It is well known that quadratic residue (QR) test should be implemented in advance of a square root (SQRT) computation. Smart algorithm, the previously known fastest algorithm for SQRT computation, only got the idea how to compute SQRT through QR test. However there is a lot of computation overlap in QR test and Smart algorithm. The essence of our proposition is thus to present a new QR test and SQRT algorithm to avoid all the overlapping computations. In this paper the authors devised a SQRT algorithm for which most of the data required have been computed in QR test. This yields many reductions in the computational time and amount. In GF(p) and GF(p^2), we implemented Smart algorithm and the proposed algorithm in C++ language on Pentium4 (2.6GHz), where p=2^<16>+1(4|p-1) and p = 2^<16>+3(4&nmid;p-1). The computer simulations showed that for p=2^<16>+1 the proposed algorithm on average accelerates the SQRT computation 2 times and 10 times faster than Smart algorithm over GF(p) and GF(p^2), respectively and for p=2^<16>+3 the proposed algorithm on average accelerates the SQRT computation 20 times and 6 times faster than Smart algorithm over GF(p) and GF(p^2), respectively.