Secret-key agreement based on biometric or physical identifiers is a promising security protocol for authenticating users or devices with small chips due to its lightweight security. In previous studies, the fundamental limits of such a protocol were analyzed, and the results showed that two auxiliary random variables were involved in the capacity region expressions. However, with these two auxiliary random variables, the complexity of computing the capacity regions may be prohibitively high. To deal with this problem, we explore classes of authentication channels that require only one auxiliary random variable in the expressions of the capacity regions. It is revealed that for the classes of degraded and less noisy authentication channels, a single auxiliary random variable is sufficient to express the capacity regions. As specific examples, we derive the closed-form expressions for binary and Gaussian sources. Also, numerical calculations for Gaussian sources are provided to show the trade-off between secret-key and privacy-leakage rates under a given storage rate, and to illustrate how the noise in the enrollment phase affects the capacity region.