| Popis: |
We consider Shamirʼs type secret sharing scheme with the secret placed as a coefficient a i of the scheme polynomial f ( x ) = ∑ i = 0 k − 1 a i x i over F q . A coalition of shareholders equipped with pairwise different public identities t j belonging to F q is called ( k , i ) -authorized if the holders using their secret shares y j = f ( t j ) ( 1 ⩽ j ⩽ n ) are able to reconstruct the secret by themselves. A ( k , i ) -authorized coalition of k − 1 or less shareholders is called a ( k , i ) -privileged coalition. We construct all ( k , i ) -privileged coalitions of k − 1 shareholders if i ≠ 0 , k − 1 . This construction allows us to deduce a new estimate for the number of such coalitions, which implies that they exist if q > 2 k − 1 . We also give a method of extending a given minimal ( k , i ) -privileged coalition of r shareholders to a coalition of n shareholders ( n ⩾ r ) each of whose ( k , i ) -privileged subcoalitions contains the given one provided q ⩾ n + r ( n − 2 k − 2 ) . |
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