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The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows: Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime.
In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing. Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial that ignores all its variables and always ...
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 probability of a composite number n passing the Fermat test approaches zero for . Specifically, Kim and Pomerance showed the following: The probability that a random odd number n ≤ x is a Fermat pseudoprime to a random base 1 < b < n − 1 {\displaystyle 1<b<n-1} is less than 2.77·10 −8 for x= 10 100 , and is at most (log x) −197 <10 ...
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 Schwartz–Zippel algorithm provides a practical probabilistic solution, by simply randomly testing inputs and checking whether the output is zero. It was the first randomized polynomial time PIT algorithm to be proven correct. [1] The larger the domain the inputs are drawn from, the less likely Schwartz–Zippel is to fail.
By testing the above conditions to several bases, one gets somewhat more powerful primality tests than by using one base alone. For example, there are only 13 numbers less than 25·10 9 that are strong pseudoprimes to bases 2, 3, and 5 simultaneously. They are listed in Table 7 of. [2] The smallest such number is 25326001.