The dealer's random bits in perfect secret sharing schemes

A secret sharing scheme permits a secret to be shared among participants of an n-element group in such a way that only qualified subsets of participants can recover the secret. If any non-qualified subset has absolutely no information on the secret, then the scheme is called perfect. The share in a scheme is the information what a participant must remember. It was known that in any perfect secret sharing scheme realizing a certain collection of qualified sets over n participant, at least one participant must use at least O(n/log n) random bits for each bit in the secret. Here we present a collection of qualified sets so that the total number of random bits used by all the participants, i.e. the dealer's random bits is at least O(n^2/log n) for each bit in the secret.