Secret sharing

Suppose a bank wants to require that two people be required to approve entry to a vault. Can they split the keys in such a way that both are required?

Easy solution: let k from (Z/2Z)ⁿ be the key that opens the door. Choose a random k₀ in (Z/2Z)ⁿ, and give out shares k₀ and k₁ = k₀ + k to the two parties.

More complicated: give access to any 3 out of 7, but no fewer.

Even more complicated: allow people to enter and leave the group.

Even more complicated: give them capability to use the keys, but don't ever bring their keys together (e.g., proactive RSA).

Even more complicated: eliminate the dealer.