Title: Design of Quantum Circuits for Galois Field Squaring and Exponentiation,
Abstract: This work presents an algorithm to generate depth, quantum gate and qubit
optimized circuits for $GF(2^m)$ 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.