Rainbow, a multivariate digital signature scheme and third round finalist in NIST's PQC standardization process, is a layered version of the unbalanced oil and vinegar (UOV) scheme. We introduce two fault attacks, each focusing on one of the secret linear transformations T and S used to hide the structure of the central map in Rainbow. The first fault attack reveals a part of T and we prove that this is enough to achieve a full key recovery with negligible computational effort for all parameter

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... sets of Rainbow. The second one unveils S, which can be extended to a full key recovery by the Kipnis-Shamir attack. Our work exposes the secret transformations used in multivariate signature schemes as an important attack vector for physical attacks, which need further protection. Our attacks target the optimized Cortex-M4 implementation and require only first-order instruction skips and a moderate amount of faulted signatures.