A large variety of encryption schemes, or modes, have been proposed to date, and some of these are known to be secure against adaptive, chosen-plaintext attacks. In this presentation, I define a joint condition on any such secure scheme and any high-performance Manipulation Detection Code (hpMDC) function, such as XOR, CRC-32, modular addition, or simply a constant, to counter adaptive chosen-message attacks, namely both adaptive chosen-plaintext and chosen-ciphertext attacks, that lead to message forgeries. I also illustrate two applications of the joint condition in practice, namely (1) the design of fast encryption-with-integrity schemes and (2) the optimal selection of a hpMDC function for a given encryption scheme.