously arose when select e-mail programs used OpenPGP with S/MIME. The new OpenPGP standard (RFC 9580) has also been criticised by the maintainer of GnuPG Werner Koch, who in response created his own speci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OpenSSL
OpenSSL is a software library for applications that provide secure communications over computer networks against eavesdropping, and identify the party at the other end. It is widely used by Internet servers, including the majority of HTTPS websites. OpenSSL contains an open-source implementation of the SSL and TLS protocols. The core library, written in the C programming language, implements basic cryptographic functions and provides various utility functions. Wrappers allowing the use of the OpenSSL library in a variety of computer languages are available. The OpenSSL Software Foundation (OSF) represents the OpenSSL project in most legal capacities including contributor license agreements, managing donations, and so on. OpenSSL Software Services (OSS) also represents the OpenSSL project for support contracts. OpenSSL is available for most Unix-like operating systems (including Linux, macOS, and BSD), Microsoft Windows and OpenVMS. Project history The OpenSSL project wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Libgcrypt
Libgcrypt is a cryptography library developed as a separated module of GnuPG. It can also be used independently of GnuPG, but depends on its error-reporting library Libgpg-error. It provides functions for all fundamental cryptographic building blocks: Libgcrypt features its own ''multiple precision arithmetic'' implementation, with assembler implementations for a variety of processors, including Alpha, AMD64, HP PA-RISC, i386, i586, M68K, MIPS 3, PowerPC, and SPARC. It also features an ''entropy gathering'' utility, coming in different versions for Unix-like and Windows machines. Usually multiple, stable branches of Libgcrypt are maintained in parallel; since 2022-03-28 this is the Libgrypt 1.10 branch as stable branch, plus the 1.8 branch as LTS ("long-term support") branch, which will be maintained at least until 2024-12-31. See also * Comparison of cryptography libraries * GNU Privacy Guard GNU Privacy Guard (GnuPG or GPG) is a free-software replacement for Sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNU Multiple Precision Arithmetic Library
GNU Multiple Precision Arithmetic Library (GMP) is a free software, free library for arbitrary-precision arithmetic, operating on Sign (mathematics), signed integers, Rational data type, rational numbers, and Floating-point arithmetic, floating-point numbers. There are no practical limits to the precision except the ones implied by the available virtual memory, memory (operands may be of up to 232−1 bits on 32-bit machines and 237 bits on 64-bit machines). GMP has a rich set of functions, and the functions have a regular interface. The basic interface is for C (programming language), C, but Wrapper function, wrappers exist for other languages, including Ada (programming language), Ada, C++, C Sharp (programming language), C#, Julia (programming language), Julia, .NET Framework, .NET, OCaml, Perl, PHP, Python (programming language), Python, R (programming language), R, Ruby (programming language), Ruby, and Rust (programming language), Rust. Prior to 2008, Kaffe, a Java virtual ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baillie–PSW Primality Test
The Baillie–PSW primality test is a probabilistic or possibly deterministic primality testing algorithm that determines whether a number is composite or is a probable prime. It is named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff. The Baillie–PSW test is a combination of a strong Fermat probable prime test to base 2 and a standard or strong Lucas probable prime test. The Fermat and Lucas test each have their own list of pseudoprimes, that is, composite numbers that pass the test. For example, the first ten strong pseudoprimes to base 2 are : 2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, and 52633 . The first ten strong Lucas pseudoprimes (with Lucas parameters (''P'', ''Q'') defined by Selfridge's Method A) are : 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519 . There is no known overlap between these lists, and there is even evidence that the numbers tend to be of different kind, in fact even with standa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
