  This article presents a new method for calculating square roots in GF(p) by
exponentiating in GF(p^3) or equivalently modulo irreducible cubic polynomials.
This algorithm is in some ways similar to the Cipolla-Lehmer algorithm which is
based on exponentiating in GF(p^2). Another less well known square root
algorithm based on quadratic sums is also given. In addition to this, several
conjectures about the output of this GF(p^3) square root algorithm are
mentioned.
