Search results
Results from the WOW.Com Content Network
A primality test is an algorithm for determining whether an input number is 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.
The first part of the book concludes with chapter 4, on the history of prime numbers and primality testing, including the prime number theorem (in a weakened form), applications of prime numbers in cryptography, and the widely used Miller–Rabin primality test, which runs in randomized polynomial time. [5]
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 of historical significance in the search for a polynomial-time deterministic ...
Probable primality is a basis for efficient primality testing algorithms, which find application in cryptography. These algorithms are usually probabilistic in nature. The idea is that while there are composite probable primes to base a for any fixed a , we may hope there exists some fixed P <1 such that for any given composite n , if we choose ...
The Miller–Rabin primality test uses the following extension of Fermat's little theorem: [14] If p is an odd prime and p − 1 = 2 s d with s > 0 and d odd > 0, then for every a coprime to p , either a d ≡ 1 (mod p ) or there exists r such that 0 ≤ r < s and a 2 r d ≡ −1 (mod p ) .
For the large primes used in cryptography, provable primes can be generated based on variants of Pocklington primality test, [3] while probable primes can be generated with probabilistic primality tests such as the Baillie–PSW primality test or the Miller–Rabin primality test.
In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm.Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography.
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 [ 1 ] (see Theorem E in the paper).