Problem $5.$ The Mafia is a secretive organization in which only the Boss knows all other members, but every member knows the Boss. Any two members are unable to recognize each other unless they have already been introduced to each other by the Boss. Every member meets the Boss at the beginning of each calendar month.

The Boss has divided the other members of the Mafia into groups. Each group has exactly k members $($k $>4)$ and all members of each group are introduced to each other. However, no one knows members from other groups. Additionally, the Boss suspects that there are a number of traitors among the members but has chosen k such that the total number of traitors is definitely less than k$.$ Therefore, each group contains at least one loyal member.

Unfortunately, there has been an uptick in $$incidents$.$ An incident is when a Mafia group crosses paths with a member of another group but kills him as they are unable to recognize him. There has been up to one incident per month!

To solve this problem, the Boss wants to give each member a secret string of a fixed length at the beginning of each month. Then, during that month, any member should be able to prove to any group of k other members that she is indeed part of the Mafia, without having to rely on the Boss. Most importantly, the proof should not work if there are only k $1$ members. Otherwise, the traitors might be able to identify loyal members.

Help the Boss solve this problem using a cryptographic protocol. Prove that your protocol is safe and satisfies the requirements above.