The Secret of Tipsy Cola is …

And so the eight directors of Tipsy Cola sit round the table. Neither of them trust each other, but they need a way to store their secret formula without anyone else finding out. “Why don’t we put it in a safe, and we can all have the key for it?”, “Well, that won’t work, as one of us could sneak in and steal it!”, “Well, why don’t we create a combination lock, and each of us can have one of the numbers?”, “Well, we will need all of the directors, but just say one of us falls under a bus, we will never be able to open the safe!”, “Ah, why don’t we accept any six numbers from eight, and we’ll be able to open the safe?”, “Yes. That’s it!”.

The method we have derived is a threshold secret sharing method, and where we have s shares and where t shares are required to built the secret. In the case of Tipsy Cola, we have eight shares and have a threshold of six. The method of threshold encryption was created by Adi Shamir, and he outlined a method where we take a mathematical function, and then derive points on this function. For example, if we have a function of:

y = mx +c

Then we can make m the secret, and then share two points on the line. Only with the knowledge of both points is it possible to derive the gradient of the line. If one point is (5,5) and the other is (6,9), the gradient is (9–5)/(6–1)=3. For most complex shares we can use higher…