In Ex5 the functionality describes:

list[i](t + 1) ← Din(t)

Din(t) is not one of the inputs for the circuit “checker”. is Din = hash(key)?

That is what I assumed as it fits the required functionality but there seems indeed to be a mistake in the document so I’ll wait for clarification too

Another thing that is not defined in the document is the output “hash”. I assume it should be hash(key) at time t but it’s also not defined

Indeed! Thanks for pointing it out.

It should be \mathit{list}[i](t + 1) \gets h \left( \mathit{key} (t) \right) where h is a hashing function (implemented by the XORing the outputs of the circuit \texttt{hashing}).

Similarly, the hash output is defined such that \mathit{hash}(t) = h\left( \mathit{key}(t) \right).

It’s fixed in the updated PDF.