'''Schoof's Algorithm''' is an an efficient algorithm to count points on [[elliptic curves]] over [[finite fields]]. The algorithm has applications in [[elliptic curve cryptography]] where it is important to know the number of points to judge the difficulty of solving the [[discrete logarithm problem]] in the [[Group (mathematics)|group]] of points on an elliptic curve.
The algorithm was published by René Schoof in 1985 and it was a theoretical break throughbreakthrough, as it was the first deterministic polynomial time algorithm for [[counting points on elliptic curves]]. Before Schoof's algorithm, approaches to [[counting points on elliptic curves]] such as the naive and [[baby-step giant-step]] algorithms were, for the most part, tedious and had an exponential running time.
This article explains Schoof's approach, laying emphasis on the mathematical ideas underlying the structure of the algorithm.
