The world of system of systems engineering (SoSE), and consequently systems engineering, is currently being reformalized so as to provide greater functionality, integration, and extensibility into designed systems of systems (SoS). Graph theory and big data techniques are in position to play substantial roles in this reformalization, and hence, this paper fills a significant need by bringing together all known current applications of graph theory to SoSE in tutorial/summary form. In this paper, we explore the applications of graph theory and known graph algorithms for system design (and SoS design), optimization techniques, complexity measures, and novel graph algorithms for real-time deployment of SoS's and managing SoS design. The work is straightforward with several examples, and is meant to be a standalone document that can be used to quickly come up to speed regarding the applications of graph theory in SoSE.

In this paper we consider tandem error control coding and cryptography in the setting of the wiretap channel due to Wyner. In a typical communications system a cryptographic application is run at a layer above the physical layer and assumes the channel is error free. However, in any real application the channels for friendly users and passive eavesdroppers are not error free and Wyner's wiretap model addresses this scenario. Using this model, we show the security of a common cryptographic primitive, i.e. a keystream generator based on linear feedback shift registers (LFSR), can be strengthened by exploiting properties of the physical layer. A passive eavesdropper can be made to experience greater difficulty in cracking an LFSR- based cryptographic system insomuch that the computational complexity of discovering the secret key increases by orders of magnitude, or is altogether infeasible. This result is shown for two fast correlation attacks originally presented by Meier and Staffelbach, in the context of channel errors due to the wiretap channel model.

In this paper, we provide secrecy metrics applicable to physical-layer coding techniques with finite blocklengths over Gaussian and fading wiretap channel models and analyze their secrecy performance over several cases of concatenated code designs. Our metrics go beyond some of the known practical secrecy measures, such as bit error rate and security gap, so as to make lower bound probabilistic guarantees on error rates over short blocklengths both preceding and following a secrecy decoder. Our techniques are especially useful in cases where application of traditional information-theoretic security measures is either impractical or simply not yet understood. The metrics can aid both practical system analysis, including cryptanalysis, and practical system design when concatenated codes are used for physical-layer security. Furthermore, these new measures fill a void in the current landscape of practical security measures for physical-layer security coding and may assist in the wide-scale adoption of physical-layer techniques for security in real-world systems. We also show how the new metrics provide techniques for reducing realistic channel models to simpler discrete memoryless wiretap channel equivalents over which existing secrecy code designs may achieve information-theoretic security.

In this paper, we fill a void between information theoretic security and practical coding over the Gaussian wiretap channel using a three-stage encoder/decoder technique. Security is measured using Kullback-Leibler divergence and resolvability techniques along with a limited number of practical assumptions regarding the eavesdropper's decoder. The results specify a general coding recipe for obtaining both secure and reliable communications over the Gaussian wiretap channel, and one specific set of concatenated codes is presented as a test case for the sake of providing simulation-based evaluation of security and reliability over the network. It is shown that there exists a threshold in signal-to-noise ratio (SNR) over a Gaussian channel, such that receivers experiencing SNR below the threshold have no practical hope of receiving information about the message when the three-stage coding technique is applied. Results further indicate that the two innermost encoding stages successfully approximate a binary symmetric channel, allowing the outermost encoding stage (e.g., a wiretap code) to focus solely on secrecy coding over this approximated channel.

Traditional cryptography assumes an eavesdropper receives an error-free copy of the transmitted ciphertext. Wyner's wiretap channel model recognizes that at the physical layer both the intended receiver and the passive eavesdropper inevitably receive an error-prone version of the transmitted message which must be corrected prior to decryption. This paper considers the implications of using both channel and cryptographic codes under the wiretap channel model in a way that enhances the information-theoretic security for the friendly parties by keeping the information transfer to the eavesdropper small. We consider a secret-key cryptographic system with a linear feedback shift register (LFSR)-based keystream generator and observe the mutual information between an LFSR-generated sequence and the received noise-corrupted ciphertext sequence under a known-plaintext scenario. The effectiveness of a noniterative fast correlation attack, which reduces the search time in a brute-force attack, is shown to be correlated with this mutual information. For an iterative fast correlation attack on this cryptographic system, it is shown that an EXIT chart and mutual information are very good predictors of decoding success and failure by a passive eavesdropper.