For the public record, I'm posting this exchange a group of us had last year, where Lorenz said (quotes' his, others' my comments that're new and original to this post):
Now the crucial thing to notice is that in the end, combining many of
these quasigroup operations in an arbitrary way will just end up being a
sum (with respect to the new group law + recovered in the script) of the
input elements twisted by compositions of A,B in various ways,
That's what basically any such entropic quasigroups boils down to - groups with independent automorphisms.
so we can linearize the group + by computing some discrete logarithms, then simply solve a linear-algebra problem to recover the secret-key operation.
Now, one thing to know is that, the composition of automorphism forms group, what Lorenz mean by here would be that, represent the automorphisms in a form that can be solved using some some algorithm for linear-algebra.
This had been the focus of some follow up discussion within the group, where another participant - Danilo Gligoroski - questioned emphasizing:
abstract existence does not imply we can also construct something efficiently.
Another thing is that, the group of automorphism composition and the group of linearized quasigroup do not form field, so the best currently known Gaussian elimination will not be capable of solving it even when it's formed.