Abstract
In this dissertation we study / review literature on the spectral properties of modern cryptographic
algorithms, with a particular focus on the Data Encryption Standard (DES).
In particular, we relate the spectrum of Cayley graphs constructed from DES substitution
boxes (S-boxes) to their corresponding Walsh spectra. We then use these spectral
characteristics to assess the strength of DES against cryptanalytic attacks.
For DES, we use S-boxes Boolean functions (BFs) to construct the Cayley graph, then
compare the resulting graph spectrum to the Boolean function’s Walsh spectrum, and
show that they are similar. This allows us to relate their properties, thereby providing a
way to use a graph to measure the cryptosystem’s resistance to attacks.
In Chapter 1 we introduce some mathematical preliminaries essential for this study. This
includes a review of some algebraic structures, linear algebra, logic, number theory, discrete
analysis, and graph theory, along with a review of cryptography and cryptanalysis.
These foundations establish the terminology and tools used throughout the dissertation.
Chapter 2 is devoted to an in-depth discussion of DES. We present its historical context
and its algorithmic structure, detailing the design and mathematical aspects of its
S-boxes. We also discuss various possible cryptanalytic attacks on DES that exposed it
to vulnerabilities over time. We review the extension of DES to Triple DES. This chapter
sets the stage for understanding the significance of spectral analysis in evaluating DES.
In Chapter 3 we shift our focus to the spectral analysis of DES. Here, we construct
the Cayley graph corresponding to the DES cipher and demonstrate that its spectrum is
equivalent to the Walsh spectrum of BFs. This chapter provides critical information on
how Cayley graphs and properties of BFs can reveal the strength of the cryptosystem.
Finally, in Chapter 4 we introduce the Advanced Encryption Standard, the successor
of DES, for the purposes of concluding with a natural question for future investigation on
the spectra of AES. We present its historical context and its algorithmic structure.
Throughout this dissertation, a combination of theoretical insights and computational
experiments is employed to validate the proposed spectral techniques.