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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 This test has been largely superseded by the Baillie–PSW primality test and the Miller–Rabin primality test, but has great historical importance in showing the practical feasibility of the RSA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] |
Robert M
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' () "fame, glory, honour, praise, renown, godlike" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin.Reaney & Wilson, 1997. ''Dictionary of English Surnames''. Oxford University Press. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe, the name entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime Integers
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Tests
A primality test is an algorithm for determining whether an input number is prime number, prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its Run-time complexity, running time is Polynomial time, polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin primality test, Miller–Rabin prove that a number is Composite number, composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is ''trial division'': given an input number, n, check whether it is divisibility, divisible by any prime number between 2 and \sqrt n (i.e., whether the division lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RP (complexity)
In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: * It always runs in polynomial time in the input size * If the correct answer is NO, it always returns NO * If the correct answer is YES, then it returns YES with probability at least 1/2 (otherwise, it returns NO). In other words, the algorithm is allowed to flip a truly random coin while it is running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if the algorithm terminates and produces YES, then the correct answer is definitely YES; however, the algorithm can terminate with NO ''regardless'' of the actual answer. That is, if the algorithm returns NO, it might be wrong. Some authors call this class R, although this name is more commonly used for the class of recursive languages. If the correct answer is YES and the algorithm is run ''n'' times with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set (mathematics), set of computational problems "of related resource-based computational complexity, complexity". The two most commonly analyzed resources are time complexity, time and space complexity, memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time complexity, time or space complexity, memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P (complexity), P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. Counting problem (complexity), counting problems and function problems) and using other models of computation (e.g. probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pocklington Primality Test
In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer. The test uses a partial factorization of N - 1 to prove that an integer N is prime. It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization of N - 1. Pocklington criterion The basic version of the test relies on the Pocklington theorem (or Pocklington criterion) which is formulated as follows: Let N > 1 be an integer, and suppose there exist natural numbers and such that Then is prime. Here i \equiv j \pmod means that after finding the remainder of division by ''k'', ''i'' and ''j'' are equal; i \vert j means that ''i'' is a divisor for ''j''; and gcd is the greatest common divisor. Note: Equation () is simply a Fermat primality test. If we find ''any'' value of , not divisible by , such that equation () is false, we may immediately conclude that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve Primality Proving
In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin in the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and , in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing (and proving) followed quickly. Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance with the advent of modern cryptography. Although many cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Jacobi Pseudoprime
In number theory, an odd integer ''n'' is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base ''a'', if ''a'' and ''n'' are coprime, and :a^ \equiv \left(\frac\right)\pmod where \left(\frac\right) is the Jacobi symbol. The Jacobi symbol evaluates to 0 if ''a'' and ''n'' are not coprime, so the test can alternatively be expressed as: :a^ \equiv \left(\frac\right) \neq 0 \pmod. If ''n'' is an odd composite integer that satisfies the above congruence, then ''n'' is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime) to base ''a''. As long as ''a'' is not a multiple of ''n'' (usually 2 ≤ ''a'' < ''n''), then if ''a'' and ''n'' are not coprime, ''n'' is definitely composite, as 1 < gcd(''a'',''n'') < ''n'' is a factor of ''n''. Properties The motivation for this definition is th ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie–Hellman key exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer (the base) is raised to the power (the exponent), and divided by a positive integer (the modulus); that is, . From the definition of division, it follows that . For example, given , and , dividing by leaves a remainder of . Modular exponentiation can be performed with a ''negative'' exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and . Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent when given , , and – is believed to be difficult. This one-way function behavior makes modular e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudocode
In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actions and conditions. Although pseudocode shares features with regular programming languages, it is intended for human reading rather than machine control. Pseudocode typically omits details that are essential for machine implementation of the algorithm, meaning that pseudocode can only be verified by hand. The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation. The reasons for using pseudocode are that it is easier for people to understand than conventional programming language code and that it is an efficient and environment-independent description of the key principles of an algorithm. It is commonly used in textbooks and scientific publications to document ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
