Analytical Model and Topology Evaluation of Quantum Secure Communication Network

Due to the intrinsic point-to-point feature of quantum key distribution system, it is inevitable to study and develop the Quantum Secure Communication Network (QSCN) technology to provide security communication service for a large-scale of nodes over a large spatial area. Considering the quality assurance and expense control, building an effective analytical model of QSCN becomes a critical task. In this paper, the first analytical model of QSCN is proposed, which is called flow-based QSCN (F-QSCN) analytical model. In addition, based on the F-QSCN analytical model, the research on QSCN topology evaluation is conducted by proposing a unique QSCN performance indicator, named Information-Theoretic Secure communication bound, and the corresponding linear programming based calculation algorithm. Plentiful experimental results based on the topologies of SECOQC network and Tokyo network validate the effectivity of the proposed analytical model and the performance indicator.


I. INTRODUCTION
W ITH the rapid development and increasing applicability of quantum key distribution (QKD) technology [1]- [4], the intrinsic feature of point-to-point (P2P) [5] has become one of the major bottlenecks which limits its application scale.In order to overcome the limitation of node quantity [6] and communication distance, it is an inevitable development trend to construct the Quantum Secure Communication Network (QSCN) with multiple QKD devices.The QSCN in this paper is defined as a network that provides secure communication service utilizing the keys generated by QKD devices [7].In order to explore the physical feasibility of QSCN networking, many practical QSCNs [8]- [13] have been constructed in recent years.The node quantity of the existing QSCNs has expanded from 6 nodes [10], [14] to 56 nodes [15], and the communication distance has extended from 19.6 kilometers [16] to 2000 kilometers [15].With the growing coverage and complexity of QSCNs, effective modeling become crucial for the functional verification, quality assurance, expense control, cycle shorten, etc. [17], [18].
Simulation model and analytical model are the two major approaches of network modeling.Unlike the traditional communication networks, the relevant research of QSCN has not drawn many attentions [7], [19]- [22].In 2017, Mehic et.al. designed a QSCN simulation model, QKDNetSim [22], based on the classical Network Simulator-version 3 (NS-3) [23] to evaluate and validate the network solution at a low Q.Li, Y. Wang, H. Mao, J. Yao and Q. Han are with the School of Computer Science and Technology, Harbin Institute of Technology, Harbin, China (email: qiongli@hit.edu.cn).
cost.Although QKDNetSim can simulate the key generation process and secure communication process, it could not reflect the practical performance of a QSCN well due to their neglect of the actual key generation capability and volatile communication demand of the QSCN.In order to reflect the practical network state, we designed a practical QSCN simulation model in our previous work [7].In the model, the actual P2P key generation capability was modeled by the GLLP theory and the volatile end-to-end (E2E) communication demand was modeled by the poisson stochastic process.Although our previous work has enhanced the accuracy of simulation to a great extent, the inherent shortcomings of simulation approach still exist, such as the empirical results, the impossible global optimal solution, and so on.
In contrast, as the general mathematical abstraction of the QSCN, an analytical model makes it possible to evaluate the performance of QSCN theoretically and obtain the global optimal solution, etc.During the modeling of traditional communication network, the problems of congestion control, congestion prevention, queue schedule, and etc. are focused.While the study results of our simulation model [7] demonstrate that the performance of a practical QSCN has neglectable relationship with those problems, but mainly depends on how its key generation capability satisfies the communication demand.Accordingly, we are motivated to study the analytical model of QSCN and its applications concentrating on this characteristic.To the best of our knowledge, this is the first time that the analytical model of QSCN is proposed.
• In this paper, the flow-based QSCN (F-QSCN) analytical model is presented.In the model, a QSCN is abstracted as a graph G = (V, E, F), with the node set V , the edge set E and the QSCN flow set F. According to the analysis of QSCN characteristics, the detailed attributes of node and edge are designed.What is more, the QSCN flow is defined referring to the generic traffic flow [24], [25], which is a unique component of a QSCN comparing to a classical network.• Based on the F-QSCN analytical model, the research on QSCN topology evaluation is conducted by proposing the indicator, Information-Theoretic Secure (ITS) communication bound, and the corresponding linear programming based calculation algorithm.The indicator is defined as the theoretically optimal quantitative performance of the QSCN topology [26].The calculation of ITS communication bound is inspired by the linear programming algorithm.
• In order to verify the validity and necessity of the proposed analytical model and performance indicator, a typical topology planning task based on the existing topologies of SECOQC network [16] and Tokyo network [14] is designed and analyzed.The experimental results demonstrate the advantages of F-QSCN analytical model and ITS communication bound.The remaining parts of this paper are organized as follows.In Section II, some related works are introduced.In Section III, the F-QSCN analytical model is presented in details.Based on the F-QSCN analytical model, a unique QSCN performance indicator and the corresponding linear programming based calculation algorithm are proposed in Section IV.In Section V, the experiments of topology evaluation based on the F-QSCN analytical model are designed, and the results are analyzed.In Section VI, some conclusions are drawn.

II. RELATED WORKS
In this section, some related works are reviewed and analyzed.Except for the construction modes, application modes and architecture models of QSCN, the generic MFP, which is usually used for the task allocation, is introduced as one of the theoretical basis of F-QSCN analytical model.

A. The Construction Modes of QSCN
The construction modes used in the existing QSCNs are mainly divided into three categories: optical switching, quantum relay and trusted relay [27]- [30].Since the optical switching device cannot break the scale limitation [31] and the core technique of quantum relay is still far from mature [32], the trusted relay is the most common construction mode at present.

B. The Application Modes of QSCN
The application modes used in the existing QSCNs mainly include key-by-key (also called as key relay) [21] and databy-data (also called as hop-by-hop) [22], where the main difference is how the communication is established.The keyby-key mode, used in the Tokyo network [14], can retain the classical network protocol as much as possible.However, in this mode, the number of key pools configured for each communication node is proportional to the number of potential communication parties.It leads to a huge memory pressure which is intolerable in a large-scale QSCN.The number of key pools for each node is only related to the degree of the node [33] in the data-by-data mode, which was used in the SECOQC network [34].It can greatly reduce the memory pressure and then enhance the availability of a large-scale QSCN.Therefore, the data-by-data is the more appropriate application mode at present.

C. The Architecture Models of QSCN
Compared to the traditional communication network, the main difference of QSCN is that its secure communication process between the E2E communication parties needs to consume the quantum keys generated by the P2P links.Therefore, the two-layers architecture model of QSCN is proposed in our previous work [7], which is shown in Fig. 1.In the model, the E2E secure communication and P2P key generation proceed in the classical layer and the quantum layer, respectively.
Because the P2P key generation capability of QKD device is extremely limited by the length of quantum channel [35], [36] and is markedly lower than the capacity of the classical channel [37], the performance of QSCN is determined by the matching degree between the communication demand and key generation capability.Referring to our previous work [7], the P2P key generation capability of quantum layer can be obtained according to the common calculation method for the secure key rate of QKD device, such as the GLLP theory [38], [39] and the universal composable framework [40].Assuming that the double decoy state protocol is adopted in QSCN, considering the use of Chernoff bound [41] to estimate the finite code length effect [42], [43], the key generation capability can be calculated as (1), where R L represents the lower bound of the key generation capability for a photon, calculated as (2), The specific description of the relevant parameters in (2) can be found in our previous work [7].

D. The Generic Maximum-Flow Problem
Let G = (V, E) be a directed graph with node set V and edge set E. The graph G = (V, E) is a flow network [44] if it has two distinguished nodes, a source s ∈ V , a sink t ∈ V , and a positive real-valued capacity c (u, v) for each edge (u, v) ∈ E. Definition 1: A traffic flow f on G is a nonnegative function, ranging over all edges (u, v) ∈ E, satisfying the following constraints [45]: The value of traffic flow, is the total difference between the flows into the sink t and the flows out of it [46], i.e., The MFP aims to compute the maximum value of [[ f ]] for a given network, and it is commonly discussed in the fields of the task assignment, the logistics network, the urban planning, and so on.
With regard to a communication network, there are usually multiple communication pairs concurrently, in the form of calls or connections [47].Therefore, the performance evaluation of a communication network is much more complicated than the MFP solving.The classical solving algorithms for MFP, such as Ford-Fulkerson [48] and Edmonds-Karp [49], cannot be directly applied.

III. FLOW-BASED ANALYTICAL MODEL
With the increase of the coverage and complexity of existing QSCNs, it is urgent to design an effective model for the functional verification, quality assurance, expense control, cycle shorten, etc.In this section, a flow-based analytical model is proposed.The "flow" does not mean the generic traffic flow, but the QSCN flow, which will be discussed in III-C.
By abstracting the communication party and trusted relay as node, communication link as edge, and the amount of traffic as QSCN flow, the definition of QSCN is given below.Definition 2: A QSCN is modeled as a graph G = (V, E, F), where V , E and F are the sets of nodes, edges, and QSCN flows respectively.

A. Node Attributes
As a communication network, the most important task of QSCN is to satisfy the communication demand between node pairs.The concept of connection is used to mathematically describe the communication demand.
indicates the communication demand between the node pair (s i ,t j ) [50], where s i is a source and t j is a sink.Let K = (s i ,t j ) |s i ∈ V,t j ∈ V denote the all desired connections in the QSCN.In general, the number of keys consumed by the communication demand is determined by the communication demand and the key consumption ratio.The node attributes are illustrated in Table I.
Communication demand d (s i ,t j ) means the average communication rate required by the connection (s i ,t j ).Moreover, the communication demand of the node s i is denoted by Key consumption ratio β (s i ,t j ) means the ratio of the key length to the plaintext length in the adopted encryption algorithm.In particular, when the value of β (s i ,t j ) is 1, it means that the one-time-pad (OTP) algorithm [51] is adopted to achieve the ITS secure communication.When the value of β (s i ,t j ) is 0, it means that the adopted encryption algorithm does not require the keys generated by QKD devices.Accordingly, the key consumption ratio of the node s i is denoted by

B. Edge Attributes
For a given channel, since there always exist upstream and downstream communication sharing the channel bandwidth [52], the edge in the QSCN is considered to be undirected.The undirected edge formed by connecting node u m ∈ V and The main peculiarity of QSCN lies on the fact that the key generation process requires the participation of quantum channel and the key generation rate is very limited by the length of quantum channel.In order to mathematically describe this characteristic, several important attributes of edge are extracted.Their symbol representations and value ranges are shown in Table II.Classical channel capacity c (u m , v n ) represents the capability of classical channel to transmit information.In particular, when the c (u m , v n ) is 0, there is not a classical channel on the edge (u m , v n ), resulting in the infeasibility of the secure communication process.Meanwhile, because the classical channel is indispensable to transmit supplementary information in the key generation process [53], the key generation process also cannot proceed on this edge.
Key generation capability r (u m , v n ) is related to the parameters of the QKD device configured on the edge (u m , v n ).Suppose DVQKD devices are configured, according to (1), the key generation capability of the edge (u m , v n ) is calculated as (6),

C. Flow Conditions
Although the concept of traffic flow is referred in this work, the flow in QSCN has many unique features due to the obvious difference between QSCN and generic flow network.For example, there exist many connections, the edge is undirected and owns two types of capacities.Definition 4: In the QSCN G = (V, E, F), a QSCN flow f ∈ F is a nonnegative function ranging over all connections (s i ,t j ) ∈ K and all edges (u m , v n ) ∈ E, which is represented by the symbol f (s i ,t j , u m , v n ).
Obviously, the QSCN flow set F can be rewritten as Since the secure transmission process is organized in packets, the value of f (s i ,t j , u m , v n ) must be the integer multiples of the packet size P, which is given as (7), where N is the set of natural numbers.
In particular, when the value of f (s i ,t j , u m , v n ) is 0, it means that there is no flow of the connection (s i ,t j ) on the edge (u m , v n ).In addition, due to the secure communication process is directed, f (s i ,t j , u m , v n ) is considered to be a directed flow.Therefore, f (s i ,t j , u m , v n ) and f (s i ,t j , v n , u m ) are different.The special conditions of QSCN flow are analyzed as below.

(i) Capacity constraint
-For all (u m , v n ) ∈ E, the total flow on the edge (u m , v n ) and its reverse edge (v n , u m ) must be non-negative and less than or equal to its classical channel capacity.
In addition, as an undirected graph, classical channel capacity is shared by upstream and downstream flows.So (8) should be satisfied.
-For all (u m , v n ) ∈ E, the total key consumption on the edge (u m , v n ) and its reverse edge (v n , u m ) must be non-negative and less than or equal to its key generation capability.Considering the key consumption ratio β (s i ,t j ), the relationship between the total flow on the edge (u m , v n ) and its reverse edge (ii) Flow conservation -For all connections (s i ,t j ) ∈ K and all of the non-source and non-sink nodes u m ∈ V − s i ,t j , the total flows into the node u m must equal to the total flows out of it, i.e., As the theoretical foundation of topology evaluation and design, routing protocol evaluation and design, QKD devices selection, expense control, the F-QSCN analytical model can not only be used for the construction of a new QSCN, but also for the optimization of the existing QSCNs.

IV. F-QSCN BASED TOPOLOGY EVALUATION
To construct a high performance QSCN, designing a precise topology evaluation scheme is one of the most important tasks.Based on the F-QSCN analytical model, the research on QSCN topology evaluation is conducted.Firstly, the ITS communication bound indicator is designed to describe the quality of QSCN topology mathematically.In addition, a linear programming based calculation algorithm is proposed to obtain the quantitative quality.

A. The Description of Topology Quality
For the sake of eliminating the influence of encryption algorithm on topology evaluation, the OTP algorithm [51], which can provide ITS communication service, is adopted in this section.In other words, for all (s i ,t j ) ∈ K, the value of β (s i ,t j ) is set to 1.The quality of QSCN topology is measured by the proposed performance indicator, ITS communication bound.Definition 5: For a QSCN with the given topology, the ITS communication bound is defined as the theoretically optimal flow assignment to achieve the maximal satisfaction degree for all connections in K.
Similar to the traffic flow, the actual value of a QSCN flow f of a given connection (s i ,t j ), [[ f (s i ,t j )]], is the total difference between the flows into the sink t j and the flows out of it, which is represented as (11), (11) Let symbol M (s i ,t j ) represent the satisfaction degree for the given connection (s i ,t j ), which means the ratio of the actual value of QSCN flow to the communication demand, i.e., M (s i ,t j ) ≥ 1 means that the communication demand is satisfied, otherwise it is not satisfied.Accordingly, the ITS communication bound B is defined as (13a), where ρ is the worst satisfaction degree of all connections.
Obviously, the communication demands of all connections are satisfied only when the value of B is greater than 1.The larger value of B means the higher satisfaction degree.Even more noteworthy is that, the gap between the simulation max x performance of a QSCN with specific network protocols and the ITS communication bound of this QSCN can be used to evaluate the performance of the network protocols.

B. The Calculation of Topology Quality
To calculate the indictor B, it is necessary to explore the optimal assignment of the QSCN flows, which is defined as multi-connection flow problem (MCFP) in this paper.Although the MCFP seems to be a combination of several MFPs, the solutions of MFP fail to work due to the interaction among multiple MFPs [54].
In ( 14), ) ∈ E are the set of decision variables.The formulation is very similar to that of the linear programming problem.However, due to the issues of non-linear objective function and non-standard data type of the decision variables, the MCFP is not a standard linear programming problem, leading to the solving difficulty.In order to transform this problem into a standard linear programming, the original decision variable f (s i ,t j , u m , v n ) is converted into a new variable x (s i ,t j , u m , v n ) by (15), Therefore, the X = x (s i ,t j , u m , v n ) | (s i ,t j ) ∈ K, (u m , v n ) ∈ E becomes the new set of decision variables.In addition, the original objective function is replace by a new objective function ρ, by adding the ρ as a new decision variable and adding two constraint conditions of ρ, which are shown as (16), where R + 0 is the set of nonnegative real numbers.According to the above operations, the MCFP is transformed into an equivalent standard mixed integer linear programming problem (MILP), which is formulated as (17).
In order to solve this MILP, a linear programming solver [55] Gurobi [56], is adopted.In the formulation, r (u m , v n ) represents the key generation capability of QKD device.It is worth to mention that all kinds of existing QKD devices can be adopted in the QSCN, the corresponding topology quality will be obtained just by changing the calculation of r (u m , v n ).

A. Experiment Design
In order to fully verify the validity and necessity of the proposed analytical model and performance indicator, two existing QSCN topologies, SECOQC network and Tokyo network, shown in Fig. 2 and Fig. 3, are selected in the experiment of topology evaluation.Fig. 2. Topology of the SECOQC network [7] In addition, in order to compare with the simulation performance of the literature [7], the same packet size P = 500 byte   III are adopted in the experiment.Moreover, the same communication scenario is considered, where the communication demands between any two different nodes are set to be the same.That is to say Considering the classical optical fiber communication technology has reached a quite high-level [57], [58], the classical channel capacities of all edges are set as 100 Mbps, i.e.,

B. Topology Evaluation
In the network topology planning, it is a typical task to find out how to enhance the network performance effectively by adding just one device to the existing topology.In the context of QSCN, the equivalent task is to find out how to enhance the QSCN performance effectively by adding a QKD device to an existing QSCN topology.When the new QKD device is placed, a modified topology is actually formed.
As we all know, the key generation process relies on the optical fiber.Therefore, the QKD device can function only by placed on the existing edge.In addition, adding a QKD device to the edge (u m , v n ) means the increase of the key generation capability of this edge and its reverse edge, r (u m , v n ) and r (v n , u m ).
In order to find out the best placement scheme, a QKD device is placed at all possible edges of SECOQC network respectively to form 8 modified topologies.To quantitatively evaluate different placement schemes, the ITS communication bound of the original SECOQC topology and 8 modified SECOQC topologies are calculated, as shown in Table IV.
As seen from the Table IV, only when the QKD device is placed on the edge e 1 , the ITS communication bound is increased and the communication demand is switched from unsatisfied to satisfied.That is to say, the edge e 1 is the bottleneck edge in the topology of SECOQC network, which is consistent with the simulation results in the literature [7].From the topology in Fig. 2, it can be seen that the length of the edge e 1 is 85 kilometers, which is markedly longer than other edges, leading to its poor key generation capability.However, as a "bridg" [59] in the topology, the traffic of edge e 1 is heavy.Therefore, the validity of the F-QSCN analytical model and the ITS communication bound indicator are verified.
Considering the conclusion of e 1 is located as the bottleneck edge in the topology of SECOQC network seems a trivial solution, another experiment based on the topology of Tokyo network is designed to further verify the necessity of the proposed analytical model and indicator.For the same topology planning task, the results of ITS communication bounds of the original Tokyo topology and 6 possible modified Tokyo topologies are calculated as shown in Table V.As seen from the Table V, when the QKD device is placed on the edge e 2 or the edge e 4 , the ITS communication bound is improved.In addition, only when the QKD device is placed on the edge e 4 , the communication demand can be satisfied.This conclusion cannot be intuitively inferred or directly obtained through simulation.The necessity of the F-QSCN analytical model and the ITS communication bound indicator are verified.

VI. CONCLUSIONS
In this paper, the analytical model of QSCN is proposed for the first time.The major contributes include: (I) The F-QSCN analytical model is proposed by modeling a QSCN as a graph with nodes, edges and QSCN flows; (II) Based on the F-QSCN analytical model, a unique QSCN performance indicator is proposed and the corresponding linear programming based calculation algorithm is designed; (III) The validity and necessity of the proposed analytical model and the performance indicator are verified through the subtly designed experiments on typical topology planning task.We will continue to study the topology design, the routing protocol design, and etc. of QSCN in our future work.Through this study, we try to explore a new possibility on the research of QSCN and promote the developing process of QSCN technology.

TABLE I ATTRIBUTES
OF NODE s iAttributes Symbol ValueCommunication demand d s i ,t j [0, +∞)

TABLE IV PERFORMANCE
COMPARISON OF THE ORIGINAL SECOQC TOPOLOGY AND MODIFIED SECOQC TOPOLOGIES (d = 50Kbps)

TABLE V PERFORMANCE
COMPARISON OF THE ORIGINAL TOKYO TOPOLOGY AND MODIFIED TOKYO TOPOLOGIES (d = 500Kbps)