Arithmetic Framework to Optimize Packet Forwarding among End Devices in Generic Edge Computing Environments

Multi-access edge computing implementations are ever increasing in both the number of deployments and the areas of application. In this context, the easiness in the operations of packet forwarding between two end devices being part of a particular edge computing infrastructure may allow for a more efficient performance. In this paper, an arithmetic framework based in a layered approach has been proposed in order to optimize the packet forwarding actions, such as routing and switching, in generic edge computing environments by taking advantage of the properties of integer division and modular arithmetic, thus simplifying the search of the proper next hop to reach the desired destination into simple arithmetic operations, as opposed to having to look into the routing or switching tables. In this sense, the different type of communications within a generic edge computing environment are first studied, and afterwards, three diverse case scenarios have been described according to the arithmetic framework proposed, where all of them have been further verified by using arithmetic means with the help of applying theorems, as well as algebraic means, with the help of searching for behavioral equivalences.

Post-Quantum and Code-Based Cryptography—Some Prospective Research Directions

Cryptography has been used from time immemorial for preserving the confidentiality of data/information in storage or transit. Thus, cryptography research has also been evolving from the classical Caesar cipher to the modern cryptosystems, based on modular arithmetic to the contemporary cryptosystems based on quantum computing. The emergence of quantum computing poses a major threat to the modern cryptosystems based on modular arithmetic, whereby even the computationally hard problems which constitute the strength of the modular arithmetic ciphers could be solved in polynomial time. This threat triggered post-quantum cryptography research to design and develop post-quantum algorithms that can withstand quantum computing attacks. This paper provides an overview of the various research directions that have been explored in post-quantum cryptography and, specifically, the various code-based cryptography research dimensions that have been explored. Some potential research directions that are yet to be explored in code-based cryptography research from the perspective of codes is a key contribution of this paper.

ModuloNET: Neural Networks Meet Modular Arithmetic for Efficient Hardware Masking

Intellectual Property (IP) thefts of trained machine learning (ML) models through side-channel attacks on inference engines are becoming a major threat. Indeed, several recent works have shown reverse engineering of the model internals using such attacks, but the research on building defenses is largely unexplored. There is a critical need to efficiently and securely transform those defenses from cryptography such as masking to ML frameworks. Existing works, however, revealed that a straightforward adaptation of such defenses either provides partial security or leads to high area overheads. To address those limitations, this work proposes a fundamentally new direction to construct neural networks that are inherently more compatible with masking. The key idea is to use modular arithmetic in neural networks and then efficiently realize masking, in either Boolean or arithmetic fashion, depending on the type of neural network layers. We demonstrate our approach on the edge-computing friendly binarized neural networks (BNN) and show how to modify the training and inference of such a network to work with modular arithmetic without sacrificing accuracy. We then design novel masking gadgets using Domain-Oriented Masking (DOM) to efficiently mask the unique operations of ML such as the activation function and the output layer classification, and we prove their security in the glitch-extended probing model. Finally, we implement fully masked neural networks on an FPGA, quantify that they can achieve a similar latency while reducing the FF and LUT costs over the state-of-the-art protected implementations by 34.2% and 42.6%, respectively, and demonstrate their first-order side-channel security with up to 1M traces.

Comparison of numbers and analysis of overflow in modular arithmetic

The method of modular arithmetic consists in operating not with a number, but with its remainders after division by some integers. In the modular number system or the number system in the residual classes, a multi-bit integer in the positional number system is represented as a sequence of several positional numbers. These numbers are the remainders (residues) of dividing the original number into some modules that are mutually prime integers. The advantage of the modular representation is that it is very simple to perform addition, subtraction and multiplication operations. In parallel execution of operations, the use of modular arithmetic can significantly reduce the computation time. However, there are drawbacks to modular representation that limit its use. These include a slow conversion of numbers from modular to positional representation; the complexity of comparing numbers in modular representation; the difficulty in performing the division operation; and the difficulty of determining the presence of an overflow. The use of modular arithmetic is justified if there are fast algorithms for calculating a number from a set of remainders. This article describes a fast algorithm for converting numbers from modular representation to positional representation based on a geometric approach. The review is carried out for the case of a comparison system with two modules. It is also shown that as a result of increasing numbers in positional calculus, they successively change in a spiral on the surface of a two-dimensional torus. Based on this approach, a fast algorithm for comparing numbers and an algorithm for detecting an overflow during addition and multiplication of numbers in modular representation were developed. Consideration for the multidimensional case is possible when analyzing a multidimensional torus and studying the behavior of the turns on its surface.

Efficient Communication Protocols for Non DHT-based Pyramid Tree P2P Architecture

In this paper, we have considered a recently reported 2-layer non-DHT-based structured P2P network. Residue Class based on modular arithmetic has been used to realize the overlay topology. At the heart of the architecture (layer-1), there exists a tree like structure, known as pyramid tree. It is not a conventional tree. A node i in this tree represents the cluster-head of a cluster of peers which are interested in a particular resource of type Ri (i.e. peers with a common interest). The cluster-head is the first among these peers to join the system. Root of the tree is assumed to be at level 1. Such a tree is a complete one if at each level j, there are j number of nodes. It is an incomplete one if only at its leaf level, say k, there are less than k number of nodes. Layer 2 consists of the different clusters. The network has some unique structural properties, e.g. each cluster has a diameter of only 1 overlay hop and the diameter of the network is just (2+2d); d being the number of levels of the layer-1 pyramid tree and d depends only on the number of distinct resources. Therefore, the diameter of the network is independent of the number of peers in the whole network. In the present work, we have used some such properties to design low latency intra and inter cluster data lookup protocols. Our choice of considering non-DHT and interest-based overlay networks is justified by the following facts: 1) intra-cluster data lookup protocol has constant complexity and complexity of inter-cluster data lookup is O(d) if tree traversal is used and 2) search latency is independent of the total number of peers present in the overlay network unlike any structured DHT-based network (as a matter fact unlike any existing P2P network, structured or unstructured). Experimental results as well show superiority of the proposed protocols to some noted structured networks from the viewpoints of search latency and complexity involved in it. In addition, we have presented in detail the process of handling churns and proposed a simple yet very effective technique related to cluster partitioning, which, in turn, helps in reducing the number of messages required to be exchanged to handle churns.

The Mystery of the Dancing Men

In this paper I describe an activity based on a 1903 Sherlock Holmes murder mystery, in which a substitution cipher is used to encrypt secret messages. The story provides a fun and interesting way to talk about frequency analysis, and can be used as a segue into mathematical constructs such as modular arithmetic and computation. The activity is accessible to ages twelve and above, and has been successfully used in mathematics outreach and popularization efforts as well as in general education and mathematics courses.