(algorithm)

**Definition:**
A *heuristic* test for prime numbers. It repeatedly checks if the number being tested, n, is pseudoprime to a randomly chosen base, a, and there are only trivial square roots of 1, modulo n. In other words, n is surely composite if a^{n-1} ≠ 1 (mod n), where 0 < a < n. Some composites may be incorrectly judged to be prime.

*Note:
For k repetitions, the chance of incorrectly judging an odd integer greater than 2 to be prime is 2 ^{-k}. For randomly chosen large integers, a small number of repetitions, say 3, is enough. After [CLR90, pages 838-844].*

Author: PEB

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Entry modified 14 December 2005.

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Cite this as:

Paul E. Black, "Miller-Rabin", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 14 December 2005. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/millerRabin.html