Verifiable secret sharing using discrete logarithms via skew polynomials

In the field of cryptography, ensuring secure collaboration among multiple parties often requires innovative methods for managing sensitive information. One such method is secret sharing, which involves breaking a secret into several portions and distributing them among participants, allowing the original secret to be reconstructed only by a group meeting specific criteria. Adi Shamir initially introduced it as a secret sharing scheme based on polynomials, where the secret is reconstructed through polynomial interpolation. This scheme requires a minimum of k participants from n participants to be able to reconstruct the secret. Over time, several challenges have arisen due to the presence of adversaries who can provide false shares to gain knowledge of the secret. Therefore, the verifiable secret sharing scheme was introduced, which is useful for verifying the shares of all participants gathered during the reconstruction process. Zhang proposed a secret sharing scheme utilizing skew polynomials in 20l0, which was later complemented by Johnson’s 20l8 exploration of an approach based on non-commutative polynomials. The use of skew polynomials can increase the complexity of the algorithm. Our prior work has explored secret sharing schemes using skew polynomials focusing on dynamic threshold management; however, these earlier schemes lacked inherent verifiability. This study will also apply skew polynomials in the verifiable secret sharing scheme, specifically using discrete logarithms. This research will compare the time complexity between the scheme designed through regular polynomials and the scheme designed through skew polynomials.