Quantum security of the Fiat-Shamir transform of commit and open protocols

Applying the Fiat-Shamir transform on identification schemes is one of the main ways of constructing signature schemes. While the classical security of this transformation is well understood, there are still many cases for which we do not know whether the quantum security holds or not. In this paper, we show that if we start from a commit-and-open identification scheme, where the prover first commits to several strings and then as a second message opens a subset of them depending on the verifier's message, then the Fiat-Shamir transform is quantum secure, for a suitable choice of commitment scheme. We use our results to prove the security of a multivariate signature scheme closely related to the post-quantum signature MQDSS submitted at the NIST competition.