Let E be an elliptic curve over the finite field Fq, P a point in E(Fq) of order n, and Q a point in the group generated by P. The DISCRETE LOGARITHM PROBLEM on E is to find the number k such that Q = kP. In this paper we reduce the DISCRETE LOGARITHM PROBLEM on E[n] to the DISCRETE LOGARITHM on the group F*q , the multiplicative group of nonzero elements of Fq, in the case where n ï q - 1, using generalized jacobian of E.

In recent years it has been trying that with regard to the question of computational complexity of DISCRETE LOGARITHM more strength and less in the elliptic curve than other hard issues, applications such as elliptic curve cryptography, a blind digital signature method, other methods such as encryption replacement DLP. In this paper, a new blind digital signature scheme based on elliptic curve cryptography system is proposed. The presented scheme, safe and efficient manner is able to meet the needs of a digital signature system meet the blind. Analysis and evaluation of the efficiency of the Protocol presented in comparison with that in the past has been raised, the superiority of the proposed method.

Network location models comprise one of the main categories of location models. These models have various applications in regional and urban planning as well as in transportation, distribution, and energy management. In a network location PROBLEM, nodes represent demand points and candidate locations to locate the facilities. If the links network is unchangeably determined, the PROBLEM will be an FLP (Facility Location PROBLEM). However, if links can be added to the network at a reasonable cost, the PROBLEM will then be a combination of facility location and NDP (Network Design PROBLEM). In previous studies, capacity of facilities was considered to be a constraint while capacity of links was not considered at all. The proposed MIP model considers capacity of facilities and links as decision variables. This approach increases the utilization of facilities and links, and prevents the construction of links and facility locations with low utilization. Furthermore, facility location cost (link construction cost) in the proposed model is supposed to be a function of the associated facility (link) capacity. Computational experiments as well as sensitivity analyses performed indicate the efficiency of the model.

Alī Muḥammad Iṣfahānī (1215/1800–1293/1876), was one of the famous scientific figures of Qajar period who passed the formative years of his scientific and professional life in Iṣfahān, i.e. before being acquainted with the European sciences through the modern institution of Dār al-Funūn. Several scientific works of Iṣfahānī were written in the style of the ancient mathematical treatises like Miftāḥ al-Ḥisāb by Kāshī or ‘Uyūn al-Ḥisāb by Muḥammad Bāghir Yazdī, whereas his other works seem to have been written in the style of the pedagogic books of Dār al-Funūn. The historians have mentioned some titles among Iṣfahānī’s works which unfortunately are not available today and therefore the attribution of some innovations to Iṣfahānī remains ambiguous. One of the innovations attributed to Iṣfahānī is the discovery of the extracting the LOGARITHM of the numbers. This attribution is repeated in some historical Persian sources such as The Diffused Articles by Abu al-Ḥasan Furūghī, The Algebra of Khayyām by Gholamhossein Mosaheb, The History of Islamic Sciences by Jalal al-Din Homaee, etc. Nevertheless, the historian of mathematics, Abu al-Ghasim Ghorbani, believes that the attribution of discovery of the LOGARITHM to Iṣfahānī could not be more than a legend, though after examining some historical facts, Abd al-Hossein Moshafi shows that this attribution is not far from reality. In this article, after introducing this mathematician and mentioning the previous debates about the subject, I will try to shed some light on these ambiguities with the help of the extant documents.

In this paper, there are two competitors in a planar market. The first competitor, called, the leader, opens p new facilities. After that, the second competitor, the follower, reacts to the leader's action and opens r new facilities. The leader and the follower have got some facilities in advance in this market. The optimal locations for leader and follower are chosen among predefined candidate locations. In this paper, it is assumed that the demand is inelastic. Considering the huff model, customers share their demand among all facilities, probabilistically. The leader's objective function is the maximization of its market share after the follower's new facilities entry. The leader and follower PROBLEMs are solved via an exact solution method.

The Multiple Facility Location PROBLEM (MFLP) is to locate certain facilities so as to serve a given set of customers, whose locations and requirements are known. When the locations have to be selected from a given set of candidate locations, the PROBLEM becomes a DISCRETE Multiple Facility Location PROBLEM (DMFLP). In this study, a special case of DMFLP is discussed where multiple facilities that are of different type are to be placed (location decision) and assign customers to these facilities (allocation or assignment). Both cases with and without interactions among new facilities are considered in this PROBLEM. In this paper, efficient lower and upper bounds are used to propose a branch-and-bound scheme as an exact method. Also, new heuristic method is provided for both cases and compared for large-job sizes. Computational results on randomly generated data in comparison with optimal solutions indicate that the heuristic method is accurate and efficient.

Railway scheduling is a complex task of rail operators that involves the generation of a conflict-free train timetable. This paper presents a DISCRETE-event simulation-based optimization approach for solving the train timetabling PROBLEM to minimize total weighted unplanned stop time in a hybrid single and double track railway networks. The designed simulation model is used as a platform for generating feasible conflict-free train timetables. It includes detailed infrastructure information, such as station characteristics, trains running time and praying intervals. The proposed approach has the capability of scheduling trains in large-scale networks subject to the capacity constraints and infrastructure characteristics. In optimization procedure, a path relinking meta-heuristic algorithm is utilized to generate near-optimal train timetables. A case study of Iran railway network is selected for examining the efficiency of the meta-heuristic algorithm. The computational result shows that the proposed approach has the capability of generating near-optimal timetable in real-sized train scheduling PROBLEMs.