Approximate message authentication codes (AMACs) for binary alphabets have been introduced recently as noise-tolerant authenticators. Different from conventional “hard” message authentications that are designed to detect even the slightest changes in messages, AMACs are designed to tolerate a small amount of noise in messages for applications where slight noise is acceptable, such as in multimedia communications. Binary AMACs, however, have several limitations. First, they do not naturally deal with messages having$N$-ary alphabets$(N≫2)$. AMACs are distance-preserving codes; i.e., the distance between two authentication tags reflects the distance between two messages. Binary representation of$N$-ary alphabets, however, may destroy the original distance information between$N$-ary messages. Second, binary AMACs lack a means to adjust authentication sensitivity. Different applications may require different sensitivities against noise. AMACs for$N$-ary alphabets are designed as a cryptographic primitive to overcome the limitations of binary AMACs.$N$-ary AMACs not only directly process messages having$N$-ary alphabets but also provide sensitivity control on the authentication of binary and of$N$-ary messages. The generalized$N$-ary AMAC algorithm and its probabilistic model are developed. A statistical analysis characterizing the behavior of$N$-ary AMACs is provided along with the simulations illustrating their properties. Security analysis under chosen message attack is also developed.