[Narrator] One useful property of XOR is that we can use it to share a secret,
and that means we can take a secret, divide it among a group of people
where each person in that group does not gain any information about this secret,
but by combining their shares they can determine the secret.
Here's an example: Alice has some secret X, and she wants to keep a backup copy of X.
What she does is first she generates some key,
selected randomly and uniformly from the key space.
Then, she computes S, which is K XOR'd with X,
so that means S of I is equal to KI XOR'd with XI.
Then, she gives K to Bob and S to Colleen.
This means by combining K and S, Bob and Colleen can produce the secret X.
Either one by themselves has no information.
K just has a sequence of random bits,
and Colleen just has X XOR'd with a sequence of random bits which provides no information about X.
This works fine as long as Alice trusts Bob and Colleen not to collude
and to give back the values of K and S when she asks for them,
but what if she's worried that they might collude?
She decides to share the secret among more people.
How many key bits will Alice need to share a 100-bit long secret among 4 people?
