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The Miller–Rabin primality test and Solovay–Strassen primality test are more sophisticated variants, which detect all composites (once again, this means: for every composite number n, at least 3/4 (Miller–Rabin) or 1/2 (Solovay–Strassen) of numbers a are witnesses of compositeness of n). These are also compositeness tests.
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 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.
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 ...
For example, Fermat pseudoprimes to base 2 tend to fall into the residue class 1 (mod m) for many small m, whereas Lucas pseudoprimes tend to fall into the residue class −1 (mod m). [1]: §6 [2]: Table 2 & §5 As a result, a number that passes both a strong Fermat base 2 and a strong Lucas test is very likely to be prime. If you choose a ...
As mentioned above, most applications use a Miller–Rabin or Baillie–PSW test for primality. Sometimes a Fermat test (along with some trial division by small primes) is performed first to improve performance. GMP since version 3.0 uses a base-210 Fermat test after trial division and before running Miller–Rabin tests.
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
Hence, the probability of failure is at most 2 −k (compare this with the probability of failure for the Miller–Rabin primality test, which is at most 4 −k). For purposes of cryptography the more bases a we test, i.e. if we pick a sufficiently large value of k , the better the accuracy of test.