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 nonqualified subset has absolutely no information on the secret, then the scheme is called perfect. The share in a scheme is the information that a participant must remember.
In  it was proved that for a certain access structure any perfect secret sharing scheme must give some participant a share which is at least 50\percent larger than the secret size. We prove that for each n there exists an access structure on n participants so that any perfect sharing scheme must give some participant a share which is at least about \(n/\log n\) times the secret size.^1 We also show that the best possible result achievable by the information-theoretic method used here is n times the secret size.
^1 All logarithms in this paper are of base 2.
Received 24 November 1993 and revised 15 September 1995
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Csirmaz, L. The Size of a Share Must Be Large . J. Cryptology 10, 223–231 (1997). https://doi.org/10.1007/s001459900029
- Key words. Secret sharing, Ideal secret sharing schemes, Polymatroid structures, Perfect security.