  Differential privacy (DP) provides a robust model to achieve privacy
guarantees for released information. We examine the protection potency of
sanitized multi-dimensional frequency distributions via DP randomization
mechanisms against homogeneity attack (HA). HA allows adversaries to obtain the
exact values on sensitive attributes for their targets without having to
identify them from the released data. We propose measures for disclosure risk
from HA and derive closed-form relationships between the privacy loss
parameters in DP and the disclosure risk from HA. The availability of the
closed-form relationships assists understanding the abstract concepts of DP and
privacy loss parameters by putting them in the context of a concrete privacy
attack and offers a perspective for choosing privacy loss parameters when
employing DP mechanisms in information sanitization and release in practice. We
apply the closed-form mathematical relationships in real-life datasets to
demonstrate the assessment of disclosure risk due to HA on differentially
private sanitized frequency distributions at various privacy loss parameters.
