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Quantum Binary Field Multiplication with Optimized Toffoli Depth and Extension to Quantum Inversion.

Kyungbae Jang1, Wonwoong Kim1, Sejin Lim1

  • 1Division of IT Convergence Engineering, Hansung University, Seoul 02876, Republic of Korea.

Sensors (Basel, Switzerland)
|March 30, 2023
PubMed
Summary
This summary is machine-generated.

This study optimizes quantum multiplication for binary elliptic curves, crucial for Shor's algorithm. The new method significantly reduces circuit depth, improving performance for quantum cryptography applications.

Keywords:
Toffoli depthbinary fieldquantum inversionquantum multiplication

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Area of Science:

  • Quantum Computing
  • Cryptography
  • Computational Mathematics

Background:

  • Shor's algorithm solves discrete logarithm problems on binary elliptic curves efficiently.
  • Quantum circuit implementation of elliptic curve arithmetic, especially multiplication, faces significant overhead.
  • Previous optimizations focused on qubit count and Toffoli gates, often neglecting circuit depth.

Purpose of the Study:

  • To optimize quantum multiplication in binary fields for enhanced Shor's algorithm performance.
  • To reduce both Toffoli depth and overall circuit depth in quantum multiplication.
  • To improve the efficiency and practicality of quantum computations involving elliptic curves.

Main Methods:

  • Utilized the Karatsuba multiplication method, a divide-and-conquer approach.
  • Focused on minimizing Toffoli depth and full circuit depth.
  • Evaluated performance using metrics like qubit count, gate count, and qubit-depth product.

Main Results:

  • Achieved a Toffoli depth of one for quantum multiplication.
  • Significantly reduced the full circuit depth compared to previous methods.
  • Demonstrated superior performance and trade-offs in qubit count, depth, and gate usage.
  • Showcased effectiveness when integrated with the Itoh-Tsujii algorithm for inversion.

Conclusions:

  • The proposed quantum multiplication method offers substantial improvements in circuit depth and overall efficiency.
  • This optimization is critical for practical implementations of Shor's algorithm and other quantum cryptographic protocols.
  • The method provides the best known trade-off for quantum multiplication resources, enhancing its applicability.