Automatic synthesis of quantum circuits for point addition on ordinary binary elliptic curves

Budhathoki, Parshuram; Steinwandt, Rainer

doi:10.1007/s11128-014-0851-6

Cited by 7 publications

(23 citation statements)

References 14 publications

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“…The quantum Galois field squaring circuit is designed with less qubit and quantum gate cost compared to existing Galois field squaring circuit design approaches [7]. Further, the proposed algorithm includes circuit depth as an optimization criterion.…”

Section: Proposed Galois Field Quantum Squaring Algorithmmentioning

confidence: 99%

“…Thus, the values in |Y will not be in sequential order but the generated squaring circuits will require no ancillae. The existing quantum squaring circuit design in [7] requires n ancillae. Thus, the proposed circuits reduce qubit cost by 50%.…”

Section: Proposed Galois Field Quantum Squaring Algorithmmentioning

confidence: 99%

“…Galois field squaring has drawn interest because it is a lower cost alternative to multiplication when both operands are the same. There are existing designs of quantum circuits for Galois field squaring such as the designs in [7]. The Galois field squaring designs in [7] are 2 input and 2 output designs.…”

Section: Introductionmentioning

confidence: 99%

“…The Feynman gate is a member of the fault tolerant Clifford+T gate set [4]. The quantum squaring circuit based on our proposed algorithm is compared and is shown to be better than the existing design of quantum Galois field squaring circuits in [7]. The comparison is performed in terms of number of gates and qubits.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, we present the quantum circuit implementation of Galois field exponentiation based on the proposed quantum Galois field squaring circuit. We calculated a significant qubit and quantum gate savings compared to identical quantum exponentiation circuit based on existing squaring circuit in [7]. The paper is organized as follows: the basics of Galois fields is presented in section II; the proposed Galois field squaring circuit algorithm and analysis are presented in sections III and IV.…”

Section: Introductionmentioning

confidence: 99%

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Design of Quantum Circuits for Galois Field Squaring and Exponentiation

Muñoz‐Coreas

Thapliyal

2017

2017 IEEE Computer Society Annual Symposium on VLSI (ISVLSI)

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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.Index Terms-quantum computing; Galois field arithmetic;978-1-5090-6762-6/17/$31.00 c 2017 IEEE arXiv:1706.05114v1 [quant-ph]

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Section: Proposed Galois Field Quantum Squaring Algorithmmentioning

confidence: 99%

Section: Proposed Galois Field Quantum Squaring Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Design of Quantum Circuits for Galois Field Squaring and Exponentiation

Muñoz‐Coreas

Thapliyal

2017

2017 IEEE Computer Society Annual Symposium on VLSI (ISVLSI)

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Quantum Cryptanalysis Landscape of Shor’s Algorithm for Elliptic Curve Discrete Logarithm Problem

Larasati

Kim

2021

Lecture Notes in Computer Science

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Depth Optimization of FLT-Based Quantum Inversion Circuit

et al. 2023

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Works on quantum computing and cryptanalysis have increased significantly in the past few years. Various constructions of quantum arithmetic circuits as one of the primary elements in the field have also been proposed. However, there have 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 Logarithm Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat's Little Theorem (FLT)-based inversion circuit for the binary finite field. In particular, we propose to 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 (i.e., CNOT count) as well as slightly reducing the T depth, which contributes to a reduced overall depth and gate count. Furthermore, we concretely verify our method and compare it with the previous work in Qiskit, a quantum computer simulation environment, by constructing both our method and the previous work from scratch and performing the resource analysis. Additionally, we propose employing the relative-phase Toffoli gate by Gidney as opposed to the standard Toffoli implementation, which yields a significantly lower T depth while further reducing the overall depth. Our approach can serve as an alternative for a time-efficient implementation.

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Automatic synthesis of quantum circuits for point addition on ordinary binary elliptic curves

Cited by 7 publications

References 14 publications

Design of Quantum Circuits for Galois Field Squaring and Exponentiation

Design of Quantum Circuits for Galois Field Squaring and Exponentiation

Quantum Cryptanalysis Landscape of Shor’s Algorithm for Elliptic Curve Discrete Logarithm Problem

Depth Optimization of FLT-Based Quantum Inversion Circuit

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