### Space-efficient quantum multiplication of polynomials for binary finite fields with sub-quadratic Toffoli gate count

Iggy van Hoof

##### Abstract

Multiplication is an essential step in a lot of calculations. In this paper we look at multiplication of 2 binary polynomials of degree at most $n-1$, modulo an irreducible polynomial of degree $n$ with $2n$ input and $n$ output qubits, without ancillary qubits, assuming no errors. With straightforward schoolbook methods this would result in a quadratic number of Toffoli gates and a linear number of CNOT gates. This paper introduces a new algorithm that uses the same space, but by utilizing space-efficient variants of Karatsuba multiplication methods it requires only $O(n^{\log_2(3)})$ Toffoli gates at the cost of a higher CNOT gate count: theoretically up to $O(n^2)$ but in examples the CNOT gate count looks a lot better.

Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. https://arxiv.org/abs/1910.02849
Keywords
post-quantum cryptographyquantum computingKaratsuba multiplication
Contact author(s)
Iggy @ lafeberhof nl
History
Short URL
https://ia.cr/2019/1170

CC BY

BibTeX

@misc{cryptoeprint:2019/1170,
author = {Iggy van Hoof},
title = {Space-efficient quantum multiplication of polynomials for binary finite fields with sub-quadratic Toffoli gate count},
howpublished = {Cryptology ePrint Archive, Paper 2019/1170},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/1170}},
url = {https://eprint.iacr.org/2019/1170}
}

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