A high-speed square root algorithm in extension fields

A square root (SQRT) algorithm in GF(p(m)) (m = r(0)r(1)center dot center dot center dot r(n-1)2(d), r(i): odd prime, d > 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF (p(2d)), where most of the computations are performed in the corresponding subfields GF(p(2d)) for 0 <= i <= d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p(m)) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p(22)), GF(P-44) and GF(P-88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF (P-22), by 45 times in GF (P-44), and by 70 times in GF(p(88)), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.