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The communication networks of low-resources applications require implementing cryptographic protocols and operations with less computational and architectural complexities. In this paper, an efficient method for high speed calculations of square (SQR) root is proposed over Galois Fields GF (2m). The method is based on using the results of certain pre-computations, and transforming the SQR root calculations into a system of linear equations. The computational complexity of our proposed method for computing the SQR root in GF (2m) is O(m) which is significantly better than existing methods such as Tonelli-Shanks and Cipolla. Our proposed method was implemented using different types of multipliers over several polynomial degrees. Software and hardware implementations were developed in NTL-C++ and VHDL, respectively. Our software experimental results show up to 38 times faster than Doliskani & Schost method. Moreover, our method is 840 times faster than Tonelli-Shanks method. In terms of hardware implementation and since Tonelli-Shanks requires less resources than Doliskani & Schost, we compare our method with Tonelli-Shanks. The hardware experimental results show that up to 50% less LUTs with a speedup of 18% that can be obtained compared to Tonelli-Shanks method.

Accelerating Square Root Computations Over Large GF (2m)