Peyravian and Zunic (2000) proposed a password transmission scheme and a password change scheme over an insecure network. Their proposed solutions do not require the use of any symmetric-key or public-key cryptosystems. However, this article points out that their schemes have several security flaws for practical applications. A slight improvement on their schemes is proposed in this paper to remove the security flaws. Key words: cryptography, password, hash function, discrete logarithm.