Paper 2015/1014 Original: eprint.iacr.org/2015/1014 ↗
Fast Fourier Orthogonalization
Abstract
The classical fast Fourier transform (FFT) allows to compute in quasi-linear time the product of two polynomials, in the {\em circular convolution ring} $\mathbb R[x]/(x^d -1)$ --- a task that naively requires quadratic time. Equivalently, it allows to accelerate matrix-vector products when the matrix is *circulant*. In this work, we discover that the ideas of the FFT can be applied to speed up the orthogonalization process of matrices with circulant blocks of size $d\times d$. We show that, when $d$ is composite, it is possible to proceed to the orthogonalization in an inductive way ---up to an appropriate re-indexation of rows and columns. This leads to a structured Gram-Schmidt decomposition. In turn, this structured Gram-Schmidt decomposition accelerates a cornerstone lattice algorithm: the nearest plane algorithm. The complexity of both algorithms may be brought down to $\Theta(d \log d)$. Our results easily extend to *cyclotomic rings*, and can be adapted to Gaussian samplers. This finds applications in lattice-based cryptography, improving the performances of trapdoor functions.
Metadata
- Available format(s)
PDF- Category
- Public-key cryptography
- Publication info
- Preprint.
- History
- 2016-05-04: last updated
- 2015-10-19: published
- Short URL
- https://ia.cr/2015/1014
- License
- https://creativecommons.org/licenses/by/4.0/
BibTeX
@misc{cryptoeprint:2015/1014,
author = {Léo Ducas and Thomas Prest},
title = {Fast Fourier Orthogonalization},
howpublished = {Cryptology {ePrint} Archive, Paper 2015/1014},
year = {2015},
url = {https://eprint.iacr.org/2015/1014}
}