I found 2 new attacks on the Rainbow signature scheme. The first attack
uses the ideas from the Kipnis-Shamir attack  and reduces the
security of Rainbow I, III and V by 7 bits, 4 bits and 19 bits
respectively. This attack also applies to the UOV scheme.
The second attack is specific to Rainbow and is also more efficient. It
reduces a key recovery to a new instance of the MinRank problem, which
can then be solved with the methods of Bardet et al. . This attack
reduces the security level of Rainbow I, III and V by 20 bits, 40 bits
and 55 bits respectively.
The paper is on ePrint: https://eprint.iacr.org/2020/1343
I would like to thank the Rainbow team for reviewing an earlier version
of the paper and providing me with helpful feedback.
All the best,