Despite several studies and attempts, in time-memory trade-off attacks on cryptoGraphic algorithms, the coverage of Hellman tables and similar methods are practically much less than half and their probability of success is low. In fact, Hellman chains are paths with given starting and end vertices on a functional Graph. In this paper, behavior of these chains is investigated with this approach. In the beginning of the paper, parameters of the functional Graph for a Random mapping are defined and based on these parameters, Hellman chains are analyzed. Our results show that the coverage of such tables can’t be high, for the following reasons: First, there exist some remarkable terminal vertices (37%) on the functional Graph such that the possible occurrence of these vertices on chains (except in the starting vertices) is zero. Secondly, appropriate parameters for constructing chains exist in Graph for about half of all hidden states of cipher function. Thirdly, for construction of noncyclic chains and collision of chains, we must pay attention to the obtained probabilities in this note. Practically, above reasons show that after some point the coverage of a Hellman table tends to zero quickly, and so construction of them will be ineffective. Our results are implemented on mAES algorithm where validate our theatrical results.

In this paper, we consider the Time-Memory-Trade-Off (TMTO) method for analysis of block ciphers and related methods. Also, we discuss some subjects including coverage in the Hellman chains, collision in chains, cycles and rings that create a block cipher function. Hellman method is analyzed by a Random Graph. The Random Graph is made of a function block cipher, which is applied to extract non-collision chains, cycles and rings. According to the unique modes and features available in the Random Graph, a new method for extraction of cycles and rings in the Random Graph entitled "agility of Graph" is offered. This meth-od extracts the cycles and rings of the lock cipher function as easily and at a very low cost. The obtained cycles and rings are used for generating non-collision chains in the block ciphers that they make a complete coverage of block ciphers in the TMTO method.

Graph structures play a vital role in modeling relationships across various domains, including social networks, knowledge bases, and biological networks. As the dimensions of these networks grow, the efficiency of proximity-based analysis methods declines, necessitating the use of Graph embedding techniques to reduce dimensionality while preserving the underlying structure. This process enhances performance in applications such as node classification and link prediction. However, traditional Graph embedding methods face challenges with capturing non-linear relationships and scaling to large networks. Additionally, in real-world networks, the essential initial and precise node features which are required by these algorithms are not always available. In this paper, we propose a novel framework called FuzzyRandomNet, which addresses these challenges by integrating fuzzy logic with Random walks. FuzzyRandomNet introduces non-linear layers and optimizes node features to provide more efficient and scalable solutions for Graph representation learning. The evaluation of the proposed method against existing techniques on standard datasets demonstrates superior performance in node classification and link prediction, exhibiting higher accuracy and flexibility in large and complex networks.

In this extended study, the focus is on advancing the generation of synthetic distribution grids (SDGs) through the introduction of a new algorithm based on the Barabási-Albert Random Graph model. The initial use of the Erdős model to create SDGs revealed limitations in size and structural adjustability beyond the number of vertices. To address these limitations and push the research forward, the new algorithm utilizes the Barabási-Albert model to provide more control over the structural features of the generated Graphs through the introduction of a novel tuning parameter known as the “richness index”. The effectiveness of both algorithms in producing SDGs of various sizes is demonstrated by generating SDGs with different sizes, confirming their ability to mimic synthetic radial distribution grids successfully. Additionally, a detailed examination of degree-based parameters and Pearson coefficients for SDGs of sizes from 20 to 1000 uncovers significant patterns. Furthermore, the proposed algorithm is examined in the terms of the variation of richness index in branching rate and μ-PMU placement, confirming the scale-free characteristic of the method. A comparison of the Erdős and Barabási-Albert models shows variations in maximum degree values, branching rates, and mixing patterns. The original Barabási-Albert model tends to have nodes with higher degrees and increased branching rates, which can be adjusted by the richness index. These findings emphasize the ability of the Barabási-Albert model to generate scale-free SDGs with diverse structures by fine-tuning the richness index.

Random walk on a Graph has large usages in various field of science. We probe some differences between classical and quantum Random walk (QRW) by explaining their circumstance (manner). We investigate absorption boundary problem on Graphs and present a reduction method to speedup in both absence and presence of bounds, and then we find that capture strength has a twofold behavior in quantum case.

The independence Graph Ind(G) of a Graph G is the Graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence Graph of Cartesian product of d copies of complete Graphs Kq, which is known as the Hamming Graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming Graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

In this paper, we find the star chromatic number of central Graph of complete bipartite Graph and corona Graph of complete Graph with path and cycle.