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

Quantum circuits for $${\mathbb {F}}_{2^{n}}$$ F 2 n -multiplication with subquadratic... 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 $${\mathbb {F}}_{2^n}$$ 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 $$T$$ T -gates being bounded by $$7\cdot n^{\log _2(3)}$$ 7 · n log 2 ( 3 ) for any given reduction polynomial of degree $$n=2^N$$ n = 2 N . If an irreducible trinomial of degree $$n$$ n exists, then a multiplication circuit with a total gate count of $${\mathcal {O}}(n^{\log _2(3)})$$ O ( n log 2 ( 3 ) ) is available. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

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

, Volume 14 (7) – May 12, 2015
14 pages

/lp/springer_journal/quantum-circuits-for-mathbb-f-2-n-f-2-n-multiplication-with-ZXot2RDDFc
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-0993-1
Publisher site
See Article on Publisher Site

### 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 $${\mathbb {F}}_{2^n}$$ 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 $$T$$ T -gates being bounded by $$7\cdot n^{\log _2(3)}$$ 7 · n log 2 ( 3 ) for any given reduction polynomial of degree $$n=2^N$$ n = 2 N . If an irreducible trinomial of degree $$n$$ n exists, then a multiplication circuit with a total gate count of $${\mathcal {O}}(n^{\log _2(3)})$$ O ( n log 2 ( 3 ) ) is available.

### Journal

Quantum Information ProcessingSpringer Journals

Published: May 12, 2015

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