In many applications of cryptography, assuring the authenticity of communications is as important as protecting their secrecy. A well known and secure method of providing message authentication is to compute a Message Authentication Code (MAC) by encrypting the message. If only one key is used to both encrypt and authenticate a message, however, the system is subject to several forms of cryptographic attack. Techniques have also been sought for combining secrecy and authentication in only one encryption pass, using a Manipulation Detection Code generated by noncryptographic means. Previous investigations have shown that a proposed MDC technique involving block-by-block Exclusive-ORing is not secure when used with the Cipher Block Chaining (CBC) mode of operation of the Data Encryption Standard (DES]. It is shown here that the Cipher Feedback (CFEI) mode of operation exhibits similar weaknesses. A linear addition modulo 264 MDC is analyzed, including discussion of several novel attack scenarios. A Quadratic Congruential Manipulation Detection Code is proposed to avoid the problems of previous schemes.