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I came across How to validate the secret of a Ring Learning with Errors (RLWE) key paper by Ding et al., which seems to provide a ZK proof that the given $p$ is of the form $as + e$ with $s, e$ small and that the prover knows the corresponding $s$ value. However, their proof seems to be interactive. I was wondering whether there exists a non-interactive PoK for this, which in addition to proving knowledge of $s$ also proves knowledge of the error vector $e$?

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