Suppose Alice shares a secret block cipher key, K_AB with Bob, and a different secret block cipher key, K_AC with Charlie.

Describe a method for Alice to encrypt an m-block message such that it can only be decrypted with the cooperation of both Bob and Charlie.

The cipher text should only be a constant size greater than m blocks.

You may assume that Bob and Charlie have a pre-established secret channel on which to communicate.

Now, suppose Alice shares a block cipher key, K_AB with Bob, a block cipher key K_AC with Charlie, and a block cipher key K_AD with David.

Describe a method for Alice to encrypt an m-block message such that any two of Bob, Charlie, and David can decrypt (for example, Bob and Charlie can decrypt), but none of them can decrypt the message themselves.

Again, the cipher text should only be a constant size greater than m blocks. How does your solution from part (b) scale as we increase the number of recipients?

In other words, suppose Alice has a secret key with each of n recipients and wants to encrypt so that any k out of n recipients can decrypt, but any k-1 cannot.

What would be the length of the header as a function of n and k?