Reducing the Depth of Quantum FLT-Based Inversion Circuit

TL;DR

This paper presents a method to reduce the depth of quantum circuits for finite field inversion using Fermat's Little Theorem, improving efficiency for quantum algorithms like Shor's algorithm.

Contribution

It introduces a complete waterfall approach to translate Itoh-Tsujii's FLT variant into quantum circuits, removing inverse squaring and lowering gate count and circuit depth.

Findings

Reduced CNOT gate count and circuit depth compared to previous methods

Implemented and analyzed the circuit using Qiskit simulator

Provides a more time-efficient quantum inversion circuit for finite fields

Abstract

Works on quantum computing and cryptanalysis has increased significantly in the past few years. Various constructions of quantum arithmetic circuits, as one of the essential components in the field, has also been proposed. However, there has only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor's algorithm for Elliptic Curve Discrete Logarith Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat's Little Theorem (FLT)-based inversion circuit for binary finite field. In particular, we propose follow a complete waterfall approach to translate the Itoh-Tsujii's variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (CNOT count), which contributes to reduced overall depth and gate count. Furthermore, compare the cost by firstly constructing our method and previous work's in Qiskit quantum computer simulator and perform the resource analysis. Our approach can serve as an alternative for a time-efficient implementation.