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.Comment: version 0.
