A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM

We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1-o(1) that takes O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW PRAM parallel model of computation. We believe this to be the first randomized sublinear time algorithm on the EREW PRAM for this problem

The Accelerated Euclidean Algorithm

We present a new GCD algorithm of two integers or polynomials. The algorithm is iterative and its time complexity is still $O(n \\log^2 n ~\\log \\log n)$ for $n$-bit inputs

A New Modular Division Algorithm and Applications

12 pagesInternational audienceThe present paper proposes a new parallel algorithm for the modular division $u/v\bmod \beta^s$, where $u,\; v,\; \beta$ and $s$ are positive integers $(\beta\ge 2)$. The algorithm combines the classical add-and-shift multiplication scheme with a new propagation carry technique. This ''Pen and Paper Inverse'' ({\em PPI}) algorithm, is better suited for systolic parallelization in a ''least-significant digit first'' pipelined manner. Although it is equivalent to Jebelean's modular division algorithm~\cite{jeb2} in terms of performance (time complexity, work, efficiency), the linear parallelization of the {\em PPI} algorithm improves on the latter when the input size is large. The parallelized versions of the {\em PPI} algorithm leads to various applications, such as the exact division and the digit modulus operation (dmod) of two long integers. It is also applied to the determination of the periods of rational numbers as well as their $p$-adic expansion in any radix $\beta \ge 2$

Worst--Case Analysis of Weber's Algorithm

11 pagesInternational audienceRecently, Ken Weber introduced an algorithm for finding the $(a,b)$-pairs satisfying $au+bv\equiv 0\pmod{k}$, with $0<|a|,|b|<\sqrt{k}$, where $(u,k)$ and $(v,k)$ are coprime. It is based on Sorenson's and Jebelean's ''$k$-ary reduction'' algorithms. We provide a formula for $N(k)$, the maximal number of iterations in the loop of Weber's GCD algorithm

Improvements on the accelerated integer GCD algorithm

6 pagesInternational audienceThe present paper analyses and presents several improvements to the algorithm for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of Weber's ''Accelerated integer GCD algorithm'' is \cO\l(\log_\phi(k)^2\r), we show that the worst-case number of iterations of the while loop is exactly \tfrac 12 \l\lfloor \log_{\phi}(k)\r\rfloor, where \phi := \tfrac 12 \l(1+\sqrt{5}\r).\par We suggest improvements on the average complexity of the latter algorithm and also present two new faster residual algorithms: the sequential and the parallel one. A lower bound on the probability of avoiding the while loop in our parallel residual algorithm is also given

When Two-Layer Federated Learning and Mean-Field Game Meet 5G and Beyond Security: Cooperative Defense Systems for 5G and Beyond Network Slicing

Cyber security for 5G and Beyond (5GB) network slicing is drawing much attention due to the increase of complex and dangerous cyber-attacks that could target the critical components of network slicing, such as radio access and core network. This paper proposes a new cyber defense approach based on two-layer Federated Learning (FL) to protect 5GB network slicing from the most dangerous network attacks and a mean-field game to safeguard the FL-enabled defense system from poisoning attacks. Our proposed distributed defense systems cooperate, intending to detect internal and external attacks targeting the critical components of 5GB network slicing and detecting infected parts in the 5GB defense system. Our experimental results show that our cooperative defense systems exhibit high accuracy detection rates against network attacks, namely (distributed) denial of service and botnets while being robust against poisoning attacks and requiring a few overheads generated by defense systems. To the best of our knowledge, we are the first to propose lightweight and accurate cooperative defense systems based on two-layer FL and non-cooperative games to enhance security against attackers in 5GB network slicing

RITA: RIsk-aware Trust-based Architecture for collaborative multi-hop vehicular communications

This is the pre-peer reviewed version of the following article: Kerrache, C. A., Calafate, C. T., Lagraa, N., Cano, J. C., & Manzoni, P. (2016). RITA: RIsk‐aware Trust‐based Architecture for collaborative multi‐hop vehicular communications. Security and Communication Networks, 9(17), 4428-4442, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/sec.1618/abstractTrust establishment over vehicular networks can enhance the security against probable insider attackers. Regrettably, existing solutions assume that the attackers have always a dishonest behavior that remains stable over time. This assumption may be misleading, as the attacker can behave intelligently to avoid being detected. In this paper, we propose a novel solution that combines trust establishment and a risk estimation concerning behavior changes. Our proposal, called risk-aware trust-based architecture, evaluates the trust among vehicles for independent time periods, while the risk estimation computes the behavior variation between smaller, consecutive time periods in order to prevent risks like an intelligent attacker attempting to bypass the security measures deployed. In addition, our proposal works over a collaborative multi-hop broadcast communication technique for both vehicle-to-vehicle and vehicle-to-roadside unit messages in order to ensure an efficient dissemination of both safety and infotainment messages. Simulation results evidence the high efficiency of risk-aware trust-based architecture at enhancing the detection ratios by more than 7% compared with existing solutions, such as T-CLAIDS and AECFV, even in the presence of high ratios of attackers, while offering short end-to-end delays and low packet loss ratios.This work was partially supported by both the Ministerio de Economia y Competitividad, Programa Estatal de Investigacion, Desarrollo e Innovacion Orientada a los Retos de la Sociedad, Proyectos I+D+I 2014, Spain, under Grant TEC2014-52690-R, and the Ministere de l'enseignement superieur et de la recherche scientifique, Programme National Exceptionnel P.N.E 2015/2016, Algeria.Kerrache, CA.; Tavares De Araujo Cesariny Calafate, CM.; Lagraa, N.; Cano Escribá, JC.; Manzoni, P. (2016). RITA: RIsk-aware Trust-based Architecture for collaborative multi-hop vehicular communications. Security and Communication Networks. 9(17):4428-4442. https://doi.org/10.1002/sec.1618S4428444291

TFDD: A trust-based framework for reliable data delivery and DoS defense in VANETs

[EN] A trust establishment scheme for enhancing inter-vehicular communication and preventing DoS attacks `TFDD¿ is proposed in this paper. Based on a developed intrusion detection module (IDM) and data centric verification, our framework allows preventing DDoS attacks and eliminating misbehaving nodes in a distributed, collaborative and instantaneous manner. In addition, a trusted routing protocol is proposed that, using context-based information such as link stability and trust information, delivers data through the most reliable way. In this study, the simulation results obtained demonstrate the effectiveness of our trust framework at detecting dishonest nodes, as well as malicious messages that are sent by honest or dishonest nodes, after a very low number of message exchanges. Furthermore, colluding attacks are detected in a small period of time, which results in network resources being released immediately after an overload period. We also show that, in a worst-case scenario, our trust-based framework is able to sustain performance levels, and outperforming existing solutions such as T-CLAIDS and AECFV.Kerrache, CA.; Lagraa, N.; Tavares De Araujo Cesariny Calafate, CM.; Lakas, A. (2017). TFDD: A trust-based framework for reliable data delivery and DoS defense in VANETs. Vehicular Communications. 9:254-267. doi:10.1016/j.vehcom.2016.11.010S254267