**Abstract:** In the context of Universal Composability, we introduce the concept of universal
environments and simulators. Then, Universal Composability is equivalent to Universal
Composability wrt. universal environments and simulators. We prove the existence
of universal environments and simulators and investigate their computational complexity.
From this, we get a number of consequences: First, we see that for polynomial-time
protocols, exponential adversarial entities are as powerful as unlimited ones.
Further, for a large class of protocols (those with bounded communication-complexity)
we can show that UC and specialised-simulator UC coincide in the case of statistical
security, i.e., that it is does not matter whether the simulator is chosen in
dependence of the environment or not. This also implies that for the Universal
Composition Theorem for polynomial-time protocols specialised-simulator UC is
sufficient. This result is the last piece needed to find all implications and
non-implications between the notions of UC, specialised-simulator UC, O(1)-bounded
and polynomially-bounded general composability for polynomial-time protocols in
the cases of perfect, statistical and polynomial security. Finally, we introduce
the notion of bounded-risk UC, which allows to give explicit security guarantees
for concrete security parameters and show that in the above case also this variant
coincides with UC.

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