# Design of Quantum Circuits for Galois Field Squaring and Exponentiation

## TL;DR

This paper introduces an optimized quantum circuit algorithm for Galois field squaring and exponentiation, achieving significant reductions in qubits and gates, which enhances resource efficiency for quantum cryptanalysis.

## Contribution

It presents the first depth-optimized quantum circuit algorithm for Galois field squaring and extends it to efficient quantum exponentiation circuits, outperforming existing methods.

## Key findings

- 50% fewer qubits required for squaring circuits
- Gates savings range from 37% to 68% for squaring
- Qubit savings between 44% and 50% for 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.