What does "constant rate" mean in universal composable commitment scheme? I'm wondering what does the "constant rate" mean in universal composable commitment scheme? I have known the rate of a commitment scheme is message length divided by the communication complexity of the scheme. What's the "constant" mean here? Must the constant be a number less than 1?
Constant rate in general means that the overhead from a non-secure method is constant. So, in a simple way, if I am committing to an $\ell$-bit message, then the size of the commitment is $O(\ell)$. In some cases, however, one also allows an additive factor that is independent of the message size. Thus, for example, it could be that to commit to a message of size $\ell$ the amount is $O(\ell)+{\rm poly}(n)$, where $n$ is the security parameter. 

Note that typically these are measured in an amortized manner. So, you have to send many commitments (or a long message) for it to be true. But, again, this depends on the exact scheme, so you'll have to read the details.