Abstract:
This paper shows that the largest possible contrast C(k,n) in a k-out-of-n secret sharing scheme is approximately 4^(-(k-1)). More precisely, we show that 4^(-(k-1)) <= C_{k,n} <= 4^(-(k-1))}n^k/(n(n-1)...(n-(k-1))). This implies that the largest possible contrast equals 4^(-(k-1)) in the limit when n approaches infinity. For large n, the above bounds leave almost no gap. For values of n that come close to k, we will present alternative bounds (being tight for n=k). The proofs of our results proceed by revealing a central relation between the largest possible contrast in a secret sharing scheme and the smallest possible approximation error in problems occuring in Approximation Theory.