how does the simulator work when the corrupted party has no input? "the only thing the simulator can do in the ideal model is to choose the corrupted parties' inputs." Yehuda Lindell said.

1- what will happen when the corrupted parties have no inputs to the protocol? e.g. in cases that the corrupted party/parties are just some computing servers with no input.

2- in these cases, is it any relationship between security against malicious and semi-honest adversaries. As I am thinking that, as the corrupted party has no input, anything that a malicious adversary is able to do is also possible for the semi-honest adversary.

if not,

is it possible in any way to conclude security against malicious adversaries in these cases when we have proved security against semi-honest adversaries?
A simulator is a thought experiment whose purpose is to:


show that the adversary learns no more about the honest party's inputs than he's supposed to,
show that the adversary's effect on the honest party's output is consistent with a legitimate input.


If the functionality takes no input from the corrupt party, #2 is moot but we still have #1. A canonical example is the zero-knowledge functionality, which takes no input from the verifier. When the verifier is corrupt, the purpose of the simulator is to demonstrate the zero-knowledge property.

There is no general implication between semi-honest and malicious security. Take any semi-honest protocol and modify it to add the behavior "if the other person sent me the message $(\textsf{asdf}, v)$ then send my input to that party and output $v$." Adding this behavior to the protocol doesn't affect semi-honest security since honest parties never send each other $\textsf{asdf}$ messages. But the effect on malicious security should be clear, and it has nothing to do with which party has input, etc.