An entropy-based demonstration of the security of Shamir's secret sharing scheme

In a k-threshold secret sharing scheme, a dealer who holds a secret s distributes parts of this secret (the shares) to n players; If k or more of these players pool their shares, they are able to determine s, but if less than k of them pool their shares, they cannot infer any information on s. In 1979, Shamir introduced such a scheme, based on polynomial interpolation in a finite field. This scheme is widely used in other cryptographic protocols, because it is simple, elegant and above all information-theoretically secure. There are a few proofs of the scheme's security, but to our knowledge, none of them is entirely based on the information-theoretical entropy function introduced by Shannon in 1948. We propose such a demonstration.