Design of Quantum Circuits for Galois Field Squaring and Exponentiation

This work presents an algorithm to generate depth, quantum gate and qubit optimized circuits for GF(2m) squaring in the polynomial basis. Further, to the best of our knowledge the proposed quantum squaring circuit algorithm is the only work that considers depth as a metric to be optimized. We compared circuits generated by our proposed algorithm against the state of the art and determine that they require 50% fewer qubits and offer gates savings that range from 37% to 68%. Further, existing quantum exponentiation are based on either modular or integer arithmetic. However, Galois arithmetic is a useful tool to design resource efficient quantum exponentiation circuit applicable in quantum cryptanalysis. Therefore, we present the quantum circuit implementation of Galois field exponentiation based on the proposed quantum Galois field squaring circuit. We calculated a qubit savings ranging between 44% to 50% and quantum gate savings ranging between 37% to 68% compared to identical quantum exponentiation circuit based on existing squaring circuits.