Efficient Multiplicative Sharing Schemes

Multiplicative threshold schemes are useful tools in threshold cryptography. For example, such schemes can be used with a wide variety of practical homomorphic cryptosystems (such as the RSA, the El Gamal and elliptic curve systems) for threshold decryption, signatures, or proofs. The paper describes a new recursive construction for multiplicative threshold schemes which makes it possible to extend the number of users of such schemes for a relatively small expansion of the share size. We discuss certain properties of the schemes, such as the information rate and zero knowledge aspects. The paper extends the Karnin-Greene-Hellman bound on the parameters of ideal secret sharing schemes to schemes which are not necessarily ideal and then uses this as a yardstick to compare the performance of currently known multiplicative sharing schemes.