Bounds on Key Appearance Equivocation for Substitution Ciphers

TL;DR

This paper extends the theoretical analysis of key appearance equivocation in substitution ciphers, providing bounds and asymptotic behavior for arbitrary key subgroups, generalizing previous results for simple substitution ciphers.

Contribution

It generalizes Dunham's bounds on key equivocation to arbitrary key subgroups, broadening understanding of cipher strength.

Findings

Derived bounds for key equivocation in general subgroups

Analyzed asymptotic behavior as cryptogram length increases

Extended previous results from simple to arbitrary key groups

Abstract

The average conditional entropy of the key given the message and its corresponding cryptogram, H(K|M,C), which is reffer 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(S}_{M}|M^L,C^L) are found and its asymptotic behaviour as a function of cryptogram length L is described for simple substitution ciphers i.e. when the key space S_{M} 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 of S_{M} and generalize Dunham's result.