An efficient square root computation in finite fields GF(p^2d)

This paper focuses on developing a square root (SQRT) algorithm in finite fields GF(p^2d)(d ≥ 0). Examining the Smart algorithm, a well-known SQRT algorithm, we can see that there is some computation overlap between the Smart algorithm and the quadratic residue (QR) test, which must be implemented before a SQRT computation. It makes the Smart algorithm inefficient. In this paper, we propose a new QR test and a new SQRT algorithm in GF(p^2d), in which not only there is no computation overlap, but also most of computations required for the proposed SQRT algorithm in GF(p^2d) can be implemented in the corresponding subfields GF(p^2d-l) for 1 ≤ i ≤ d, which yields many reductions in the computational time and complexity. The computer simulation also shows that the proposed SQRT algorithm is much faster than the Smart algorithm.