New Space-Efficient Quantum Algorithm for Binary Elliptic Curves using the Optimized Division Algorithm

• URL: http://arxiv.org/abs/2303.06570v1
• Date: Sun, 12 Mar 2023 05:00:46 GMT
• Title: New Space-Efficient Quantum Algorithm for Binary Elliptic Curves using the Optimized Division Algorithm
• Authors: Hyeonhak Kim, Seokhie Hong
• Abstract summary: We suggest a new quantum division algorithm on the binary field which uses a smaller number of qubits. For elements in a field of $2n$, we can save $lceil n/2 rceil - 1$ qubits instead of using $8n2+4n-12+(16n-8)lfloorlog(n)rfloor$ more Toffoli gates.
• Score: 1.2183405753834562
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In previous research, quantum resources were concretely estimated for solving Elliptic Curve Discrete Logarithm Problem(ECDLP). In [1], the quantum algorithm was optimized for the binary elliptic curves and the main optimization target was the number of the logical qubits. The division algorithm was mainly optimized in [1] since every ancillary qubit is used in the division algorithm. In this paper, we suggest a new quantum division algorithm on the binary field which uses a smaller number of qubits. For elements in a field of $2^n$, we can save $\lceil n/2 \rceil - 1$ qubits instead of using $8n^2+4n-12+(16n-8)\lfloor\log(n)\rfloor$ more Toffoli gates, which leads to a more space-efficient quantum algorithm for binary elliptic curves.

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