On error distributions in ring-based LWE

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 O∨ K ⊂ K called the codifferent or ‘dual’, rather than from the ring of integers OK 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 |∆K | with ∆K the discriminant of K. As a main result, we provide, for any ε > 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 |∆K |.