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Michael O. Rabin analyzed a randomized algorithm for primality testing in his 1980 paper. The Miller-Rabin primality test checks whether a candidate integer satisfies conditions obeyed by primes under modular exponentiation. A failed check certifies compositeness; passing a check gives a probable prime, not a proof of primality.^[ref]

For each fixed odd composite candidate, at most one quarter of the admissible bases are strong liars. Independent uniformly chosen bases therefore reduce the probability of accepting that composite to at most $4^{-k}$ after $k$ rounds. This probability is over the test's choices for a fixed input; it is not a posterior probability that an arbitrary accepted integer is prime.

By Michael O. Rabin.

Primary source: math.nyu.edu/~goodman/teaching/Algebra_2_2021/MillerRabinTest.pdf

Sections 1–2, strong probable-prime testing and one-quarter bound.