(Difficult) The shamir threshold scheme was a little confusing to me. I think I understand how each pair (x,y) is chosen, but im not sure I understand why we know the determinant of V will be non-zero. My other concern is whether or not this method works only once. In other words, do we need to replace the polynomial s(x) with new random values s1, s2, etc. each time this method is used?
(Reflective) There are two different types of secret sharing in the examples from 12.1. The first case is one person giving a shared secret to various people. The second case, involves a secret where no single person, (including the inventor of the secret code), should be able to generate the whole code by themselves. The methods in the book all require that the code be constructed first, then the parts of the code are distributed out amongst multiple people. I tried to find a way to construct such a code so that no one would have access to the actual code. Is there a way to make a function where no one, (not even those distributing each part of the code), would have complete knowledge of the code? Would the secret code M be encrypted and then stored somewhere?
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