**The closest vector problem in tensored root lattices of type A and in their duals**

*Léo Ducas and Wessel P.J. van Woerden*

**Abstract: **In this work we consider the closest vector problem (CVP) ---a problem also known as maximum-likelihood decoding--- in the tensor of two root lattices of type A ($A_m \otimes A_n$), as well as in their duals ($A^*_m \otimes A^*_n$). This problem is mainly motivated by {\em lattice based cryptography}, where the cyclotomic rings $\mathbb Z[\zeta_c]$ (resp. its co-different $\mathbb Z[\zeta_c]^\vee$) play a central role, and turn out to be isomorphic as lattices to tensors of $A^*$ lattices (resp. $A$ root lattices). In particular, our results lead to solving CVP in $\mathbb Z[\zeta_c]$ and in $\mathbb Z[\zeta_c]^\vee$ for conductors of the form $c = 2^\alpha p^\beta q^\gamma$ for any two odd primes $p,q$.

For the primal case $A_m \otimes A_n$, we provide a full characterization of the Voronoi region in terms of simple cycles in the complete directed bipartite graph $K_{m+1,n+1}$. This leads ---relying on the Bellman-Ford algorithm for negative cycle detection--- to a CVP algorithm running in *polynomial time*. Precisely, our algorithm performs $O(l\ m^2 n^2 \min\{m,n\})$ operations on reals, where $l$ is the number of bits per coordinate of the input target. For the dual case, we use a gluing-construction to solve CVP in sub-exponential time $O(n m^{n+1})$.

**Category / Keywords: **public-key cryptography / Lattice based cryptography, Cyclotomic lattices, Tensored root lattices, Closest vector problem, Maximum likelihood decoding.

**Date: **received 19 Sep 2016, last revised 21 Sep 2016

**Contact author: **ducas at cwi nl

**Available format(s): **PDF | BibTeX Citation

**Version: **20160921:191802 (All versions of this report)

**Short URL: **ia.cr/2016/910

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