Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Guide to Elliptic Curve Cryptography
Guide to Elliptic Curve Cryptography
An Efficient Square Root Computation in Finite Fields GF(p2d)
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
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A square root (SQRT) algorithm in GF(pm) (m = r0r1⋯rn−−1 2d, ri: odd prime, d 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in $GF(p^{2^d})$, where most of the computations are performed in the corresponding subfields $GF{(p^{2^{i}})}$ 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(pm) 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(p22), GF(p44) and GF(p88) 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(p22), by 45 times in GF(p44), and by 70 times in GF(p88), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.