Revisiting Interval Graphs for Network Science

The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear structure in the system. The generalization is an embedding of a graph onto a multi-dimensional Euclidean space and it was used by scientists to study the multi-relational complexity of ecology. However the research went out of fashion in the 1980s and was not revisited when Network Science recently expressed interests with multi-relational networks known as multiplexes. This paper studies interval graphs from the perspective of Network Science

On the topology Of network fine structures

Multi-relational dynamics are ubiquitous in many complex systems like transportations, social and biological. This thesis studies the two mathematical objects that encapsulate these relationships --- multiplexes and interval graphs. The former is the modern outlook in Network Science to generalize the edges in graphs while the latter was popularized during the 1960s in Graph Theory. Although multiplexes and interval graphs are nearly 50 years apart, their motivations are similar and it is worthwhile to investigate their structural connections and properties. This thesis look into these mathematical objects and presents their connections. For example we will look at the community structures in multiplexes and learn how unstable the detection algorithms are. This can lead researchers to the wrong conclusions. Thus it is important to get formalism precise and this thesis shows that the complexity of interval graphs is an indicator to the precision. However this measure of complexity is a computational hard problem in Graph Theory and in turn we use a heuristic strategy from Network Science to tackle the problem. One of the main contributions of this thesis is the compilation of the disparate literature on these mathematical objects. The novelty of this contribution is in using the statistical tools from population biology to deduce the completeness of this thesis's bibliography. It can also be used as a framework for researchers to quantify the comprehensiveness of their preliminary investigations. From the large body of multiplex research, the thesis focuses on the statistical properties of the projection of multiplexes (the reduction of multi-relational system to a single relationship network). It is important as projection is always used as the baseline for many relevant algorithms and its topology is insightful to understand the dynamics of the system.Open Acces

Applying Time-Memory-Data Trade-Off to Meet-in-the-Middle Attack

In this paper, we present several new attacks on multiple encryption block ciphers based on the meet-in-the-middle attack. In the first attack (GDD-MTM), we guess a certain number of secret key bits and apply the meet-in-the-middle attack on multiple ciphertexts. The second attack (TMTO-MTM) is derived from applying the time-memory trade-off attack to the meet-in-the-middle attack on a single ciphertext. We may also use rainbow chains in the table construction to get the Rainbow-MTM attack. The fourth attack (BS-MTM) is defined by combining the time-memory-data trade-off attack proposed by Biryukov and Shamir to the meet-in-the-middle attack on multiple ciphertexts. Lastly, for the final attack (TMD-MTM), we apply the TMTO-Data curve, which demonstrates the general methodology for multiple data trade-offs, to the meet-in-the-middle attack. GDD-MTM requires no pre-processing, but the attack complexity is high while memory requirement is low. In the last four attacks, pre-processing is required but we can achieve lower (faster) online attack complexity at the expense of more memory in comparison with the GDD-MTM attack. To illustrate how the attacks may be used, we applied them in the cryptanalysis of triple DES. In particular, for the BS-MTM attack, we managed to achieve pre-computation and data complexity which are much lower while maintaining almost the same memory and online attack complexity, as compared to a time-memory-data trade-off attack by Biryukov et al. at SAC 2005. In all, our new methodologies offer viable alternatives and provide more flexibility in achieving time-memory-data trade-offs

Generalized Correlation Analysis of Vectorial Boolean Functions

Abstract. We investigate the security of n-bit to m-bit vectorial Boolean functions in stream ciphers. Such stream ciphers have higher throughput than those using single-bit output Boolean functions. However, as shown by Zhang and Chan at Crypto 2000, linear approximations based on composing the vector output with any Boolean functions have higher bias than those based on the usual correlation attack. In this paper, we introduce a new approach for analyzing vector Boolean functions called generalized correlation analysis. It is based on approximate equations which are linear in the input x but of free degree in the output z = F (x). Based on experimental results, we observe that the new generalized correlation attack gives linear approximation with much higher bias than the Zhang-Chan and usual correlation attacks. Thus it can be more effective than previous methods. First, the complexity for computing the generalized nonlinearity for this new attack is reduced from 2 2m ×n+n to 2 2n. Second, we prove a theoretical upper bound for generalized nonlinearity which is much lower than the unrestricted nonlinearity (for Zhang-Chan’s attack) or usual nonlinearity. This again proves that generalized correlation attack performs better than previous correlation attacks. Third, we introduce a generalized divide-and-conquer correlation attack and prove that the usual notion of resiliency is enough to protect against it. Finally, we deduce the generalized nonlinearity of some known secondary constructions for secure vector Boolean functions. Keywords. Vectorial Boolean Functions, Unrestricted Nonlinearity, Resiliency.