# American Institute of Mathematical Sciences

eISSN:
2577-8838

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## Mathematical Foundations of Computing

November 2018 , Volume 1 , Issue 4

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2018, 1(4): 311-330 doi: 10.3934/mfc.2018015 +[Abstract](380) +[HTML](335) +[PDF](994.72KB)
Abstract:

Privacy in online applications has drawn tremendous attention in recent years. With the development of cloud-based applications, protecting users' privacy while guaranteeing the expected service from the server has become a significant issue. This paper surveyed the most popular cryptographic algorithms in privacy-preserving online applications to provide a tutorial-like introduction to researchers in this area. Specifically, this paper focuses on introduction to homomorphic encryption, secret sharing, secure multi-party computation and zero-knowledge proof.

2018, 1(4): 331-348 doi: 10.3934/mfc.2018016 +[Abstract](336) +[HTML](216) +[PDF](1992.06KB)
Abstract:

In the last two decades, a lot of scientific fields have experienced a huge growth in data volume and data complexity, which brings data miners lots of opportunities, as well as many challenges. With the advent of the era of big data, applying data mining techniques on assembling data from multiple parties (or sources) has become a leading trend. However, those data mining tasks may divulge individuals' privacy, which leads to the increased concerns in privacy preserving. In this work, a Privacy Preserving feature selection method (PPFS-IFW) and Multiclass Classification method (PPM2C) are proposed. Experiments had been conducted to validate the performance of the proposed approaches. Both PPFS-IFW and PPM2C were tested on six benchmark datasets. The testing results demonstrate PPFS-IFW's capability in enhancing the classification performance at the level of accuracy by selection informative features. PPFS-IFW can not only preserve private information but also outperform some other state-of-the-art feature selection approaches. Experimental results also show that the proposed PPM2C method is workable and stable. Particularly, It reduces the risk of over-fitting when compared with the regular Support Vector Machine. In the meantime, by employing the Secure Sum Protocol to encrypt data at the bottom layer, users' privacy is preserved.

2018, 1(4): 349-368 doi: 10.3934/mfc.2018017 +[Abstract](372) +[HTML](207) +[PDF](649.23KB)
Abstract:

Firefly and cuckoo search algorithms are two of the most widely used nature-inspired algorithms due to their simplicity and inexpensive computational cost when they applied to solve a wide range of problems. In this article, a new hybrid algorithm is suggested by combining the firefly algorithm and the cuckoo search algorithm to solve constrained optimization problems (COPs) and real-world engineering optimization problems. The proposed algorithm is called Hybrid FireFly Algorithm and Cuckoo Search (HFFACS) algorithm. In the HFFACS algorithm, a balance between the exploration and the exploitation processes is considered. The main drawback of the firefly algorithm is it is easy to fall into stagnation when the new solution is not better than its previous best solution for several generations. In order to avoid this problem, the cuckoo search with Lèvy flight is invoked to force the firefly algorithm to escape from stagnation and to avoid premature convergence. The proposed algorithm is applied to six benchmark constrained optimization problems and five engineering optimization problems and compared against four algorithms to investigate its performance. The numerical experimental results show the proposed algorithm is a promising algorithm and can obtain the optimal or near optimal solution within a reasonable time.

2018, 1(4): 369-382 doi: 10.3934/mfc.2018018 +[Abstract](310) +[HTML](166) +[PDF](429.11KB)
Abstract:

Despite the extensive study on relay selection in mobile social networks (MSNs), few work has taken both transmission latency (i.e. efficiency) and information leakage probability (i.e. security) into consideration. Therefore we target on designing an efficient and secure relay selection algorithm to enable communication among legitimate users while reducing the information leakage probability to other users. In this paper, we propose a novel mobility model for MSN users considering both the randomness and the sociality of the movements, based on which the social relationship among users, i.e. the meeting probabilities among the users, are predicted. Taken both efficiency and security into consideration, we design a network formation game based relay selection algorithm by defining the payoff functions of the users, designing the game evolving rules, and proving the stability of the formed network structure. Extensive simulation is conducted to validate the performance of the relay selection algorithm by using both synthetic trace and real-world trace. The results show that our algorithm outperforms other algorithms by trading a balance between efficiency and security.

2018, 1(4): 383-392 doi: 10.3934/mfc.2018019 +[Abstract](270) +[HTML](201) +[PDF](320.76KB)
Abstract:

In the one-way trading problem, a seller has some product to be sold to a sequence of buyers in an online fashion, i.e., buyers come one after another. Each buyer has the accepted unit price which is known to the seller on his arrival. To maximize the total revenue, the seller has to carefully decide the amount of products to be sold to each buyer at the then-prevailing prices. In this paper, we study the unbounded one-way trading, i.e., the highest unit price among all buyers is positive and unbounded. We assume that the highest prices of buyers follow some distribution with monotone hazard rate, which is a well-adopted assumption. We investigate two variants, (1) the distribution is on the highest price among all buyers, and (2) the distribution of the highest price of each buyer is independent and identically distributed. To measure the performance of the algorithms, the expected competitive ratios, \begin{document}$\mathrm{E}[OPT]/\mathrm{E}[ALG]$\end{document} and \begin{document}$\mathrm{E}[OPT/ALG]$\end{document}, are considered. If the distributions satisfy the monotone hazard rate, for both of the above two variants, constant-competitive algorithms can be achieved.

2018, 1(4): 393-414 doi: 10.3934/mfc.2018020 +[Abstract](1033) +[HTML](321) +[PDF](902.32KB)
Abstract:

The past three years have seen the rapid increase of Bitcoin difficulty, which has led to a substantial variance in solo mining. As a result, miners tend to join a large open pool to get a more stable reward. Nowadays, mining pools take up over 98% of Bitcoins total computation power. In a sense, this is a manifestation of Bitcoin that tends to be centralized. Thus, researchers have shown an increased interest in pool mining payoff and security. The purpose of this paper is to review and summarize recent research in Bitcoin pool mining system. We first introduce several common reward distribution schemes, and analyze their advantages and disadvantages with some improvement mechanisms; In the second section, to address pool security problems, we examined the practical utility of some existing and potential attack strategies. To study those malicious attack in details, several defense methods are collected. Finally, we make an outlook on Bitcoin future.