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and any 2k 1 such matrices (with distinct αi’s) generate all the others. This can be argued using again Vandermonde’s determinants. It follows from Proposition 2.6.6 that, provided that 2k 1 n, Shamir’s scheme is multi- plicative. If in addition 2k 1 n (k 1), i.e. n 3(k 1) + 1, then Shamir’s scheme is (k 1)-strongly multiplicative. In particular this allows us to construct a scheme which attains the bound given in Proposition 2.6.9. We will show in Section 4.3.1 that the converse also holds, i.e. that a secret sharing scheme which attains the bound of Proposition 2.6.9 is necessarily based on a Reed-Solomon code.
