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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 ...
The Miller–Rabin and the Solovay–Strassen primality tests are simple and are much faster than other general primality tests. One method of improving efficiency further in some cases is the Frobenius pseudoprimality test ; a round of this test takes about three times as long as a round of Miller–Rabin, but achieves a probability bound ...
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), 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.
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]
A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them " pseudoprimes ". Unlike the Fermat pseudoprimes , for which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers ), there are no ...
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). This result may be deduced from Fermat's little theorem by the fact that, if p is an odd prime, then the integers modulo p form a finite field , in which ...
Alabama has scheduled a second execution with nitrogen gas, months after the state became the first to put a person to death with the previously untested method. Alabama Gov. Kay Ivey set a Sept ...
The BigInteger class in standard versions of Java and in open-source implementations like OpenJDK has a method called isProbablePrime. This method does one or more Miller–Rabin tests with random bases. If n, the number being tested, has 100 bits or more, this method also does a non-strong Lucas test that checks whether U n+1 is 0 (mod n).