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Quantum circuits for \({\mathbb {F}}_{2^{n}}\)-multiplication with subquadratic gate count

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Abstract

One of the most cost-critical operations when applying Shor’s algorithm to binary elliptic curves is the underlying field arithmetic. Here, we consider binary fields \({\mathbb {F}}_{2^n}\) in polynomial basis representation, targeting especially field sizes as used in elliptic curve cryptography. Building on Karatsuba’s algorithm, our software implementation automatically synthesizes a multiplication circuit with the number of \(T\)-gates being bounded by \(7\cdot n^{\log _2(3)}\) for any given reduction polynomial of degree \(n=2^N\). If an irreducible trinomial of degree \(n\) exists, then a multiplication circuit with a total gate count of \({\mathcal {O}}(n^{\log _2(3)})\) is available.

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Notes

  1. As usual, by a \(T\) -gate we mean the matrix \(\left( \begin{array}{cc}1&{}0\\ 0&{}\hbox {e}^{i\pi /4}\end{array}\right) \), and for the resource analysis we do not distinguish between \(T\)- and \(T^\dagger \)-gates.

  2. The somewhat unusual indexing will become clear in a moment.

  3. Performing the same multiplication with constant \(T\)-depth would be possible, the trade-off being additional wires—our Sage implementation can be adapted to optimize the number of qubits for a given constraint on the \(T\)-depth.

  4. The processing of \(X\) and \(Y\) is identical up to relabeling, so the length of \(L\) is half of the total number of CNOT gates.

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Acknowledgments

The authors thank Richard Cleve, Stephen Locke, and Dmitri Maslov for helpful discussions, and an anonymous referee for making us aware of [19]. RS is supported by NATO’s Public Diplomacy Division in the framework of “Science for Peace,” Project MD.SFPP 984520.

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  1. Florida Atlantic University, 777 Glades Rd, Boca Raton, FL, 33431, USA

    Shane Kepley & Rainer Steinwandt

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  1. Shane Kepley
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  2. Rainer Steinwandt
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Correspondence to Shane Kepley.

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Kepley, S., Steinwandt, R. Quantum circuits for \({\mathbb {F}}_{2^{n}}\)-multiplication with subquadratic gate count. Quantum Inf Process 14, 2373–2386 (2015). https://doi.org/10.1007/s11128-015-0993-1

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