Short, Invertible Elements in Partially Splitting Cyclotomic Rings and Applications to Lattice-Based Zero-Knowledge Proofs

Abstract

When constructing practical zero-knowledge proofs based on the hardness of the Ring-LWE or the Ring-SIS problems over polynomial rings \(\mathbb {Z}_p[X]/(X^n+1)\), it is often necessary that the challenges come from a set \(\mathcal {C}\) that satisfies three properties: the set should be large (around \(2^{256}\)), the elements in it should have small norms, and all the non-zero elements in the difference set \(\mathcal {C}-\mathcal {C}\) should be invertible. The first two properties are straightforward to satisfy, while the third one requires us to make efficiency compromises. We can either work over rings where the polynomial \(X^n+1\) only splits into two irreducible factors modulo p, which makes the speed of the multiplication operation in the ring sub-optimal; or we can limit our challenge set to polynomials of smaller degree, which requires them to have (much) larger norms.