International Association
for Cryptologic Research

The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev

(Eurocrypt 2010), has been steadily finding many uses in numerous

cryptographic applications. Still, the Ring-LWE problem defined

in [LPR10] involves the

fractional ideal $R^\\vee$, the dual of the ring $R$, which is the

source of many theoretical and implementation technicalities. Until

now, getting rid of $R^\\vee$, required some relatively complex

transformation that substantially increase the magnitude of the

error polynomial and the practical complexity to sample it.

It is only for rings $R=\\Z[X]/(X^n+1)$ where $n$ a power of

$2$, that this transformation is simple and benign.

In this work we show that by applying a different, and much simpler

transformation, one can transfer the results from [LPR10] into an ``easy-to-use\'\' Ring-LWE setting ({\\em i.e.} without the dual ring $R^\\vee$), with only a very

slight increase in the magnitude of the noise coefficients.

Additionally, we show that creating the correct noise distribution

can also be simplified by generating a Gaussian distribution over a

particular extension ring of $R$, and then performing a reduction

modulo $f(X)$. In essence, our results show that one does not need

to resort to using any algebraic structure that is more complicated

than polynomial rings in order to fully utilize the hardness of the

Ring-LWE problem as a building block for cryptographic applications.