# 17.1 Secret Splitting

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.