No abstract

Due to the global increase in mobile devices and cellular networks such as 3G, 4G, or LTE-A networks, the concept of the interpenetration of peer-to-peer (P2P) overlay networks in cellular environments has gained immense popularity among mobile users. P2P networks allow users to share their computing resources efficiently with the advantages of fault tolerance, robust connectivity, load balancing, and easy maintenance. The ability of a peer to connect to another peer depends on the distinguishing features of the overlay networks. For connection, network routing tables are maintained by each peer in the network to construct the P2P overlay on the Internet. This construction renders the P2P networks vulnerable to the existing attacks using the Internet. Some of these attack models have been proposed in the past using botnets. However, models using botnets do not accommodate the advanced attacks that can infect the majority of the entries in the routing table. In this paper, we propose a model to analyze the impacts of routing table poisoning attacks on the P2P botnet (rBot) architecture. We present the security analysis of the proposed epidemic model for state-of-the-art rBots. The obtained results were effective for attacks on the unstructured P2P static and mobile environments. INDEX TERMS Decentralized botnet, epidemic model, P2P overlay, routing table poisoning, mobility.

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

Let M be a given set of positive integers. A set S of nonnegative integers is said to be an M -set if a, b ∈ S implies a−b / ∈ M . In an unpublished problem collection, Motzkin asked to find maximal upper asymptotic density, denoted by µ(M ), of Msets. The first published work on µ(M ) is due to Cantor and Gordon in 1973, in which, they found the exact value of µ(M ) when |M | ≤ 2. In fact, this is the only general case, in which, we have a closed formulae for µ(M ). If |M | ≥ 3, then the exact value of µ(M ) is not known for the general set M . In the past six decades or so, several attempts have been given to study µ(M ) but µ(M ) has been found exactly or estimated only in very few cases. In this paper, we study µ(M ) for the families M = {a, a + 1, x} and M = {a, a + 1, x, y}, where y − x ≤ 2 and y > x > a + 1. Our results in the case of M = {a, a + 1, x} also give counterexamples to a conjecture of Carraher. Although, different counterexamples to this conjecture, were already given by Liu and Robinson in 2020. We also relate our results with the already know results for the families M = {1, 2, x, x + 2} and M = {2, 3, x, x + 2}.