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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.
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 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.
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 ...
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 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.
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 ...