The first situation that we present is the simplest. Consider the case where you have a message , represented as an integer, that you would like to split between two people Alice and Bob in such a way that neither of them alone can reconstruct the message . A solution to this problem readily lends itself: Give Alice a random integer and give Bob . In order to reconstruct the message , Alice and Bob simply add their pieces together.
A few technical problems arise from the fact that it is impossible to choose a random integer in a way that all integers are equally likely (the sum of the infinitely many equal probabilities, one for each integer, cannot equal 1). Therefore, we choose an integer larger than all possible messages that might occur and regard and as numbers mod . Then there is no problem choosing as a random integer mod ; simply assign each integer mod the probability .
Now let us examine the case where we would like to split the secret among three people, Alice, Bob, and Charles. Using the previous idea, we choose two random numbers and mod and give to Alice, to Bob, and to Charles. To reconstruct the message , Alice, Bob, and Charles simply add their respective numbers.
For the more general case, if we wish to split the secret among people, then we must choose random numbers mod and give them to of the people, and to the remaining person.