Bounds on Key Appearance Equivocation for Substitution Ciphers

The average conditional entropy of the key given the message and its corresponding cryptogram, ff(K|M, C), which is refer as a key appearance equivocation, was proposed as a theoretical measure of the strength of the cipher system under a known plaintext attack by Dunham in 1980. In the same work (among other things), lower and upper bounds for H(SM |M^LC^L) are found and its asymptotic behavior as a function of cryptogram length L is described for simple substitution ciphers, i.e., when the key space S_Lambda1, Is the symmetric group acting on a discrete alphabet M. In the present paper we consider the same problem when the key space is an arbitrary subgroup K < S,_M and generalize Dunham´s result.