Zero-Knowledge Proof (ZKP) scheme is a type of cryptographic technique that is being commonly applied in practice, such as blockchain technology, authentication systems, and electronic voting systems. The Fiat-Shamir ZKP scheme on elliptic curves has been standardized in RFC 8235. In this paper, we will present some weaknesses that exist in the Fiat-Shamir ZKP scheme on elliptic curves, such as being affected by replay attacks and errors caused in the pseudorandom number generator. The goal of the article is to propose security enhancements for the Fiat-Shamir ZKP scheme to combat these limitations. Our method is analyzing, comparing the computational cost, and experimental results of the proposed solution with the original Fiat-Shamir ZKP scheme, which is mentioned in RFC 8235, and providing a user authentication application in the Client-Server model.