[lampington]

Thank you! I was confused by the ℎ∼𝑈[−1/2,1/2], thinking that it meant either that they both have a set of random numbers (in which case, how does the receiver know which one of them the sender used?) or that they both had the same single random number in that range (in which case it looks like it's basically the same as the randomised rounding case).

Your explanation that they are both going through the same list at the same rate clarifies that. They have n random numbers and are both using the same one for each transmission/receipt event.

I guess that in practice, synchronising that would be a pain. Either they sync on packets sent/received, which means that one dropped packet messes everything up, or they sync on time, in which case there's clock skew to worry about. Or they could have sync information in the data sent ("I'm using the 1234th item in the shared randomness"), which makes a bit of a nonsense of the whole "one bit" thing.

Am I missing something there?