Bounds for Message Authentication with Partially Leaked Secret Key Using Conditional Rényi Entropy

In message authentication, we consider a situation where a sender sends messages 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 legitimate messages as legitimate. In Igawa's study, the upper and lower bounds of the product of the success probabilities of the attacks are derived under the condition that each party observes correlated i.i.d. sequences as secret information, and it is shown that the upper and lower bounds are asymptotically equal in certain conditions. In Shikata's study, the success probabilities of attacks are evaluated in terms of the Rényi entropy in the situation where the sender and receiver share a secret key of finite length. In this study, we evaluate the success probability of attacks using the conditional Rényi entropy for the case where each party observes correlated information of finite length and the sender and receiver observe the same, which is one of the open questions in Igawa's study. We can also think of such a situation as one where the secret key is partially leaked. First, we extend the definitions of the attack success probabilities to our situation. Next, we evaluate these probabilities using the conditional Rényi entropy by dividing them into cases by the size of the alphabet of the secret key.