Not all perfect extrinsic secret sharing schemes are ideal.

TLDR: A perfect extrinsic secret sharing scheme is constructed for any case in which a set of participants can gain access to the secret if and only if the set contains a pair of members from some given list of pairs.

Abstract: We construct a perfect extrinsic secret sharing scheme for any case in which a set of participants can gain access to the secret if and only if the set contains a pair of members from some given list of pairs. A secret sharing scheme is a way by which a dealer rnay distribute secret information to individuals (call participants). There is associated '-<U.lh'.lcH"""Fof subsets of a set of can if they constitute a member of the access structure. usual to demand that the access structure be monotone: If A contains a subset which to the structure, then A itself must also belong to the structure. A secret scheme is called perfect if a set of pa.rticipants can no information by their shares unless the set is member of the access structure. An structure is called extrinsic if each participant '8 share contains the same amount of information. Simmons [4] has pointed out the desirability of a scheme which is and the access structure of which extrinsic. Observe that in such scheme the amount of information in the secret will equal at most the amount of information in each share. Brickell a somewhat stronger concept than a perfect extrinsic structure. He asks for what he calls an ideal scheme, which he defines to be a perfect scheme in which the number of possible secrets is equal to the number of possible shares available to each participant. Such a scheme is clearly extrinsic. We define a set L of subsets of the participants to be a basis for the access structure r if r consists precisely of those sets of participants with a subset belonging to L . It is usually convenient to use a minimal basis, but this is not essential we even admit r a basis for itself. VYe now outline a secret sharing scheme for a set P of w participants PI, Pz, ,PW' The structure r has a basis consisting of a number of 2-sets from P. The secret to be determined is an integer q in the range 2 S q S n. The secret information distributed to Pi is a sequence of n( w 1) symbols a}k' in the IThis research was supported in part by the National Security Agency under Grant No. MDA90489-H-2048 and by the Office of Research and Development Administration of Southern Illinois University under a Summer Research Grant. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon. Australasian Journal of Combinatorics 2(1990) pp. 237-238 where k ranges from 1 to nand j ranges from 1 to w except that case j = i is omitted. The symbols chosen for the w participants should all be different, except for the rules: (1) ail for all i,j, The nature of the symbols is not important a sufficiently large set of lIlteg;en" in random order, could be used. {Pi, P j } is in vVhen pool their they find two cornmon elements. To determine q, they count off the sequence elements from one common element to the next, inclusive. If {Pi, Pj} IS not in , they gain no information about q. So the is perfect, and extrinsic. We have THEOREM: Any access structure a basis entirely of 2-sets can be realized a extrinsic secret sharing system. # Benaloh and Leichter [1] show that there exist monotone access structures for which no ideal scheme is possible. In fact their proof involves showing that there is no ideal realization of the structure with basis {A, B}, {B,C}, {C,D}. we have proven that a perfect extrinsic realization does exist. So the problem of whether all monotone access structures can be realized a extrinsic scheme is still open. Finally, Ive observe that our construction can be broadened to include the possibility of certain individuals who can determine the secret alone. One simply adds a new participant Po, and adds to the set all the pairs {po, pd where Pi alone can determine the secret. Finally, Po's share is made public knowledge.