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This work presents an algorithm to generate depth, quantum gate and qubitoptimized circuits for $GF(2^m)$ squaring in the polynomial basis. Further, tothe best of our knowledge the proposed quantum squaring circuit algorithm isthe only work that considers depth as a metric to be optimized. We comparedcircuits generated by our proposed algorithm against the state of the art anddetermine that they require $50 \%$ fewer qubits and offer gates savings thatrange from $37 \%$ to $68 \%$. Further, existing quantum exponentiation arebased on either modular or integer arithmetic. However, Galois arithmetic is auseful tool to design resource efficient quantum exponentiation circuitapplicable in quantum cryptanalysis. Therefore, we present the quantum circuitimplementation of Galois field exponentiation based on the proposed quantumGalois 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 squaringcircuits.

Design of Quantum Circuits for Galois Field Squaring and Exponentiation