On error distributions in ring-based LWE

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${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$with$\unicode[STIX]{x1D6E5}_{K}$the discriminant of$K$. As a main result, we provide, for any$\unicode[STIX]{x1D700}>0$, 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${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$.