Verifiable threshold sharing of a large secret safe-prime

Verifiable secret sharing schemes (VSS) are schemes for the purpose of ensuring that the players are sharing a unique secret and this secret is the secret originally distributed by the dealer if the dealer was honest. However, such schemes do not ensure that the shared secret has any special characteristics (such as being a prime, safe prime or being with a specific bit-length). In this paper, we introduce a secret sharing scheme to allow a set of players to have confidence that they are sharing a large secret prime. Next, we introduce another scheme that allows the players to have confidence that they are sharing a large secret safe prime. Finally we give a subroutine that allows the players to ensure that the shared primes are of the appropriate bit-length. What we have in mind is to add fault-tolerance property to the recent all honest RSA function sharing protocol as presented in M. H. Ibrahim et al. (2004).