Date of Award

2025

Document Type

Thesis (Master's)

Department or Program

Computer Science

First Advisor

Christophe Hauser

Second Advisor

Sergey Bratus

Third Advisor

Sean Smith

Abstract

This thesis presents a comprehensive and chronological overview of cryptographic techniques designed to break Enigma, beginning in 1932 and culminating in the creation of the Turing-Welchman Bombe. We discuss the mathematical theory and electromechanical implements used to decode one of history's greatest ciphers.

Reexamining the Bombe through the lens of modern group theory, we critique Alan Turing's estimation of the number of "stops" that the Bombe produces for various plaintext-ciphertext pairing structures. To address its limitations, we introduce a new framework for estimating the number of stops by extending John Dixon's theorem concerning the probability that uniformly distributed elements of Sn generate a transitive subgroup. Our formulation generalizes this result to compute the probability of transitivity when permutations are sampled from arbitrary distributions over conjugacy classes.

All results are supported by extensive simulation, and a full suite of implementations of various cryptographic techniques are made available via an open source repository to provide researchers with new tools to study the methods discussed herein.

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