Coding theorems for secret-key authentication systems

This paper provides Shannon theoretic coding theorems on the impersonation attack and the substitution attack against authentication systems constructed by secret key cryptography. Though several lower bounds on the success probability of the impersonation attack and the substitution attack have been developed, their upper bounds are rarely discussed. This paper treats an extended authentication system including blocklength K and permits the decoding error probability tending to zero as K→∞. It is shown that 2^-KI(W:E) is the smallest attainable upper bound of the success probability of the impersonation attack, where I(W;E) denotes the mutual information between cryptogram W and key E. A relationship between the success probability of the substitution attack and H(E|W) is also characterized, where H(E|W) denotes the conditional entropy of E given W