For example there's an easy randomized algorithm to determine a quadratic nonresidue modulo a prime with success probability 1/2 per Legendre symbol evaluation. In expectation this is 2 iterations, and that's independent of the prime. The best known deterministic algorithm takes log(p)^2 such evaluations in the worst case, even assuming generalized Riemann hypothesis.
For example there's an easy randomized algorithm to determine a quadratic nonresidue modulo a prime with success probability 1/2 per Legendre symbol evaluation. In expectation this is 2 iterations, and that's independent of the prime. The best known deterministic algorithm takes log(p)^2 such evaluations in the worst case, even assuming generalized Riemann hypothesis.