A non-interactive modular verifiable secret sharing scheme

We prove that the verifiable secret sharing scheme of C. Asmuth and J. Bloom (see IEEE Trans. Inform. Theory, vol.IT-29.547, p.208-10, 1983) is not realizable although their basic modular secret sharing scheme is efficient and perfect. An efficient non-interactive modular verifiable secret sharing scheme is proposed, based on their basic scheme. The verification needs about (2|q|n+(n-1))r multiplications. The secret recovery takes O(r) operations while A. Shamir's scheme (1979) takes O(r log/sup 2/ r) operations. It is also shown that the scheme is a perfect secret sharing scheme.