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http://dx.doi.org/10.7471/ikeee.2019.23.4.1321

An Efficient Hardware Implementation of Square Root Computation over GF(p)  

Choe, Jun-Yeong (School of Electronic Engineering, Kumoh National Institute of Technology)
Shin, Kyung-Wook (School of Electronic Engineering, Kumoh National Institute of Technology)
Publication Information
Journal of IKEEE / v.23, no.4, 2019 , pp. 1321-1327 More about this Journal
Abstract
This paper describes an efficient hardware implementation of modular square root (MSQR) computation over GF(p), which is the operation needed to map plaintext messages to points on elliptic curves for elliptic curve (EC)-ElGamal public-key encryption. Our method supports five sizes of elliptic curves over GF(p) defined by the National Institute of Standards and Technology (NIST) standard. For the Koblitz curves and the pseudorandom curves with 192-bit, 256-bit, 384-bit and 521-bit, the Euler's Criterion based on the characteristic of the modulo values was applied. For the elliptic curves with 224-bit, the Tonelli-Shanks algorithm was simplified and applied to compute MSQR. The proposed method was implemented using the finite field arithmetic circuit with 32-bit datapath and memory block of elliptic curve cryptography (ECC) processor, and its hardware operation was verified by implementing it on the Virtex-5 field programmable gate array (FPGA) device. When the implemented circuit operates with a 50 MHz clock, the computation of MSQR takes about 18 ms for 224-bit pseudorandom curves and about 4 ms for 256-bit Koblitz curves.
Keywords
ECC; EC-ElGamal; Modular square root; Euler's Criterion; Tonelli-Shanks algorithm;
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