Cryptology ePrint Archive: Report 2015/1014

Fast Fourier Orthogonalization

Léo Ducas and Thomas Prest

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.

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

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Version: 20160504:100306 (All versions of this report)

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