Normally when you use OP_CHECKMULTISIG you're basically setting up something like x-of-y, where any x of the y key holders can carry out a transaction.
I want to know, if you can repurpose this operation in some way, so that its not just any x, but one of those x-keys needs be a specific party. How could you define a scriptPubKey such that the specific party is required and any 1 one of the remaining parties.
That is possible, you could use a custom P2SH contract with redeem script like:
<key1.public_key>
OP_CHECKSIGVERIFY
OP_1
<key2.public_key>
<key3.public_key>
OP_2
OP_CHECKMULTISIG
This script always requires a signature from key1, and a signature from either key2 or key3, making this a 1 AND 1/2 scheme. It would be spent using:
OP_0
<key2.signature.all_outputs>
<key1.signature.all_outputs>
<redeem_script>
but it could be extended to any L AND M/N, as long as it can fit inside Script VM limits.
