  Ciminion and Hydra are two recently introduced symmetric key Pseudo-Random
Functions for Multi-Party Computation applications. For efficiency both
primitives utilize quadratic permutations at round level. Therefore, polynomial
system solving-based attacks pose a serious threat to these primitives. For
Ciminion we construct a quadratic degree reverse lexicographic (DRL) Gr\"obner
basis for the iterated polynomial model via affine transformations. For Hydra
we provide a computer-aided proof in SageMath that a quadratic DRL Gr\"obner
basis is already contained within the iterated polynomial system for the Hydra
heads after affine transformations and a linear change of coordinates.
  Our Ciminion DRL Gr\"obner basis simplifies cryptanalysis, since one does not
need to impose genericity assumptions, like being regular or semi-regular,
anymore to derive complexity estimates on key recovery attacks.
  In the Hydra proposal it was claimed that $r_\mathcal{H} = 31$ rounds for the
heads are sufficient to achieve $128$ bits of security against Gr\"obner basis
attacks for key recovery. However, for $r_\mathcal{H} = 31$ standard term order
conversion to a lexicographic (LEX) Gr\"obner basis for our Hydra DRL Gr\"obner
basis requires just $126$ bits. Moreover, via the Eigenvalue Method up to
$r_\mathcal{H} = 33$ rounds can be attacked below $128$ bits.
