Although it is not clear, it appears that definition 5 of this paper,
page 10 of this paper, and page 6 of this paper, each assume
that the honest parties will all use the same security parameter.
Are there any known constructions of concurrent non-malleable
commitment that do not assume a common security parameter?
(If I'm wrong about the first sentence in this post, then
one of those three papers might be such a construction.)
I notice that, by rounding to a value of the form $2^\left(2^n\right)\hspace{-0.02 in}$,$\:$ one can assume without
loss of generality that at most $\:O\hspace{.01 in}(1)\:$ different security parameters will be used.