This paper demonstrates that \textit{A. Shamir}'s scheme [Commun. ACM 22, 612-613 (1979; Zbl 0414.94021)] is not secure against certain forms of cheating. A small modification to his scheme retains the security and efficiency of the original, is secure against these forms of cheating, and preserves the property that its security does not depend on any unproven assumptions such as the intractability of computing number- theoretic functions.