  The problem of secret-key based authentication under a privacy constraint on
the source sequence is considered. The identifier measurements during
authentication are assumed to be controllable via a cost-constrained "action"
sequence. Single-letter characterizations of the optimal trade-off among the
secret-key rate, storage rate, privacy-leakage rate, and action cost are given
for the four problems where noisy or noiseless measurements of the source are
enrolled to generate or embed secret keys. The results are relevant for several
user-authentication scenarios including physical and biometric authentications
with multiple measurements. Our results include, as special cases, new results
for secret-key generation and embedding with action-dependent side information
without any privacy constraint on the enrolled source sequence.
