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.Category / Keywords: public-key cryptography / Fast Fourier Transform, Gram-Schmidt Orthogonalization, Nearest Plane Algorithm, Lattice Algorithms, Lattice Trapdoor Functions. Date: received 17 Oct 2015, last revised 4 May 2016 Contact author: thomas prest at ens fr Available format(s): PDF | BibTeX Citation Version: 20160504:100306 (All versions of this report) Short URL: ia.cr/2015/1014 Discussion forum: Show discussion | Start new discussion