Information-theoretical secure verifiable secret sharing with vector space access structures over bilinear groups and its applications

• We present an information-theoretical secure VSS scheme with vector space access structure over bilinear groups.
• We convert the scheme to a modified one with improved efficiency.
• We show the application of our schemes in Boneh and Franklin’s identity-based encryption scheme.
• We show how to use our scheme to realize distributed key generation and distributed decryption in bilinear ElGamal encryption system.

As a basic tool, Verifiable Secret Sharing (VSS) has wide applications in distributed cryptosystems as well as secure multi-party computations. A number of VSS schemes for sharing a secret from a finite field, both on threshold access structures and on general access structures, have been available. In this paper, we investigate the verifiably sharing of a secret that is a random element from a bilinear group on vector space access structures. For this purpose, we present an information-theoretical secure VSS scheme, and then convert it to a modified one with improved efficiency. The performance and the security of the proposed schemes are analyzed in detail. Two examples are given to illustrate the applications of our proposed VSS schemes. One is the secure sharing of an organization’s private key in Boneh and Franklin’s identity-based encryption system, and the other is the distributed key generation and distributed decryption for bilinear ElGamal encryption system, both with vector space access structures.