New space-efficient quantum algorithm for binary elliptic curves using the optimized division algorithm

Abstract In the previous research on solving elliptic curve discrete logarithm problem, quantum resources were concretely estimated. Banegas et al. (IACR Trans Cryptogr Hardw Embed Syst 2021(1):451–472, 2020. https://doi.org/10.46586/tches.v2021.i1.451-472 ), algorithm was optimized for binary curves, with main optimization target being number of logical qubits. The division primarily in (2020) since every ancillary qubit is used algorithm. this paper, we propose a new field that uses fewer Specifically, elements $$2^n$$ 2 n , our saves $$n - 3\lfloor \log {n} \rfloor 2$$ - 3 ⌊ log ⌋ qubits instead using $$n^2 64n\lfloor (n) + O(n)$$ 64 ( ) + O more Toffoli gates, which leads to space-efficient curves. For small size n 16, 127, 163, 233, 283 and 571, both gates are actually reduced. When reduction amounts approximately 23% compared