Bulletin of the
Korean Mathematical Society BKMS

We present a new square root algorithm in finite fields which is a variant of the Pocklington-Peralta algorithm. We give the complexity of the proposed algorithm in terms of the number of operations (multiplications) in finite fields, and compare the result with other square root algorithms, the Tonelli-Shanks algorithm, the Cipolla-Lehmer algorithm, and the original Pocklington-Peralta square root algorithm. Both the theoretical estimation and the implementation result imply that our proposed algorithm performs favorably over other existing algorithms. In particular, for the NIST suggested field P-224, we show that our proposed algorithm is significantly faster than other proposed algorithms.

Keywords: Square root algorithm, finite field, Pocklington-Peralta algorithm, Tonelli-Shanks algorithm, Cipolla-Lehmer algorithm

MSC numbers: 11T06, 11Y16, 68W40

Supported by: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No.~2016R1A5A1008055). Namhun Koo was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.~2021R1C1C2003888). Soonhak Kwon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.~2019R1F1A1058920 and No.~2021R1F1A1050721).