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


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.


Quantum circuits Finite field arithmetic Cryptography Circuit synthesis 



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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