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 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 a circulant matrix. We show that, when $n$ is composite, it is possible to proceed to the orthogonalization in an inductive way, leading to a structured Gram-Schmidt decomposition. In turn, this structured Gram-Schmidt decomposition accelerates a cornerstone lattice algorithm: the Nearest Plane algorithm. The 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

Contact author: thomas prest at ens fr

Available format(s): PDF | BibTeX Citation

Version: 20151019:205917 (All versions of this report)

Short URL: ia.cr/2015/1014

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