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arXiv · 2501.16136

Quantum binary field multiplication with subquadratic Toffoli gate count and low space-time cost

Vivien Vandaele
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Abstract

Multiplication over binary fields is a crucial operation in quantum algorithms designed to solve the discrete logarithm problem for elliptic curve defined over $GF(2^n)$. In this paper, we present an algorithm for constructing quantum circuits that perform multiplication over $GF(2^n)$ with $\mathcal{O}(n^{\log_2(3)})$ Toffoli gates. We propose a variant of our construction that achieves linear depth by using $\mathcal{O}(n\log_2(n))$ ancillary qubits. This approach provides the best known space-time trade-off for binary field multiplication with a subquadratic number of Toffoli gates. Additionally, we demonstrate that for some particular families of primitive polynomials, such as trinomials, the multiplication can be done in logarithmic depth and with $\mathcal{O}(n^{\log_2(3)})$ gates.

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Vivien Vandaele. 2025-01-27. Quantum binary field multiplication with subquadratic Toffoli gate count and low space-time cost. https://arxiv.org/abs/2501.16136

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