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This paper demonstrates that Shamir's scheme ("How to share a secret", Communications of the ACM, vol. 22, no. 11, November 1979, 612-613) is not secure against cheating. A small modification to his scheme retains the security and efficiency of the original, is secure against cheating, and preserves the property that its security does not depend on any unproven assumptions such as the intractability of computing number-theoretic functions.