Combinatorial Lower Bounds for Secret Sharing Schemes

In a perfect secret sharing scheme, it holds that log 2 j ^ V i j H(S), where S denotes the secret and ^ V i denotes the set of the share of user i. On the other hand, it is well known that log 2 j ^ Sj > H(S) if S is not uniformly distributed, where ^ S denotes the set of secrets. In this case, log 2 j ^ V i j H(S) < log 2 j ^ Sj : Then, which is bigger, j ^ V i j or j ^ Sj ? We rst prove that j ^ V i j j ^ Sj for any distribution on S by using a combinatorial argument. This is a more sharp lower bound on j ^ V i j for not uniformly distributed S. Our proof makes it intuitively clear why j ^ V i j must be so large. Next, we extend our technique to show that max i log 2 j ^ V i j 1:5 log 2 j ^ Sj for some access structure.