VLEO-LEO satellite network quantum key distribution method based on MILP

Abstract

The invention relates to the technical field of satellite communication, in particular to a VLEO-LEO satellite network quantum key distribution method based on MILP. The method comprises the following steps: S1, establishing a satellite-ground link model containing atmospheric attenuation characteristics of a VLEO satellite and an LEO satellite, and analyzing a coupling relationship between a key generation rate and link stability; s2, constructing an MILP-ARA algorithm based on time slot division, realizing optimal path selection under multiple service priorities in the discretization topology by introducing a flow conservation constraint, a key capacity constraint and an end-to-end time delay constraint, and dynamically balancing the link quality and the key utilization rate by using an adaptive weight factor; and S3, large-scale constellation simulation verification is passed. According to the VLEO-LEO satellite network quantum key distribution method based on MILP provided by the invention, the heterogeneous constellation cooperative relay model is constructed, and the static topology of time slot division and the dynamic key management mechanism are combined, so that core challenges such as inter-satellite link high dynamics, key resource limitation and link attenuation are solved.

Description

Technical Field

[0001] The present invention relates to the field of satellite communication technology, and in particular to a quantum key distribution method for a VLEO-LEO satellite network based on MILP. Background Art

[0002] The spatial deployment of quantum key distribution (QKD) is essential for building a global quantum security network. The feasibility of QKD on low-Earth orbit (LEO) satellites has been demonstrated through experiments such as the Micius mission. With the explosive growth of near-Earth space traffic, traditional LEO orbits face inherent limitations such as the high number of inter-satellite link hops and high end-to-end latency. Extending the free-space quantum key distribution network to very low Earth orbit (VLEO) can significantly improve the timeliness of key transmission and provide greater bandwidth.

[0003] A heterogeneous constellation architecture consisting of low-Earth orbit and very low-Earth orbit satellites is constructed. LEO satellites provide stable relay services, while VLEO satellites enable low-latency key transmission. The combination of these two can simultaneously meet the multi-dimensional requirements of wide coverage, large capacity, low latency, and high reliability. However, in actual deployment, this quantum network for polar-orbiting satellites faces core challenges such as frequent reconfiguration of inter-satellite links, drastic fluctuations in the key generation rate (KGR) of the satellite-to-ground link, and the strong coupling between resource allocation and path optimization under the concurrency of multiple services. Innovative dynamic key distribution strategies are urgently needed to advance space quantum communications from experimental verification to large-scale application.

[0004] Free-space quantum key distribution (FSKD) technology has gradually evolved from single-satellite verification to large-scale constellation networking research. Early research focused on feasibility verification of single-satellite systems. Between 2008 and 2018, several research teams, including those at the University of Padova in Italy, systematically explored the impact of different orbital altitudes—LEO, MEO, and GEO—on quantum signal transmission. Furthermore, Pan Jianwei's team at the University of Science and Technology of China achieved a significant breakthrough in thousand-kilometer-scale satellite-to-ground quantum key distribution using the Micius satellite, providing crucial experimental support for the subsequent development and application of satellite-to-ground quantum communication technology.

[0005] As single-satellite technology matures, researchers are turning to the exploration of constellation network architecture. [Vergoossen T,Loarte S,Bedington R,et al.Modelling of satellite constellations for trusted node QKD networks[J].A cta Astronautica,2020,173:164-171]A groundbreaking satellite relay model based on a key buffer was proposed. This solution, through onboard key storage and a dynamic scheduling mechanism, enables low-latency symmetric key distribution between ground stations. To further improve system performance, the academic community has proposed a variety of innovative architectures.

[0006] In terms of heterogeneous architecture, the literature [Huang D,Zhao Y,Yang T,et al.Quantum key distribution over double-layer qua ntum satellite networks[J].IEEE Access,2020,8:16087-16098] A dual-layer QKD network architecture combining GEO satellites and LEO satellites is proposed, and a new joint GEO / LEO routing and key distribution algorithm is proposed for the new architecture.

[0007] In terms of hybrid QKD systems, the literature [Vu M Q,Le H D,Pham T V,et al.Design of satellite-based fso / qkd system s using geo / leos for multiple wireless users[J].IEEE Photonics Journal,2023,15(4):1-14] For the dual architecture, a global FSO / QKD network based on GEO satellites as key sources and LEO satellites as relay nodes is further proposed. The continuous variable QKD (CV-QKD) protocol is used in combination with a dual threshold / direct detection (DT / DD) receiver. The impact of atmospheric attenuation, optical loss and security attacks on the system is analyzed, and the key rate performance is verified through Monte Carlo simulation.

[0008] At the system optimization level, researchers have proposed a variety of innovative methods. [Grillo M,Dowhuszko AA,Khalighi M A ,et al.Resource allocation in a quantum key distribution network with LEO and GEO trusted-repeaters[C] / / 2021 17th In ternational Symposium on Wireless Communication Systems(ISWCS).IEEE,2021:1-6] Aiming at the resource allocation problem in the QKD network of GEO / LEO satellites, an optimization algorithm is proposed to maximize the minimum key exchange amount and minimize the key consumption between ground stations by modeling the QKD network as a graph and converting it into a linear programming problem. The key exchange amount and resource consumption are simultaneously optimized through the linear programming method.

[0009] literature [De GrossiF,Alberico S,Circi C.Orbit design of satellite quantum key distribution constellation s in different ground stations networks[J].Advances in Space Research,2024,73(11):5446-5463] From the perspective of orbital mechanics, a systematic analysis of the impact of different geographically distributed ground station networks on orbit design was conducted, confirming that multi-orbital plane configuration combined with inter-satellite links (ISL) can significantly improve key distribution capabilities and the average daily key generation volume. Regarding the problem of how to provide keys for remote multicast services, He et al. [He X,Li L,Han D,et al.Routing and secret key assign ment for secure multicast services in quantum satellite networks[J].Journal of Optical Communications and Network ing,2022,14(4):190-203] A quantum satellite network with a special node structure is designed to realize point-to-multipoint key distribution, and a point-to-multipoint satellite relay solution is proposed, which effectively improves the success rate of key distribution for remote multicast services.

[0010] At present, research on quantum key distribution in VLEO orbits is still blank, and the utilization of space resources within the polar orbit altitude range is still in its early stages.

[0011] To this end, a VLEO-LEO satellite network quantum key distribution method based on MILP was designed to provide a technical solution to the above technical problems. Summary of the Invention

[0012] Based on this, it is necessary to provide a VLEO-LEO satellite network quantum key distribution method based on MILP to address the above technical problems, so as to solve the technical problems raised in the above background technology.

[0013] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0014] The VLEO-LEO satellite network quantum key distribution method based on MILP has the following steps:

[0015] S1: Establish a satellite-to-ground link model that includes the atmospheric attenuation characteristics of VLEO and LEO satellites, and analyze the coupling relationship between key generation rate and link stability;

[0016] S2: Build a slot-based MILP-ARA algorithm. By introducing traffic conservation constraints, key capacity constraints, and end-to-end delay constraints, it achieves optimal path selection under multiple service priorities in a discretized topology and dynamically balances link quality and key utilization using adaptive weight factors.

[0017] S3: Verified through large-scale constellation simulation.

[0018] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step S1, the steps are as follows:

[0019] A. Build a QKD network architecture for VLEO-LEO satellite networking;

[0020] B. Calculate the key generation rate based on quantum key distribution using decoy states;

[0021] C. Calculate the attenuation of space-to-Earth and inter-satellite links.

[0022] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step A, the steps are as follows:

[0023] The network consists of LEO satellites and VLEO satellites at very low orbit altitudes;

[0024] Each layer of the satellite network contains p orbital planes, and each orbit has q satellites evenly distributed. l represents the layer where the satellite is located, and p l and q l They represent the number of orbital planes and the number of satellites per orbit in this layer respectively;

[0025] N l =p l q l Indicates the total number of satellites in this layer, Indicates the total number of satellites in all layers before the current layer, and serves as the starting number of this layer. Each satellite is numbered as follows:

[0026]

[0027] Among them, v is the track surface index, u is the track internal index, and l' is used to represent a layer index variable.

[0028] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step B, the steps are as follows:

[0029] Assuming that the phase θ of each signal is random, the number of photons in the signal state follows a Poisson distribution with parameter μ, which represents the intensity of the signal state, then

[0030] The probability that Alice generates a pulse consisting of n photons is expressed as follows:

[0031] P n (μ)=(μ n e -μ ) / n! ;

[0032] The lower limit of the key generation rate is expressed as:

[0033] R bb84 ≥q{-Q μ f(E μ )H2(E μ )+Q1[1-H2(e1)]},;

[0034] Among them, Q μ (Q v ) and E μ (E v ) are the gain and quantum bit error rate (QBER) of the signal (weak decoy) state μ(v), while Q1 and e1 are the gain and error rate of the single-photon state, respectively. q = 1 / 2 is the BB84 protocol efficiency, and f(x) = 1.22 is the two-way error correction efficiency of the cascaded protocol.

[0035] and get:

[0036] H2(x)=-xlog2(x)-(1-x)log2(1-x);

[0037] For the coherent state, the gain and QBER of the signal state are expressed as:

[0038]

[0039] Where Y n =Y0+δ n -Y0δ n ≈Y0+δ n is the probability that Bob's measurement is deterministic when Alice emits a pulse of n photons;

[0040] At the same time, the error rate and attenuation of the n-photon pulse signal are obtained, and the expressions are as follows:

[0041] e n =Y0 / (2Y n ),δ n =1-(1-δ) n ;

[0042] When using the vacuum state plus weak decoy state method, the gain and error rate of the single-photon state verify the lower limit L and upper limit U respectively:

[0043]

[0044] After optimization, we get:

[0045]

[0046] Among them, the background gain Y0 can be calculated by the gain of the vacuum decoy state; and the background bit error rate e0 occurs randomly due to the dark count.

[0047] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step C, the steps are as follows:

[0048] When the optical receiver is located in the far field of the transmitter, the attenuation caused by diffraction in the FSO link is expressed as:

[0049]

[0050] Where λ is the wavelength, D T and D R are the diameters of the transmitting and receiving telescopes, T T and T R are the transmission coefficients of the transmitting and receiving telescopes, L P is the pointing loss due to misalignment between transmitter and receiver;

[0051] The divergence angle of the transmitting telescope is θ T ≈λ / D T , and the additional divergence angle caused by atmospheric turbulence is θ atm=λ / r0, where r0 is the atmospheric coherence constant, r0 can be expressed as:

[0052]

[0053] Where k = 2π / λ is the wave number of light, is the turbulent refractive index structure constant, which describes the atmospheric turbulence intensity, and L is the path length of light transmission;

[0054] When the receiving spot diameter is larger than the aperture of the receiving telescope, part of the transmitted power cannot be collected by the receiver, resulting in energy loss. The total attenuation experienced by the optical wireless link is:

[0055] δ=δ diff ×δ atm ×δ rec ,δ atm =δ abs ×δ scat ×δ turb ;

[0056] Among them, δ rec represents the attenuation due to efficiency loss in the photon detection process, δ atm represents the attenuation in the atmosphere, including the absorption caused by gases and particles in the atmospheric composition δ abs and scattering δ scat , and the atmospheric turbulence effect δ caused by random fluctuations in the refractive index turb .

[0057] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step S2, the steps are as follows:

[0058] 1) Discrete routing selection and dynamic key traffic allocation by encoding integer variables and continuous variables;

[0059] 2) A hybrid optimization architecture based on traffic conservation constraints, key capacity constraints, and end-to-end delay constraints.

[0060] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step 1), discrete routing selection and continuous variable dynamic key flow are encoded by integer variables, and the expression is as follows:

[0061]

[0062] in, For business satisfaction items; is the satisfaction ratio of service k in time slot t; w k Represents the business priority weight; is the path cost term;

[0063] is the adaptive weight coefficient of the path cost, where α0 is the baseline weight of the path cost term;

[0064] is the comprehensive cost coefficient of the link, is the attenuation cost, 0.5e u is the error cost, For switching penalties;

[0065] is the resource utilization item, is the adaptive weight coefficient of key utilization, is to maximize the utilization of key resources in the entire network, where β0 is the benchmark weight of the resource utilization term.

[0066] As a preferred embodiment of the MILP-based VLEO-LEO satellite network quantum key distribution method provided by the present invention, in step 2), the steps are as follows:

[0067] The constraint ensures that the transmission of quantum keys in the network complies with the "source-relay-destination" key conservation, which is expressed as follows:

[0068]

[0069] in, represents the total amount of keys for business k sent from node i, represents the total amount of keys for service k received from node i;

[0070] when When represents the source node, the right formula is Indicates the key injection amount of business k; on the contrary, when When represents the destination node, the right formula is Indicates the key extraction amount of business k; when When represents a relay satellite node;

[0071] Generative Capacity With pre-stored keys The mixed supply capacity is expressed as follows:

[0072]

[0073] in, R QKD is the ideal key rate, η atm is the atmospheric channel transmittance, η point is the beam alignment efficiency; the link is not activated when it is unavailable due to obstruction or turbulence, Forced to 0;

[0074] Ensure that the total end-to-end delay of service k does not exceed its maximum tolerance value The expression is as follows:

[0075]

[0076] It can be seen without a doubt that the above-mentioned technical solution of this application can definitely solve the technical problem to be solved by this application.

[0077] At the same time, through the above technical solutions, the present invention has at least the following beneficial effects:

[0078] The MILP-based quantum key distribution method for VLEO-LEO satellite networks provided by the present invention solves core challenges such as high dynamics of inter-satellite links, limited key resources, and link attenuation by constructing a heterogeneous constellation collaborative relay model and combining a static topology with time slot division and a dynamic key management mechanism. As a result, the overall end-to-end delay of the heterogeneous constellation collaborative relay model can reach approximately 55ms under the MILP-ARA algorithm, which is an average reduction of approximately 20ms compared with the Dijkstra and SA-MFP algorithms. Under high load, its key satisfaction rate is improved by approximately 29.35% compared with the SA-MFP algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following is a brief introduction to the drawings required for use in the description of the embodiments. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0080] Figure 1 Schematic diagram of a simplified model of the VLEO-LEO trusted relay satellite network QKD of the present invention;

[0081] Figure 2 This is a schematic diagram of the SDN centralized control network architecture of the present invention;

[0082] Figure 3 Schematic diagram of the quantum satellite QKD implementation method based on trusted relay of the present invention;

[0083] Figure 4 Schematic diagram of key generation rate of different types of quantum links under 24 time slots of the present invention;

[0084] Figure 5 Schematic diagram of a curve showing the key distribution success rate changing over time according to the present invention;

[0085] Figure 6Schematic diagram of average end-to-end delay of each algorithm under different service priorities of the present invention;

[0086] Figure 7 Schematic diagram of the number of path hops of each algorithm under different service priorities of the present invention;

[0087] Figure 8 Schematic diagram showing changes in key satisfaction rates of services of various priorities as the service volume increases;

[0088] Figure 9 Schematic diagram of the curve showing the key utilization rate versus key generation rate in the present invention. DETAILED DESCRIPTION

[0089] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0090] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0091] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features and technical solutions therein may be combined with each other.

[0092] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, it does not require further definition or explanation in subsequent drawings.

[0093] Reference Figure 1-Figure 3 , quantum key distribution method for VLEO-LEO satellite network based on MILP.

[0094] 1. System Model

[0095] 1.1 QKD Network Architecture for VLEO-LEO Satellite Networking

[0096] Figure 1A simplified model of a satellite QKD network combining VLEO and LEO trusted relays is presented. The network consists of LEO satellites and very low-orbiting VLEO satellites, which serve globally distributed ground stations (GSs) and devices such as submarines that lack a terrestrial base station network. Similar to terrestrial fiber-based QKD networks, each node in the satellite QKD network is equipped with a quantum communication channel for key generation and a classical communication channel for transmitting QKD protocol signaling and forwarding encryption keys between nodes without direct connections. Each LEO (VLEO) satellite has two inter-satellite links (ISLs) and two free-space optical (FSO) transmission links for quantum key transmission from inter-satellite nodes to the ground stations. The service area of the entire QKD network is divided into several non-overlapping zones, each of which is managed by a ground station. The ground stations distribute keys to their associated users via the terrestrial fiber network. Each GS is equipped with three free-space optical (FSO) transceivers for connecting to VLEO and LEO satellites.

[0097] Each layer of the satellite network contains p orbital planes, and each orbit has q satellites evenly distributed. l represents the layer where the satellite is located (VLEO=0, LEO=1), and p l and q l Respectively represent the number of orbital planes and the number of satellites per orbit in this layer. l =p l q l Indicates the total number of satellites in this layer, Represents the total number of satellites in all layers before the current layer and serves as the starting number of this layer. The number of each satellite can be expressed as shown in formula (1):

[0098]

[0099] Among them, v is the track surface index, u is the track internal index, and l' is used to represent a layer index variable.

[0100] like Figure 2 As shown in the figure, the QKD network model adopts SDN centralized resource management. The management system can obtain the status information of the entire network in real time, including the key storage status of each node, the satellite network topology and the key generation rate of each FSO link, and use the control link to send the information to the QKD node, thereby calculating the optimal key distribution path to efficiently exchange keys between remote ground stations.

[0101] 1.2 Quantum Key Distribution Based on Trapped States

[0102] In the downlink from space to the ground, QKD transmits the key from the VLEO / LEO satellite to the ground station. The QKD transmitter on the satellite uses weak coherent laser pulses to implement the decoy state BB84 protocol, which effectively resists the photon number division attack. Assume that the BB84 protocol is implemented by combining the vacuum state and the weak decoy state. Alice can prepare and transmit a weak coherent state Assuming that the phase θ of each signal is random, the number of photons in the signal state follows a Poisson distribution with parameter μ, which represents the intensity of the signal state. Under this premise, the probability that Alice generates a pulse containing n photons is:

[0103] P n (μ)=(μ n e -μ ) / n! (2)

[0104] Under this premise, we can assume that Alice can adjust the intensity of each individual pulse and prepare any mixed state with a Poisson-distributed number of photons. In this case, the lower bound on the rate at which the key can be generated is expressed as:

[0105] R bb84 ≥q{-Q μ f(E μ )H2(E μ )+Q1[1-H2(e1)]}, (3)

[0106] In formula (3), Q μ (Q v ) and E μ (E v ) are the gain and quantum bit error rate (QBER) of the signal (weak decoy) state μ(v), while Q1 and e1 are the gain and error rate of the single-photon state, respectively. In addition, q = 1 / 2 is the BB84 protocol efficiency, and f(x) = 1.22 is the two-way error correction efficiency of the cascaded protocol. Among them, we get:

[0107] H2(x)=-xlog2(x)-(1-x)log2(1-x) (4)

[0108] is the binary Shannon entropy. For the coherent state, the gain and QBER of the signal state are given by:

[0109]

[0110] Where Y n =Y0+δ n -Y0δ n ≈Y0+δ n is the probability that Bob's measurement is decisive when Alice emits a pulse of n photons. At the same time, we have:

[0111] e n =Y0 / (2Y n ),δ n =1-(1-δ) n (6)

[0112] is the error rate and attenuation of a pulse signal consisting of n photons.

[0113] Q μ (Q v ) and E μ (E v ) can be estimated directly from equation (5), while the value of Q1 cannot be determined in closed form and needs to be defined based on other gains. [Ma X,Qi B,Zhao Y,et al.Practical decoy state for quantum key distribution[J].Physical Review A—Atomic,Molecular ,and Optical Physics,2005,72(1):012326] , when using the vacuum state plus weak decoy state method, the gain and error rate of the single-photon state verify the lower limit (L) and upper limit (U) respectively:

[0114]

[0115] Among them, E v and Q v It is obtained by replacing v with μ in (5). At the same time, we get:

[0116]

[0117] The background gain Y0 can be calculated from the gain of the vacuum decoy state. The background bit error rate e0 is calculated as e0 = 1 / 2 because dark counts occur randomly. This is because in this case, dark counts cause the detector to trigger randomly, so there is a 50% probability that a photon will trigger the correct detector.

[0118] 1.3 Attenuation of space-to-Earth and inter-satellite links

[0119] The total attenuation experienced by an FSO link is defined as the ratio of the average transmitted power to the received power at the entrance and exit points of the transmitter and receiver telescopes. When , the attenuation caused by diffraction in the FSO link is given by [Pfennigbauer M,Leeb W,Aspelmeyer M,et al.Free-space optical quantum key distribution using intersatellite links[ M].na,2003] gives:

[0120]

[0121] Where λ is the wavelength, D T and D R are the diameters of the transmitting and receiving telescopes, T T and T R are the transmission coefficients of the transmitting and receiving telescopes, LP is the pointing loss due to misalignment between the transmitter and receiver. The divergence angle of the transmitting telescope is θ T ≈λ / D T , and the additional divergence angle caused by atmospheric turbulence is θ atm =λ / r0, where r0 is the atmospheric coherence constant, r0 can be expressed as:

[0122]

[0123] Where k = 2π / λ is the wave number of light, is the turbulent refractive index structure constant, which describes the atmospheric turbulence intensity, and L is the path length of light transmission.

[0124] The attenuation caused by diffraction originates from the natural expansion of the beam during light propagation. When the receiving spot diameter is larger than the aperture of the receiving telescope, part of the transmitted power cannot be collected by the receiver, resulting in energy loss. In addition to the geometric loss mentioned in formula (9), other losses need to be considered when estimating the key generation rate in the quantum channel. The total attenuation experienced by the optical wireless link is:

[0125] δ=δ diff ×δ atm ×δ rec ,δ atm =δ abs ×δ scat ×δ turb (11)

[0126] Among them, δ rec represents the attenuation due to efficiency loss in the photon detection process, δ atm represents the attenuation in the atmosphere, which is caused by absorption by gases and particles in the atmosphere (δ abs ) and scattering (δ sca t), and atmospheric turbulence effects caused by random fluctuations in refractive index (δ turb ).

[0127] 1.4QKD Key Relay Network

[0128] In order to realize that quantum satellites can be used as trusted repeaters to generate keys between remote nodes that do not share a QKD link, Figure 3 The quantum satellite QKD implementation method based on trusted relay is demonstrated. Through quantum channel and classical channel communication, GS1 and QS1 share the key K a , QS1 generates a random key K and shares K with QS2, QS3 and GS2 in turn b , K c and K d In classical channels, QS1 calculation Send to GS1 and calculate Send to QS2. QS2 recovers K through the inverse XOR operation and then calculates Send to QS3. QS3 further calculates It is sent to GS2, and finally GS2 recovers the key K. During the whole process, each relay node only processes the key encrypted by XOR, and this XOR operation consumes the quantum key available in each link.

[0129] The low quantum key generation rate can lead to an insufficient supply of quantum keys when the network is busy and there are a large number of security service requests. To this end, each LEO-to-GS, VLEO-to-GS, and LEO-to-VLEO link must continuously generate quantum keys and store them in a quantum key pool associated with each link in the QKD network. A centralized control system optimizes routing and key distribution, effectively utilizing the limited resources of the QKD network.

[0130] 2. Adaptive Routing Algorithm Based on Mixed Integer Linear Programming (MILP-AKA)

[0131] Refer to Table 1 for an explanation of the compliance and Table 2 for an explanation of the decision variables.

[0132] Table 1: Model symbols

[0133]

[0134] Table 2: Decision variables

[0135]

[0136] 2.1 Satellite Topology Time Slot Division

[0137] Since satellites have strong dynamic characteristics and their topology changes over time, it is difficult to accurately describe the real-time updated satellite network topology. Considering the connection relationship of satellite topology, the satellite network can be divided into a static topology of n time slots. Among them G t is a set of n sub-topologies, is the partitioned n topology.

[0138] Through the set E of all available links in time slot t t , and establish the link key matrix based on the adjacency matrix of the satellite topology The quantum key information of each quantum link used to update and store in time slot t is as shown in formula (12):

[0139]

[0140] 2.2 Objective Function

[0141] In response to the three major challenges faced by VLEO / LEO satellite QKD networks, namely, periodic intersatellite link on-off, attenuated key generation rate, and imbalanced key supply and demand under multiple service requests, this paper proposes a MILP model. The core of this model is to dynamically allocate key traffic by encoding discrete routing with integer variables and dynamic allocation of continuous variables. Its objective function is shown in Equation (13):

[0142]

[0143] Where, It is a business satisfaction item that maximizes the priority-weighted business request satisfaction rate. is the satisfaction ratio of service k in time slot t. k Represents the service priority weight, which prioritizes key services through differentiated weights and automatically downgrades low-priority services when the network is congested. is the path cost term to minimize the total transmission cost of the entire network. is the path cost adaptive weight coefficient, where α0 is the baseline weight of the path cost term. is the comprehensive cost coefficient of the link, is the attenuation cost (diffraction loss, atmospheric loss), 0.5e u is the error cost, For switching penalties. is the resource utilization item, is the adaptive weight coefficient of key utilization, is to maximize the utilization of key resources in the entire network, where β0 is the benchmark weight of the resource utilization term.

[0144] 2.3 Hybrid Optimization Architecture

[0145] (1) Traffic conservation constraint: To ensure that the transmission of quantum keys in the network complies with the "source-relay-destination" key conservation, for each node i and service k, the constraint in equation (14) must be satisfied in time slot t.

[0146]

[0147] in, represents the total amount of keys for business k sent from node i, represents the total amount of keys for service k received from node i. When represents the source node, the right formula is Indicates the key injection amount of business k; on the contrary, when When represents the destination node, the right formula is Indicates the key extraction amount of business k. When represents a relay satellite node.

[0148] (2) Key capacity constraint, in order to limit the sum of key traffic of all services on a single link to not exceed the actual carrying capacity of the link. Unlike traditional communications, quantum key traffic is limited by two independent factors: timely generation capability. With pre-stored keys Take the minimum value of the two to ensure generation capacity With pre-stored keys The mixed supply capacity is shown in formula (15).

[0149]

[0150] Where, R QKD is the ideal key rate, η atm is the atmospheric channel transmittance, η point is the beam alignment efficiency. At the same time, when the link is unavailable due to obstruction or turbulence, the link is not activated. Forced to 0.

[0151] (3) Service path continuity constraint, to ensure the physical feasibility and key distribution continuity of global QKD. In the first inequality, service k can only choose the currently activated link This eliminates invalid links caused by satellite movement or atmospheric obstruction. In the second inequality, the traffic of a single service does not exceed its total demand. And only when The traffic distribution is only run when the key is received, ensuring that each relay node must receive the complete key before forwarding it.

[0152]

[0153] (6) End-to-end delay constraint, ensuring that the total end-to-end delay of service k does not exceed its maximum tolerance value The following constraints need to be made:

[0154]

[0155] (4) Link utilization constraint, in order to quantify the efficiency of link key resource utilization, as shown in formula (18), is the actual amount of keys used, is the maximum available key amount for the link. Indicates that key resources are fully utilized. Indicates idle resources. By prioritizing links with high key generation rates, the load on each link is balanced to avoid local congestion.

[0156]

[0157] (5) Dynamic update constraints of the key pool to describe the time domain evolution of the key pool state. represents the replenishment amount of the key pool in time slot t (existing inventory + newly generated keys), represents the key consumption in time slot t.

[0158]

[0159] 2.4 Implementation of Adaptive Routing Algorithm Based on Mixed Integer Linear Programming

[0160] Refer to Table 3 to further detail the implementation of the MILP-AKD method, presenting its dynamic optimization process in pseudocode. The algorithm uses time slots as units and completes the following core steps within each time step:

[0161] (1) Network status update. Update the topology map G in real time based on the satellite motion model and link attenuation characteristics. t , key pool inventory and the key generation rate

[0162] (2) Mixed integer programming modeling. ① Variable definition: Introducing binary variables Indicates link-service correlation, continuous variable Denotes the key traffic distribution. ②Objective function: maximize the priority-weighted service satisfaction rate while minimizing the path cost and optimizing the key utilization rate (13).

[0163] (3) Constraint integration. ① Flow conservation formula (14) ensures lossless transmission of the key from the source node to the destination node via the relay satellite. ② Capacity constraint formula (15) ensures that the link key flow is subject to the dual constraints of real-time generation capability and the amount of pre-stored keys. ③ Path continuity formula (16) prevents link interruptions caused by satellite movement from affecting business continuity. ④ Link delay constraint formula (17) avoids large communication delays. ⑤ Link utilization formula (18) balances the load of each link to avoid local congestion.

[0164] (4) Dynamic key management: The key pool state is recursively updated through formula (19) to achieve cross-slot resource collaboration.

[0165] (5) Solution and output: Call the MILP solver to generate the optimal routing strategy and update the global key distribution path table.

[0166] Table 3: MILP-ARA algorithm

[0167]

[0168]

[0169] Example 2

[0170] refer to Figure 4-Figure 9 , based on the above embodiment 1, a specific simulation experiment is disclosed.

[0171] 1 Simulation environment and parameter settings

[0172] The simulation experiment of the present invention is carried out on the Ubuntu 20.04 system and the NVIDIA RTX 4090 hardware platform. A QKD constellation model consisting of 120 VLEO and 60 LEO satellites is constructed based on STK11.6. The wavelengths of 850nm (satellite-to-ground link) and 1550nm (inter-satellite link) are used, combined with the BB84 weak coherent state protocol (pulse frequency 100MHz) and the dynamic atmospheric turbulence model, and the simulation verification is carried out using the MILP-ARA algorithm.

[0173] The satellite simulation parameters, FSO link parameters and BB84-based weak coherent state parameters are shown in Tables 4, 5 and 6.

[0174] Table 4: Satellite parameters

[0175]

[0176] Table 5: FSO link parameters

[0177]

[0178] Table 6: Parameters of weak coherent states based on BB84

[0179]

[0180] 2 Analysis of simulation experiment results

[0181] Figure 4 Figure 2 shows the key generation rate trends for five link types over 24 consecutive time slots. Overall, the KGR of each link shows a pattern of first decreasing and then recovering, primarily due to the impact of changes in solar incidence angle and atmospheric turbulence during the diurnal cycle on link performance. The VLEO-VLEO link, due to its intersatellite communication nature, is least affected by external disturbances, maintaining a consistently high key rate between 1.6 and 1.8 kbps. The LEO-LEO link is second, with a slightly longer transmission distance and a slightly lower KGR, but with less fluctuation. In contrast, the satellite-to-ground links (VLEO-GS and LEO-GS) are significantly affected by atmospheric disturbances, with a particular rate dip occurring around the 14th time slot, indicating a significant impact of atmospheric conditions on link attenuation during this period. The LEO-GS link's rate is generally lower than that of the VLEO-GS link, primarily due to its longer transmission distance. As a cross-layer intersatellite link, the VLEO-LEO link has the longest slant range and is significantly affected by attitude changes, resulting in the lowest KGR, dropping to approximately 0.3 kbps.

[0182] In order to verify the effectiveness of the MILP-ARA algorithm, Figure 5 The study demonstrates the temporal evolution of the request success rates of the MILP-ARA algorithm, the adaptive maximum flow algorithm (SA-MFP), and the shortest path algorithm (Dijkstra) in a quantum key distribution network at different time slots. It can be observed that the request success rates of all algorithms show a significant decline between time slots 10 and 18, followed by a gradual recovery. This phenomenon is primarily due to link degradation caused by increased solar illumination and atmospheric turbulence. During this period, the increased solar background light intensity significantly increases the bit error rate of single-photon detectors, disrupting the stable reception of qubits and reducing link availability and key generation efficiency. Comparative results show that despite periods of channel degradation, MILP-ARA maintains a relatively high request success rate throughout the entire time span, with its lowest point (time slot 14) still exceeding that of the other two algorithms, achieving improvements of approximately 13.5% and 7.6% over Dijkstra and SA-MFP, respectively. Its effect mainly comes from the introduction of a joint perception mechanism of link quality factor and service priority under multi-dimensional resource constraints. Unreliable quantum links in the link will not be activated, and link attenuation perception and path fault tolerance optimization are achieved through the dynamic weight coefficient in the objective function.

[0183] Furthermore, MILP-ARA combines key link capacity limitations, and can more reasonably complete link resource allocation and task path selection when facing atmospheric disturbances and link jitter, thereby improving the service success rate of the overall system.

[0184] Figure 6Figure 3 shows the end-to-end delay distribution of different algorithms under various service priorities (high Pri H, medium Pri M, and low Pri L). Overall, all three algorithms exhibit distinct delay stratification, but MILP-ARA demonstrates lower latency for high-priority services (PriH). Specifically, MILP-ARA's average latency at all priority levels is lower than or close to that of the comparison algorithms. In particular, for high-priority services, MILP-ARA's latency distribution is more concentrated, demonstrating its superior delay controllability. The MILP-ARA algorithm introduces a priority weighted scheduling mechanism and an objective function designed to minimize weighted path hop count. This gives high-priority services higher key scheduling priority during resource allocation and comprehensively considers hop count and link status in path selection, preventing services from waiting for extended periods in congested or degraded areas. In contrast, while the Dijkstra algorithm can quickly select the shortest path, it lacks link status and key resource awareness, resulting in high-priority services being blocked or experiencing large latency fluctuations when resources are limited or links are degraded. As a heuristic algorithm, SA-MFP has certain link adaptability, but it lacks global resource optimization in scenarios where multiple priority services are concurrent, resulting in large fluctuations in latency distribution.

[0185] Figure 7 The end-to-end path hop count distribution of the three algorithms for services of different priority levels is presented. Overall, MILP-ARA effectively controls path hop count while ensuring transmission reliability, significantly outperforming SA-MFP and exhibiting lower hop count variance for medium- and high-priority services. Specifically, MILP-ARA achieves lower average hop counts than Dijkstra and SA-MFP for Pri H and Pri M services, respectively, demonstrating its ability to select relatively optimal paths and maintain high transmission efficiency under conditions of high key resource pressure and link degradation. By modeling the joint minimization of hop count and resource cost in the objective function, the algorithm not only considers key resource availability during path selection but also guides paths as close to the lower bound of hop count as possible, effectively avoiding key consumption and delay accumulation caused by redundant hops. While the Dijkstra algorithm has the theoretical advantage of the shortest path in terms of hop count, it lacks resource awareness and is prone to unstable behavior, such as rerouting, when some links are unavailable or resources are scarce. Although the SA-MFP algorithm has certain heuristic search capabilities, its path selection is more inclined to local resource optimization, resulting in the overall path hop control ability being inferior to MILP-ARA.

[0186] Figure 8The key satisfaction rates for different priority services, under the MILP-ARA and SA-MFP algorithms, are shown as the request volume increases from 1,000 to 10,000. Experimental results show that MILP-ARA outperforms SA-MFP across all priority levels, with the highest performance being in high-priority (Pri H) service scenarios. When the request volume reaches 5,000, MILP-ARA achieves a key satisfaction rate of approximately 0.82 for Pri H services, while SA-MFP achieves only 0.65, a 26.1% improvement. Under extremely high load conditions, with the request volume reaching 10,000, MILP-ARA maintains a key satisfaction rate of approximately 0.49, while SA-MFP drops to 0.35. MILP-ARA also demonstrates stronger key satisfaction rates for medium and low priority services: at 8,000 requests, the key satisfaction rates for Pri M and Pri L services are approximately 46.3% and 20.6% higher than SA-MFP, respectively. We can further observe that when the number of service requests ranges from 1000 to 8000, SA-MFP achieves a higher key satisfaction rate at Pri H than MILP-ARA at Pri M and Pri L. However, when the number of requests reaches 9000, it begins to gradually decline compared to MILP-ARA at Pri M. This indicates that the MILP-ARA algorithm is more robust to high loads, and its overall key satisfaction rate is approximately 29.35% higher than that of the SA-MFP algorithm.

[0187] Figure 9 The key utilization (consumption / generation) performance of MILP-ARA and the Dijkstra algorithm under different key generation rate conditions is demonstrated for high attenuation (ATTH), medium attenuation (ATTM), and low attenuation (ATTL) links. MILP-ARA outperforms Dijkstra at all attenuation levels. Its key utilization efficiency under low attenuation conditions steadily surpasses the efficiency threshold (0.8) when the key generation rate exceeds 1.0 kbps, reaching a maximum utilization efficiency of 0.95, while Dijkstra never breaks this efficiency limit. Specifically, under link attenuation (ATTH), MILP-ARA achieves approximately 92.7% resource utilization at a key generation rate of 2.0 kbps, a 23.2% improvement over Dijkstra's 75.3%. At medium and low link attenuation, MILP-ARA's utilization also exceeds Dijkstra by approximately 18.5% and 20.6%, respectively, significantly improving the overall utilization efficiency of link key resources. At the same time, the three curves of MILP-ARA tend to converge in the high generation rate segment, indicating that the algorithm can fully utilize link resources and improve global scheduling efficiency under high key generation capability.

[0188] 3. Summary

[0189] 1. Due to the enormous potential of very low Earth orbit satellites, this paper proposes a quantum key distribution network for VLEO-LEO satellite networks for the first time. To address the frequent changes in inter-satellite links, unstable key generation rates in satellite-to-ground links, and resource competition under multiple concurrent services in polar orbit quantum networks, an adaptive routing algorithm (MILP-ARA) is proposed that combines a heterogeneous constellation collaborative relay mechanism with a mixed integer linear programming (MILP) optimization framework. This algorithm achieves efficient key distribution and resource scheduling for multi-priority services. By constructing a link model with atmospheric attenuation characteristics, the relationship between key generation rate and link stability is analyzed. Multiple constraints are introduced under a discretized topology to achieve a dynamic balance between link quality and key resources. Simulation results demonstrate that the proposed algorithm outperforms traditional methods in terms of request success rate, end-to-end latency, and key resource utilization. Under high load conditions, the overall request success rate is improved by approximately 29.35% compared to the SA-MFP algorithm, and the communication latency is kept to around 55ms, significantly reducing average latency. Future work will further explore hybrid routing strategies that integrate multi-objective optimization models and machine learning to cope with more complex and changeable spatial quantum communication environments.

[0190] 2. The present invention first establishes a satellite-to-ground link model that incorporates the atmospheric attenuation characteristics of VLEO and LEO satellites, analyzing the coupling relationship between key generation rate and link stability. Secondly, a slot-based MILP-ARA algorithm is designed. By introducing flow conservation constraints, key capacity constraints, and end-to-end delay constraints, optimal path selection under multiple service priorities is achieved in a discretized topology, and adaptive weight factors are used to dynamically balance link quality and key utilization. Finally, through large-scale constellation simulation verification, the proposed MILP-ARA algorithm improves the request success rate by approximately 29.35% compared to the baseline algorithm at each service priority level. In terms of end-to-end delay, the overall delay of services of different priorities in this heterogeneous constellation collaborative relay model is concentrated around 55ms under the MILP-ARA algorithm, effectively reducing the communication delay by approximately 20ms compared to the Dijkstra and SA-MFP algorithms.

[0191] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to the specific embodiments described. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A quantum key distribution method for VLEO-LEO satellite networks based on MILP, characterized in that: Here are the steps: S1: Establish a satellite-to-ground link model that includes the atmospheric attenuation characteristics of VLEO and LEO satellites, and analyze the coupling relationship between key generation rate and link stability; S2: Build a slot-based MILP-ARA algorithm. By introducing traffic conservation constraints, key capacity constraints, and end-to-end delay constraints, it achieves optimal path selection under multiple service priorities in a discretized topology and dynamically balances link quality and key utilization using adaptive weight factors. S3: Verified through large-scale constellation simulation.

2. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 1, characterized in that: In step S1, the steps are as follows: A. Build a QKD network architecture for VLEO-LEO satellite networking; B. Calculate the key generation rate based on quantum key distribution using decoy states; C. Calculate the attenuation of space-to-Earth and inter-satellite links.

3. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 2, characterized in that: In step A, the steps are as follows: The network consists of LEO satellites and VLEO satellites at very low orbit altitudes; Each layer of the satellite network contains p orbital planes, and each orbit has q satellites evenly distributed. l represents the layer where the satellite is located, and p l and q l They represent the number of orbital planes and the number of satellites per orbit in this layer respectively; N l =p l q l Indicates the total number of satellites in this layer, Indicates the total number of satellites in all layers before the current layer, and serves as the starting number of this layer. Each satellite is numbered as follows: Among them, v is the track surface index, u is the track internal index, and l' is used to represent a layer index variable.

4. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 2, characterized in that: In step B, the steps are as follows: Assuming that the phase θ of each signal is random, the number of photons in the signal state follows a Poisson distribution with parameter μ, which represents the intensity of the signal state, then The probability that Alice generates a pulse consisting of n photons is expressed as follows: P n (μ)=(μ n e -μ ) / n!; The lower limit of the key generation rate is expressed as: R bb84 ≥q{-Q μ f(E μ )H2(E μ )+Q1[1-H2(e1)]},; Among them, Q μ (Q v ) and E μ (E v ) are the gain and quantum bit error rate (QBER) of the signal (weak decoy) state μ(v), while Q1 and e1 are the gain and error rate of the single-photon state, respectively. q = 1 / 2 is the BB84 protocol efficiency, and f(x) = 1.22 is the two-way error correction efficiency of the cascaded protocol. and get: H2(x)=-xlog2(x)-(1-x)log2(1-x); For the coherent state, the gain and QBER of the signal state are expressed as: Where Y n =Y0+δ n -Y0δ n ≈Y0+δ n is the probability that Bob's measurement is deterministic when Alice emits a pulse of n photons; At the same time, the error rate and attenuation of the n-photon pulse signal are obtained, and the expressions are as follows: e n =Y0 / (2Y n ),d n =1-(1-d) n ; When using the vacuum state plus weak decoy state method, the gain and error rate of the single-photon state verify the lower limit L and upper limit U respectively: After optimization, we get: Among them, the background gain Y0 is calculated by the gain of the vacuum decoy state; and the background bit error rate e0 occurs randomly due to the dark count.

5. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 2, characterized in that: In step C, the steps are as follows: When the optical receiver is located in the far field of the transmitter, the attenuation caused by diffraction in the FSO link is expressed as: Where λ is the wavelength, D T and D R are the diameters of the transmitting and receiving telescopes, T T and T R are the transmission coefficients of the transmitting and receiving telescopes, L P is the pointing loss due to misalignment between transmitter and receiver; The divergence angle of the transmitting telescope is θ T ≈λ / D T , and the additional divergence angle caused by atmospheric turbulence is θ atm =λ / r0, where r0 is the atmospheric coherence constant, r0 can be expressed as: Where k = 2π / λ is the wave number of light, is the turbulent refractive index structure constant, which describes the atmospheric turbulence intensity, and L is the path length of light transmission; When the receiving spot diameter is larger than the aperture of the receiving telescope, part of the transmitted power cannot be collected by the receiver, resulting in energy loss. The total attenuation experienced by the optical wireless link is: d=d diff ×d atm ×d rec ,d atm =d abs ×d scat ×d turb ; Among them, δ rec represents the attenuation due to efficiency loss in the photon detection process, δ atm represents the attenuation in the atmosphere, including the absorption caused by gases and particles in the atmospheric composition δ abs and scattering δ scat , and the atmospheric turbulence effect δ caused by random fluctuations in the refractive index turb .

6. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 1, characterized in that: In step S2, the steps are as follows: 1) Discrete routing selection and dynamic key traffic allocation by encoding integer variables and continuous variables; 2) A hybrid optimization architecture based on traffic conservation constraints, key capacity constraints, and end-to-end delay constraints.

7. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 6, characterized in that: In step 1), discrete routing selection and continuous variable dynamic allocation of key traffic are encoded by integer variables, and the expression is as follows: in, For business satisfaction items; is the satisfaction ratio of service k in time slot t; w k Represents the business priority weight; is the path cost term; is the adaptive weight coefficient of the path cost, where α0 is the baseline weight of the path cost term; is the comprehensive cost coefficient of the link, is the attenuation cost, 0.5e u is the error cost, For switching penalties; is the resource utilization item, is the adaptive weight coefficient of key utilization, is to maximize the utilization of key resources in the entire network, where β0 is the benchmark weight of the resource utilization term.

8. The VLEO-LEO satellite network quantum key distribution method based on MILP according to claim 6, characterized in that: In step 2), the steps are as follows: The constraint ensures that the transmission of quantum keys in the network complies with the "source-relay-destination" key conservation, which is expressed as follows: in, represents the total amount of keys for business k sent from node i, represents the total amount of keys for service k received from node i; when When represents the source node, the right formula is Indicates the key injection amount of business k; on the contrary, when When represents the destination node, the right formula is Indicates the key extraction amount of business k; when When represents a relay satellite node; Generative Capacity With pre-stored keys The mixed supply capacity is expressed as follows: in, R QKD is the ideal key rate, η atm is the atmospheric channel transmittance, η point is the beam alignment efficiency; Forced to 0; Ensure that the total end-to-end delay of service k does not exceed its maximum tolerance value The expression is as follows:

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