This paper presents a non-interactive, information-theoretically secure verifiable secret sharing scheme. The scheme allows a dealer to distribute a secret among \( n \) participants such that any \( k \) participants can recover the secret, while fewer than \( k \) participants receive no Shannon information about the secret. The information rate of the scheme is \(\frac{1}{2}\), and the distribution and verification processes require approximately \( 2k \) modular multiplications per bit of the secret. The scheme combines Shamir's polynomial sharing method with a commitment scheme, ensuring that the dealer cannot distribute incorrect shares even if they have unlimited computational power. The paper also demonstrates how the shareholders can compute linear combinations of shared secrets and how \( l \) participants can democratically select and distribute a secret without knowing it. The scheme is efficient and secure, but it relies on the assumption that the dealer cannot find discrete logarithms before the distribution is completed.This paper presents a non-interactive, information-theoretically secure verifiable secret sharing scheme. The scheme allows a dealer to distribute a secret among \( n \) participants such that any \( k \) participants can recover the secret, while fewer than \( k \) participants receive no Shannon information about the secret. The information rate of the scheme is \(\frac{1}{2}\), and the distribution and verification processes require approximately \( 2k \) modular multiplications per bit of the secret. The scheme combines Shamir's polynomial sharing method with a commitment scheme, ensuring that the dealer cannot distribute incorrect shares even if they have unlimited computational power. The paper also demonstrates how the shareholders can compute linear combinations of shared secrets and how \( l \) participants can democratically select and distribute a secret without knowing it. The scheme is efficient and secure, but it relies on the assumption that the dealer cannot find discrete logarithms before the distribution is completed.