Do black box secret sharing schemes [DF90, DF94] have any advantages over the better-known secret sharing schemes like Shamir’s [Sham79]?

[DF90]: Threshold cryptosystems, by Desmedt, Yvo and Frankel, Yair, in CRYPTO' 89, 1990

[DF94]: Homomorphic Zero-Knowledge Threshold Schemes over any Finite Abelian Group, by Desmedt, Yvo G. and Frankel, Yair, in SIAM Journal on Discrete Mathematics, 1994, [URL]

[Sham79]: How to Share a Secret, by Shamir, Adi, in Commun. ACM, 1979, [URL]


2 Answers 2


Yes, they do!

To quote [CF02], "A black-box secret sharing scheme for the threshold access structure $T_{t,n}$ is one which works over [secrets from] any finite Abelian group $G$.

In contrast, Shamir secret sharing works only over secrets from a finite field $\mathbb{F}_q$.

[CF02]: Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups, by Ronald Cramer and Serge Fehr, in Cryptology ePrint Archive, Paper 2002/036, 2002, [URL]

  • 1
    $\begingroup$ You should use $F_q$ for the finite field, since a ring $Z_q$ is not a field unless $q$ is prime. Furthermore, since the "secrets" are encodings of some real world information, it would be great if you could elaborate if there are any other advantages to black box secret sharing--there may well be. $\endgroup$
    – kodlu
    Jun 17, 2022 at 23:25
  • $\begingroup$ Fixed, thank you! Will have to think a bit about other advantages. $\endgroup$ Jun 19, 2022 at 0:32
  • $\begingroup$ Shamir secret sharing can also work over more general rings, even non-commutative ones eprint.iacr.org/2021/1025. $\endgroup$
    – Daniel
    Jul 20, 2022 at 13:03

Based on [DF94], a way of sharing trapdoor functions is exploited. See [SDFY94].

[SDFY94]: How to share a function securely, by Alfredo De Santis, Yvo Desmedt, Yair Frankel, and Moti Yung, in Proceedings of the twenty-sixth annual ACM symposium on Theory of Computing (STOC '94). https://doi.org/10.1145/195058.195405


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