Definition & Betydelse | Engelska ordet PRIMALITY

Definition av PRIMALITY

1. primalitet

Exempel på hur man kan använda PRIMALITY i en mening

• Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
• In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a.
• There are several known primality tests that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.
• Sophie Germain primes and safe primes have applications in public key cryptography and primality testing.
• Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography.
• Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is , where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
• This is unfortunately false for weak probable primes, because there exist Carmichael numbers; but it is true for more refined notions of probable primality, such as strong probable primes (P = 1/4, Miller–Rabin algorithm), or.
• Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input).
• The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test.
• It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.
• In practice, Wilson's theorem is useless as a primality test because computing (n − 1)! modulo n for large n is computationally complex, and much faster primality tests are known (indeed, even trial division is considerably more efficient).
• Pocklington primality test, an improved version of this test which only requires a partial factorization of n − 1.
• In addition to his significant contributions to number theory algorithms for multiprecision integers, such as factoring, Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjecture and participated in the Cunningham project.
• Prime95 tests numbers for primality using the Fermat primality test (referred to internally as PRP, or "probable prime").
• The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P".
• Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns.
• Solovay on the Solovay–Strassen primality test, the first method to show that testing whether a number is prime can be performed in randomized polynomial time and one of the first results to show the power of randomized algorithms more generally.
• In 1975 John Brillhart, Derrick Henry Lehmer, and Selfridge developed a method of proving the primality of p given only partial factorizations of p − 1 and p + 1.
• Hence the chance of the algorithm failing in this way is so small that the (pseudo) prime is used in practice in cryptographic applications, but for applications for which it is important to have a prime, a test like ECPP or the Pocklington primality test should be used which proves primality.
• He created the AKS primality test with Neeraj Kayal and Nitin Saxena, for which he and his co-authors won the 2006 Fulkerson Prize, and the 2006 Gödel Prize.

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