Does that violate the perfect secrecy in any way?
Yes, obviously. Restricting the message space doesn't hurt in any way (the attacker can't get any additional information about the message, even if he knows quite a lot about it already); restricting the key does.
OTP uses a group operation to combine the message and the key; one thing that it needs to ...
What I want is, that Alice and Bob cannot read the message, only their
combined knowledge would allow them to decipher it.
"Committing with partial knowledge of group order" scenario was presented at the CECC 2010.
A scheme was designed such that order of an RSA-like multiplicative group is split into two parts shared by two parties such that both ...