Yes. But I don’t see why that’s a problem? Which preimage a given x would be assigned to is random. The hope is that repeated trials would give the same preimage frequently enough for it to be a meaningful partition of the input space. How well it would work depends on the details of the ECC but I suspect it would work reasonably well in many cases. You could also just apply the decoder directly to the string x but I thought that might be a bit more unnatural since in reality Bob will never see the full string.
Yes. But I don’t see why that’s a problem? Which preimage a given x would be assigned to is random. The hope is that repeated trials would give the same preimage frequently enough for it to be a meaningful partition of the input space. How well it would work depends on the details of the ECC but I suspect it would work reasonably well in many cases. You could also just apply the decoder directly to the string x but I thought that might be a bit more unnatural since in reality Bob will never see the full string.