Dissertation (MSc Mathematics)Computing square roots over finite fields is a problem of interest, especially to understanding which algorithm is efficient, and how it works well. There are several known algorithms that computes square roots over finite fields, of all of them the shank’s algorithm is known to be the most efficient. The objective of this dissertation is to survey the square root computing algorithms over finite fields, particularly we consider the the Shank’s algorithm for computing square roots over finite fields. We will write the conceptual explanation and general explanations of the whole algorithm (Shank’s) and finaly show how or why the algorithm works efficiently well.