New Space-Efficient Quantum Algorithm for Binary Elliptic Curves using the Optimized Division Algorithm
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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 ⌈ n/2 ⌉ - 1 qubits instead of using 8n^2+4n-12+(16n-8)⌊log(n)⌋ more Toffoli gates, which leads to a more space-efficient quantum algorithm for binary elliptic curves.
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