**Recovering Short Generators of Principal Ideals in Cyclotomic Rings**

*Ronald Cramer and Léo Ducas and Chris Peikert and Oded Regev*

**Abstract: **A handful of recent cryptographic proposals rely on the conjectured
hardness of the following problem in the ring of integers of a
cyclotomic number field: given a basis of a principal ideal that is
guaranteed to have a ``rather short'' generator, find such a
generator. Recently, Bernstein and Campbell-Groves-Shepherd sketched
potential attacks against this problem; most notably, the latter
authors claimed a \emph{polynomial-time quantum} algorithm.
(Alternatively, replacing the quantum component with an algorithm of
Biasse and Fieker would yield a \emph{classical subexponential-time}
algorithm.) A key claim of Campbell \etal\ is that one step of their
algorithm---namely, decoding the \emph{log-unit} lattice of the ring
to recover a short generator from an arbitrary one---is classically
efficient (whereas the standard approach on general lattices takes
exponential time). However, very few convincing details were provided
to substantiate this claim.

In this work, we clarify the situation by giving a rigorous proof that the log-unit lattice is indeed efficiently decodable, for any cyclotomic of prime-power index. Combining this with the quantum algorithm from a recent work of Biasse and Song confirms the main claim of Campbell \etal\xspace Our proof consists of two main technical contributions: the first is a geometrical analysis, using tools from analytic number theory, of the standard generators of the group of cyclotomic units. The second shows that for a wide class of typical distributions of the short generator, a standard lattice-decoding algorithm can recover it, given any generator.

By extending our geometrical analysis, as a second main contribution we obtain an efficient algorithm that, given any generator of a principal ideal (in a prime-power cyclotomic), finds a $2^{\tilde{O}(\sqrt{n})}$-approximate shortest vector in the ideal. Combining this with the result of Biasse and Song yields a quantum polynomial-time algorithm for the $2^{\tilde{O}(\sqrt{n})}$-approximate Shortest Vector Problem on principal ideal lattices.

**Category / Keywords: **public-key cryptography / Ideal Lattices, Cryptanalysis

**Original Publication**** (in the same form): **IACR-EUROCRYPT-2016

**Date: **received 6 Apr 2015, last revised 25 Feb 2016

**Contact author: **ducas at cwi nl

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

**Note: **Moved appendices.

**Version: **20160225:160043 (All versions of this report)

**Short URL: **ia.cr/2015/313

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