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2

Just to complement prof. Lindell's answer, although one cannot have a formal description of the case in which round complexity will matter more than communication or computation, a colleague of mine did some estimations two years ago, for a paper we were writing on round-efficient primitives for zero-knowledge. It's just a particular case, but having figures ...

5

First note that all polynomial-time functions can be securely computed with a constant number of rounds (Yao and BMR families) and all can be securely computed with protocols that have rounds dependent on the depth of the circuit computing the function (GMW families). The question is when is one type better than another. The answer to this question is not ...

1

The Encryption is not likely Compression where one can choose Normal, High or Low or something from scale of 0-100 as you mentioned. If you really want to do it, you can think of it in two ways Reduce the keyspace, as already mentioned in an above answer Reduce the Rounds. But these both will lead to a situation where you may end up in a breakthrough (...

2

If you are only given $a^{\lambda(n)} = 1$, then it is obviously impossible to recover $a$, since this is true for all $a$. To take a simpler example, say I tell you that the square of a number is $4$, and ask you to find what the number is. This is impossible, because you are not given enough information to tell whether the number is $2$ or $-2$.

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