Three party coin flipping protocol with only 2 active participants Is there a coin flipping protocol where:


Alice, Bob and Carol don't trust each other, and
The coin is flipped only by Alice and Bob, and
The result must be trusted by all three participants.
Your requirements are not terribly precise, so here is what I think you mean:


"The result must be trusted by all three participants" ==> Even if Alice &amp; Bob are both malicious &amp; colluding, the output of Carol should be uniform. Also, all 3 should get the same output.
"The coin is flipped only by Alice and Bob" ==> Alice &amp; Bob do all the work. At the most Carol should just send some information to them beforehand, and get some information from them afterwards.


In that case, it's impossible. Alice &amp; Bob can collude and do the following to force the coin toss to be whatever they want. Suppose they want it to come up heads:


Receive the initial message from Carol.
Run the protocol between the two of them until they are just about to send the final messages to Carol. By our assumption, Carol is not involved in this part. 
At this point Alice &amp; Bob can compute what the output is going to be. They both get output eventually (by correctness, it must be the same output Carol will get). But they will receive no more messages from Carol so they must be able to compute their output without her.
If the output is going to be tails, repeat part 2 fresh. Since Carol has no input to the protocol during step 2, correctness of the protocol requires that doing this is undetectable. This must result in a completely fresh outcome, otherwise there is an attack where Carol can fix the outcome of the coin.
If the output is going to be heads, send the final message to Carol.