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Quantum Information Processing

, Volume 14, Issue 7, pp 2373–2386 | Cite as

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

  • Shane Kepley
  • Rainer Steinwandt
Article

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.

Keywords

Quantum circuits Finite field arithmetic Cryptography Circuit synthesis 

Notes

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Florida Atlantic UniversityBoca RatonUSA

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