Paper 2015/1014

Fast Fourier Orthogonalization

Léo Ducas and Thomas Prest


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.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Fast Fourier TransformGram-Schmidt OrthogonalizationNearest Plane AlgorithmLattice AlgorithmsLattice Trapdoor Functions.
Contact author(s)
thomas prest @ ens fr
2016-05-04: last of 3 revisions
2015-10-19: received
See all versions
Short URL
Creative Commons Attribution


      author = {Léo Ducas and Thomas Prest},
      title = {Fast Fourier Orthogonalization},
      howpublished = {Cryptology ePrint Archive, Paper 2015/1014},
      year = {2015},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.