Quantum Oblivious LWE Sampling and Insecurity of Standard Model Lattice-Based SNARKs

The Learning With Errors (𝖫𝖶𝖤) problem asks to find 𝐬 from an input of the form (𝐀, 𝐛 = 𝐀𝐬+𝐞) ∈ (ℤ/qℤ)^m × n× (ℤ/qℤ)^m, for a vector 𝐞 that has small-magnitude entries. In this work, we do not focus on solving 𝖫𝖶𝖤 but on the task of sampling instances. As these are extremely sparse in their range, it may seem plausible that the only way to proceed is to first create 𝐬 and 𝐞 and then set 𝐛 = 𝐀𝐬+𝐞. In particular, such an instance sampler knows the solution. This raises the question whether it is possible to obliviously sample (𝐀, 𝐀𝐬+𝐞), namely, without knowing the underlying 𝐬. A variant of the assumption that oblivious 𝖫𝖶𝖤 sampling is hard has been used in a series of works constructing Succinct Non-interactive Arguments of Knowledge (SNARKs) in the standard model. As the assumption is related to 𝖫𝖶𝖤, these SNARKs have been conjectured to be secure in the presence of quantum adversaries. Our main result is a quantum polynomial-time algorithm that samples well-distributed 𝖫𝖶𝖤 instances while provably not knowing the solution, under the assumption that 𝖫𝖶𝖤 is hard. Moreover, the approach works for a vast range of 𝖫𝖶𝖤 parametrizations, including those used in the above-mentioned SNARKs.