**On Error Distributions in Ring-based LWE**

*Wouter Castryck and Ilia Iliashenko and Frederik Vercauteren*

**Abstract: **Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the Ring Learning With Errors problem (Ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But for a given modulus $q$ and degree $n$ number field $K$, generating Ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod $q$ of a certain fractional ideal $\mathcal{O}_K^\vee \subset K$ called the codifferent or `dual', rather than from the ring of integers $\mathcal{O}_K$ itself. This has led to various non-dual variants of Ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by $|\Delta_K|^{1/2n}$ with $\Delta_K$ the discriminant of $K$. As a main result, we provide for any $\varepsilon > 0$ a family of number fields $K$ for which this variant of Ring-LWE can be broken easily as soon as the errors are scaled up by $|\Delta_K|^{(1-\varepsilon)/n}$.

**Category / Keywords: **public-key cryptography /

**Date: **received 3 Mar 2016, last revised 31 May 2016

**Contact author: **wouter castryck at gmail com

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

**Version: **20160531:062934 (All versions of this report)

**Short URL: **ia.cr/2016/240

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