Impossibility Bounds Using Conditional Rényi Entropy in Authentication Codes with Partially Leaked Key

In message authentication, we consider a situation where a sender transmits a message to a receiver through an insecure channel. In the insecure channel, there is a risk of impersonation or substitution by an adversary. Message authentication is a scheme to detect such attacks and to accept the message sent by the sender as legitimate. One of the research topics in message authentication is the estimation of limits on how small the probabilities of a successful attack can be. Previous research has shown impossibility bounds using Rényi entropy in the case where a shared key may be non-uniform. In this study, we consider a situation where the adversary has side-information that is correlated with a possibly non-uniform shared key, and investigate the impossibility using conditional Rényi entropy. In particular, we show that, in contrast to the previous research, impossibility bounds using conditional min-entropy do not hold in general. In addition, the success probabilities can be bounded in general using conditional collision entropy, and we show that the bound is the tightest in terms of conditional Rényi entropy.