CN112039658A

CN112039658A - Quantum key distribution method using orbital angular momentum coding

Info

Publication number: CN112039658A
Application number: CN202010771909.2A
Authority: CN
Inventors: 尚涛; 孙海正; 张源境; 唐瑶; 刘然; 刘建伟
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2020-08-04
Filing date: 2020-08-04
Publication date: 2020-12-04

Abstract

The invention discloses a quantum key distribution method using orbital angular momentum encoding, which comprises the following steps: step 1, constructing a QKD protocol model which is irrelevant to a general measuring device, wherein Alice and Bob are both parties of legal communication needing to establish secure key sharing, and Charlie is an untrusted third-party measuring device; step 2, the third party completes Bell state measurement of the auxiliary photons and sends the measurement result to Alice and Bob through a classical public channel; step 3, the two communication parties obtain a key seed according to the measurement result; step 4, repeating the steps 1-3 to obtain a key seed with enough length; step 5, generating a Fibonacci key matrix with the same rank; and 6, verifying whether the key matrix is correct or not. The invention has the advantages that: (1) no adjustment of the system reference frame is required. (2) No flipping of the qubits is required. (3) And a high-capacity coding mode is used, so that the entangled photon coding capacity is improved. (4) With a high data transmission rate.

Description

Quantum key distribution method using orbital angular momentum coding

Technical Field

The invention relates to a quantum key distribution method using orbital angular momentum coding, and belongs to the technical field of communication networks.

Background

The Quantum Key Distribution (QKD) technology is a branch developed and matured in Quantum communication, and is used for designing and testing a small-sized QKD network with limited node number, especially a Quantum satellite-based QKD network, so that the communication distance can be greatly increased. Although quantum satellites overcome the problem of long-distance photon loss, the problems of real-time omnibearing coverage and multi-node construction of communication still need to be solved, and the practical application range of the quantum satellites is limited. In addition, the establishment of a complete quantum satellite communication network in space requires a great deal of manpower, material resources and time. In recent years, researchers begin to research low-altitude airborne QKD platforms, the network deployment mode has the remarkable characteristics of simple structure, convenience in operation, low cost and the like, and the network deployment mode has wide application prospects in future quantum communication networks.

In a QKD system based on fiber channel, birefringence and attenuation effects present in the fiber limit the safe distance for key distribution, which typically can only reach around a hundred kilometers. Other methods need to be used to achieve quantum key distribution over greater distances. In the free space channel, the birefringence phenomenon is substantially negligible for the information-carrying optical signal, and the decoherence effect is small. In addition, the optical signal wave band applied in the experiment has good transmission property and low loss in free space, and the related detector technology is gradually mature. However, the near-surface free space QKD is still subject to atmospheric turbulence, weather conditions, and terrain, and as such, no longer safe transmission distances can be achieved. With the research on aircraft technology and quantum satellites, researchers plan to utilize earth satellites or other space platforms as relay nodes to implement QKD networks over greater distances and even worldwide, especially in vacuum environments, where optical signals can be transmitted almost without loss. The use of Orbital Angular Momentum (OAM) to encode quantum information has two advantages: one is that the OAM state has rotational invariance in the transmission direction, so the sender and receiver do not have to adjust the reference system in real time. Secondly, the OAM theoretically has an infinite dimension eigen state, and the coding efficiency of the system can be greatly improved by using high-dimension coding.

In recent years, orbital angular momentum technology has been rapidly developed, such as key technologies for generating, controlling, and interfacing with other systems. Furthermore, Yu et al propose the use of a planar plasma interface to generate a single OAM-state beam. Studies have also shown that the unique scattering resonance in nanoplasmon-helix arrays can support photonic bandgaps with band-edge modes with multiple OAM values distributed between fibonacci numbers. Therefore, research and development on high-capacity quantum key distribution schemes are started, and certain progress is made. The study of the high-capacity quantum key distribution scheme by developing a high-dimensional hilbert space has two main advantages: in one aspect, multiple bits of a shared key may be encoded on a quantum. On the other hand high dimensional systems are more robust to certain types of noise. In 2013, Simon et al proposed a quantum key distribution scheme in a high-capacity, high-efficiency form. They use specially designed OAM entangled state and Fibonacci sequence of light to realize quantum key distribution, and the transmission capacity of single quantum information of the scheme is further increased, but the high capacity characteristic of the scheme is still limited by difficult realization and inflexible coding. This scheme has a problem mainly because increasing the information capacity depends on OAM having a larger bandwidth and a method for encoding, and thus it is impossible to satisfy both a longer transmission distance and a lower error rate due to a limitation of a practical bandwidth. Secondly, although theoretical analysis of the OAM-QKD protocol of Simon et al has been proposed, due to the defects of the hardware devices of the practical application system, photon number separation attacks, blindness attacks, and the like may still be suffered. In 2012, Lo et al proposed a method of using measurement device independence to solve the problem of measurement-side vulnerabilities. In a Measurement device-independent QKD (MDI-QKD), both legal communication parties (Alice and Bob) send quantum key information to a third party (Charlie) to complete Bell Measurement, and even if Charlie is not authentic, a safe QKD process can be completed. Also in this system, the transmission distance of the information of Alice or Bob is only the distance from Alice or Bob to the detector, and therefore, the communication distance between Alice and Bob is twice the actual transmission distance in QKD.

Quantum OAM by a nanoplasmonic spiral array can produce OAM values having a plurality of distributions between fibonacci numbers. The invention provides a scheme of OAM-MDI-QKD (operation administration and maintenance-digital-rendering-digital-description-descriptive) encoded by utilizing the relationship between Fibonacci number series and Lucas number series based on the above characteristics of OAM, and the information transmission capacity of single quantum key distribution is improved on the premise of ensuring that the transmission distance of a safe key is as long as possible.

Disclosure of Invention

The technical problem of the invention is solved: in QKD systems using fiber optic channels, the presence of birefringence and attenuation effects in the fiber can limit the safe distance for key distribution, while in free space channels, the birefringence is essentially negligible for the optical signal carrying the information, and the decoherence effect is small. In addition, the optical signal wave band has good transmission property and low loss in free space, and related detector technology is gradually mature. However, near-surface free space QKD is subject to atmospheric turbulence, weather conditions, and terrain, and as such, greater safe transmission distances are not achievable. How to increase the transmission capacity of single quantum information without increasing extra bandwidth is a key problem that must be solved to realize the practicality of QKD technology.

The technical scheme adopted by the invention is as follows: a quantum key distribution method using orbital angular momentum encoding, comprising the steps of:

step 1, a general measurement equipment independent QKD protocol model is constructed, wherein Alice and Bob are both legal communication parties needing to establish secure key sharing and are responsible for preparing photon information. Charlie is an untrusted third party measurement device responsible for detecting entangled photons transmitted by Alice and Bob. Alice encodes OAM entangled photon pairs using Fibonacci (Fibonacci) values, denoted

Bob encodes OAM entangled photon pairs using Lucas (lucas) values, denoted as | φ>. Two photon states existing in the OAM entangled state are respectively named as a signal photon state and an auxiliary photon state with Fibonacci values, wherein the signal photon state is marked as | F_k>_sThe auxiliary photon state is denoted as | F_k>_iWherein k is a positive integer. Alice and Bob leave the entangled pair signal photons to themselves, respectively. Meanwhile, Alice and Bob respectively send the modulated auxiliary photons to third-party measuring equipment Charlie through a free space channel.

Step 2. in the Charlie measurement equipment, two static optical OAM sorters (sorters) and one lens are used first to filter out the stack states

And

and prevents any OAM states other than Fibonacci values from passing. L is_C1、D_C1、L_C2And D_C2Is four single photon detectors for detecting OAM states of Fibonacci values, wherein L_C1For detecting OAM state | F_k-2>/|F_k-1>；L_C2For detecting OAM state | F_k-1>/|F_k+1>；D_C1For detecting OAM stack states

D_C2For detecting OAM stack states

Then, if and only if L_C1And L_C2、L_C1And D_C2、D_C1And L_C2、D_C1And D_C2When two detectors under any one of the four conditions are triggered simultaneously, the measurement is finishedAnd (4) working. Finally, Charlie completes the Bell state measurement of the secondary photons and sends the measurement results to Alice and Bob through the classical open channel.

And 3, according to the measurement result, reserving the condition that the corresponding Charlie is successfully measured in all the results by Alice and Bob. Then, Alice transmits the locally reserved signal photon after modulating the signal photon to the local L_AIn the detector, Bob also modulates and transmits locally reserved signal photons to L of the local side_BIn the detector. Alice and Bob use OAM sorter L, respectively_AOr L_BThe signal photon state is detected. If Alice or Bob finds that the results issued by Charlie do not match the results they detected, the two communicating parties will terminate the communication. Otherwise, they will perform the following operations. When one party detects the determined Fibonacci value, the other party still cannot determine the detected Fibonacci value. When Alice detects a certain Fibonacci value, it can be based on the Fibonacci encoded OAM entanglement status formula

And Charlie measurements, using formula F_k＝F_k-1+F_k-2K is more than or equal to 2, k is an integer, and a Fibonacci value for generating key information is obtained and is recorded as F_k. Meanwhile, when Bob detects a certain Fibonacci value, the value can be according to the Lucas coded OAM entanglement state formula | phi>And Charlie measurements, using formula L_k＝F_k+1+F_k-1K is not less than 2, k is an integer, and a lucas value for generating key information is obtained and recorded as L_k. Then, Alice and Bob can simultaneously obtain F according to the Fibonacci sequence and the Lucas sequence_kAnd L_kFurther obtain a key seed F_2k＝F_k×L_k. The above process does not require exchanging classical information nor bit flipping.

And 4.Alice and Bob repeat the steps 1-3 until a sufficiently long key seed is obtained.

Step 5, according to the result of the Charlie detector, Alice and Bob obtain a key seed F of the Fibonacci key matrix_2k. Then, they generate Fibonacci key matrices with the same rank using a random number generator.

And 6, enabling Alice and Bob to pass through a determinant det (D) corresponding to the Fibonacci key matrix_2k) And (1) verifying whether the key matrix is correct. If the rank of the determinant is not equal to 1, the communication is aborted. Otherwise, they can encrypt the digital message by matrix multiplication using the Fibonacci diagonal key matrix.

The quantum key distribution method based on the orbital angular momentum coding has the advantages that the characteristic of high dimensionality of the quantum orbital angular momentum is fully exerted, the quantum information is coded by the Fibonacci number and the Lucas number, the key information is decoded through the relationship between the Fibonacci number sequence and the Lucas number sequence, and the information capacity of single quantum key distribution is improved.

Compared with the prior art, the invention has the advantages that:

(1) the present invention does not require adjustment of the system reference frame. In the QKD system based on OAM coding, because the OAM state has rotation invariance in the transmission direction, the system does not need to monitor and adjust the reference system in real time.

(2) The invention does not require flipping of the qubits. The two communication parties Alice and Bob can select the sorting machine L according to the local party_AAnd L_BThe key seed is obtained and the result of the measurement is known from Charlie, not the bit flipping method used in the original MDI-QKD protocol.

(3) The present invention uses a high capacity coding scheme. The obtained Fibonacci value is used as a seed for the Fibonacci block diagonal matrix, and the sequence of digital messages can then be encrypted using matrix multiplication. In Simon et al, each Fibonacci value is used to represent a three-bit binary string. In the invention, the same Fibonacci number is used for constructing a Fibonacci diagonal matrix, and a Fibonacci block diagonal matrix is further constructed with the Lucas number from Bob, and the generated key is obviously longer than the key coded into a three-bit binary character string.

(4) The invention has high data transmission rate. Will be divided intoSelector L_AAnd L_BRespectively arranged at the Alice end and the Bob end, and two pairs of detectors, namely L, are arranged in the measuring equipment of Charlie_C1And D_C1、L_C2And D_C2. Unlike the original MDI-QKD protocol proposed by Lo et al, each probe will generate key information and neither case will be discarded. That is, in the Charlie measurement apparatus, when L is_C1And L_C2、L_C1And D_C2、D_C1And L_C2、D_C1And D_C2When any one of the detectors responds, both Alice and Bob can obtain the key information.

Drawings

FIG. 1 is an experimental schematic diagram of the OAM-MDI-QKD scheme based on Fibonacci coding of the present invention;

the symbols in the figures are as follows:

alice and Bob are both legal communication parties;

charlie is an untrusted third party measurement device;

L_C1、D_C1、L_C2and D_C2Four single photon detectors;

the Sorter is a photon OAM state sorting machine;

BS is a beam splitter;

L_Aand L_BOAM state classifiers of an Alice end and a Bob end respectively;

Decoy-IM is a Decoy state intensity modulator;

the SLM is a spatial light modulator;

l is a quantum superposition state generated by photon OAM and having Fibonacci number series characteristics and lucas number series characteristics;

|F_k-1>_A、|F_k-2>_Aand | F_k+1>_B、|F_k-1>_BSignal photon states, left locally (Alice or Bob);

|F_k-2>_B、|F_k-1>_Band | F_k-1>_A、|F_k+1>_ATo aid the photon state, it is sent to a third party (Charlie) for measurement.

Detailed Description

The quantum key distribution method using orbital angular momentum coding provided by the invention needs to solve the following two problems: (1) in a classical quantum key distribution protocol, the transmission capacity of single quantum information is mostly 1-bit classical information, the information capacity cannot be effectively improved by selecting horizontal and vertical basis vectors to encode the quantum information under polarization encoding, and how to change the encoding mode of the quantum information so as to improve the transmission capacity of the single quantum information in key distribution is a first problem to be solved; (2) the quantum key distribution scheme cannot quickly enter a practical stage, and one main reason is the uncertainty of a quantum measurement process, so that the efficiency of the overall scheme is low, and in an entanglement-type protocol (Ekert91 protocol), an efficiency factor q is 1/2. The problem of how to improve the efficiency of the quantum key distribution scheme is a key technology for the practical application of the quantum key distribution scheme.

The main realization idea of the invention is as follows: the high-dimensional characteristic of quantum OAM is fully exerted, the Fibonacci number sequence is utilized to encode quantum information, and a quantum key distribution method using orbital angular momentum encoding is designed.

Fibonacci and Lucas number series

Fibonacci number series: f_k＝F_k-1+F_k-2K is not less than 2, and k is an integer; lucas number series: l is_k＝L_k-1+L_k-2K is not less than 2, and k is an integer.

The relationship that exists between them:

L_k＝F_k+1+F_k-1,F_2k＝F_k×L_k。

constructing a Fibonacci matrix:

constructing a Fibonacci diagonal key matrix:

det(D_2k)＝(-1)^k(k+1)。

the invention relates to a quantum key distribution method using orbital angular momentum coding, which comprises the following specific implementation steps:

step 1, a general measurement equipment independent QKD protocol model is constructed, wherein Alice and Bob are both legal communication parties needing to establish secure key sharing and are responsible for preparing photon information. Charlie is an untrusted third party measurement device responsible for detecting entangled photons transmitted by Alice and Bob. Alice encodes OAM entangled photon pairs using Fibonacci (Fibonacci) values, denoted as

Bob encodes OAM entangled photon pairs using Lucas (lucas) values, denoted as | φ>. Two photon states existing in the OAM entangled state are respectively named as a signal photon state and an auxiliary photon state with Fibonacci values, wherein the signal photon state is marked as | F_k>_sThe auxiliary photon state is denoted as | F_k>_iWherein k is a positive integer. The OAM entanglement states for the Fibonacci and lucas values are as follows:

alice and Bob leave the entangled pair signal photons to themselves, respectively. Meanwhile, Alice and Bob respectively send the modulated auxiliary photons to third-party measuring equipment Charlie through a free space channel.

Step 2. in the Charlie measurement device, two static optical OAM sorters (sorters) and one transparent optical OAM Sorter are used firstMirror filtering out the superimposed states

And

D_C2For detecting OAM stack states

Then, if and only if L_C1And L_C2、L_C1And D_C2、D_C1And L_C2、D_C1And D_C2When two detectors under any one of the four conditions are triggered simultaneously, the measurement is successful. Finally, Charlie completes the Bell state measurement of the secondary photons and sends Alice and Bob the measurement results over the classical open channel.

TABLE 1 possible measurement results of key information in OAM-MDI-QKD scheme

In Table 1, when Charlie performs Bell state measurement, T represents the detector L_C1And L_C2Responding at the same time; u denotes a detector L_C1And D_C2Responding at the same time; v denotes a detector D_C1And L_C2Responding at the same time; w denotes a detector D_C1And D_C2And respond at the same time.

Step 3. Alice and Bob retain the possible measurements given in Table 1All results correspond to the case of successful Charlie measurement (T, U, V and W). Then, Alice transmits the locally reserved signal photon after modulating the signal photon to the local L_AIn the detector, Bob also modulates and transmits locally reserved signal photons to L of the local side_BIn the detector. Alice and Bob use OAM sorter L, respectively_AOr L_BThe signal photon state is detected. If Alice (or Bob) finds that the results issued by Charlie do not match the results they detected, both parties to the communication will terminate the communication. Otherwise, they continue to perform the following operations.

Table 2 possible output results of this scenario

When one of the two communication parties detects a definite Fibonacci value, the Fibonacci value detected by the other party still cannot be determined. By analyzing the OAM entanglement status and Charlie measurement results (as shown in table 2) of fig. 1, both communicating parties can obtain a certain Fibonacci value. For example, if Alice and Bob detect a result of | F respectively_k-1>And | F_k-1>And the detector producing the response in Charlie is L_C1And L_C2Measured as | F_k-2>And | F_k+1>Corresponding to the first possible scenario in table 2. The Fibonacci value owned by Alice party is F_k-1According to OAM entangled state

The transmitted auxiliary photon is known to have a value of F_k-2In contrast to the measurement results and the responding probe published by Charlie, the communication is normal. Alice according to formula F_k＝F_k-1+F_k-2The Fibonacci value F can be obtained when k is not less than 2_k. Similarly, Bob has Fibonacci value of F_k-1According to OAM entangled state

The transmitted auxiliary photon is known to have a value of F_k+1In contrast to the measurement results and the responding probe published by Charlie, the communication is normal. Bob according to formula L_k＝F_k+1+F_k-1The Lucas value L can be obtained when k is more than or equal to 2_k. Then, Alice and Bob can simultaneously obtain F according to the Fibonacci sequence and the Lucas sequence_kAnd L_kFurther obtain a key seed F_2k＝F_k×L_k. The above process does not require exchanging classical information nor bit flipping. When the results detected by Alice and Bob are other, as shown in table 2, the two parties of communication can also obtain the key seed F by using the same method as above_2k。

And 4, repeating the steps 1-3 by Alice and Bob until a key seed with enough length is obtained (for example, according to Shannon's one-time pad encryption theory, unconditionally safe is realized, the key generated by quantum key distribution is as long as the plaintext, and different keys are ensured to be used for plaintext encryption each time).

The invention relates to an MDI-QKD improved scheme based on OAM coding, which is carried out on the basis of an original MDI-QKD protocol, and the improved scheme is different from the original scheme in two points: 1) in the improved scheme, photon states with different values of l are used for encoding quantum information, the method does not weaken the safety of the protocol, and the safety certification of OAM applied to quantum communication is described in the literature of Simon et al. In addition, the OAM coding can avoid the dependency problem of the base, improve the performance of the photon preparation stage and ensure that the key rate is not influenced by the defect. 2) The second point is different at the detector end, and the corresponding measuring device and method are different due to different arrangement of the base. The original MDI-QKD protocol can eliminate eavesdropping on the detector side mainly because it does not rely on measuring equipment, the measuring device only publishes the measurement result, and the generation of the security key also requires operations such as bit flipping. The function of the measuring device of the invention is the same as the original MDI-QKD protocol of Lo et al, and the analysis shows that the core of the safety of the MDI-QKD protocol lies in post-selection (post-select) and bit-flip (bit-flip), and the post-selection is not changed in the improved scheme, so the improved scheme does not weaken the safety performance of the original MDI-QKD protocol in terms of protocol principle.

In the practical application of QKD system, because an ideal single-photon source is difficult to implement, other light sources are generally used for substitution, and when a non-ideal light source is applied, it may be intercepted by means such as photon splitting attack, etc., so a decoy state technology also needs to be introduced. The MDI-QKD polarization coding scheme proposed by Lo et al uses weak coherent light sources and decoy states in practical implementation, from which some bases for security, such as GLLP, Shor-presekill, mutual unbiased bases, etc., can be found. If non-ideal light sources and trap state technology are adopted in the MDI-QKD protocol scheme based on OAM coding, the safety of the MDI-QKD protocol scheme can be ensured by GLLP, Shor-Preskill and the like, and the safety of the MDI-QKD protocol scheme is equivalent to that of the original scheme.

Those skilled in the art will appreciate that the invention may be practiced without these specific details.

The foregoing is only a preferred embodiment of the quantum key distribution method using orbital angular momentum encoding according to the present invention, and it should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principle of the quantum key distribution method using orbital angular momentum encoding according to the present invention, and these improvements and modifications should also be considered as the protection scope of the quantum key distribution method using orbital angular momentum encoding according to the present invention.

Claims

1. A quantum key distribution method using orbital angular momentum encoding, characterized by: the method comprises the following steps:

step 1, constructing a QKD protocol model which is irrelevant to general measuring equipment, wherein Alice and Bob are both legal communication parties needing to establish secure key sharing and are responsible for preparing photon information; charlie is an untrusted third party measurement device responsible for detecting entangled photons transmitted by Alice and Bob; alice encodes OAM entangled photon pairs using Fibonacci values, noted

Bob encodes OAM entangled photon pairs using Lucas values, denoted as φ>(ii) a Two photon states existing in the OAM entangled state are respectively named as a signal photon state and an auxiliary photon state of Fibonacci value, wherein the signal photon state is marked as | F_k>_sThe auxiliary photon state is denoted as | F_k>_iWherein k is a positive integer; respectively reserving the entanglement centering signal photons for Alice and Bob; meanwhile, Alice and Bob respectively send the modulated auxiliary photons to third-party measuring equipment Charlie through a free space channel;

step 2. in the Charlie measurement equipment, two static optical OAM sorters and a lens are used to filter out the superimposed state

And

and preventing any OAM state of non-Fibonacci values from passing; l is_C1、D_C1、L_C2And D_C2Is four single photon detectors for detecting OAM states of Fibonacci values, wherein L_C1For detecting OAM state | F_k-2>/|F_k-1>；L_C2For detecting OAM state | F_k-1>/|F_k+1>；D_C1For detecting OAM stack states

D_C2For detecting OAM superimposed state

Then, if and only if L_C1And L_C2、L_C1And D_C2、D_C1And L_C2、D_C1And D_C2When two detectors under any one of the four conditions are triggered simultaneously, the measurement is successful; finally, Charlie completes Bell state measurement of the auxiliary photons and sends the measurement result to Alice and Bob through a classical public channel;

step 3, according to the measurement result, Alice and Bob reserve the successful measurement condition of corresponding Charlie in all the results; then, Alice transmits the locally reserved signal photon after modulating the signal photon to the local L_AIn the detector, Bob also modulates and transmits locally reserved signal photons to L of the local side_BIn the detector; alice and Bob use OAM sorter L, respectively_AOr L_BDetecting the signal photon state; if Alice or Bob finds that the result issued by Charlie does not match the result detected by the Charlie, the two communication parties terminate the communication; otherwise, they will perform the following operations; when one party detects the determined Fibonacci value, the other party still cannot determine the detected Fibonacci value; when Alice detects a certain Fibonacci value, it can be based on the Fibonacci encoded OAM entanglement status formula

And Charlie measurements, using formula F_k＝F_k-1+F_k-2K is more than or equal to 2, k is an integer, and a Fibonacci value for generating key information is obtained and is recorded as F_k(ii) a Meanwhile, when Bob detects a certain Fibonacci value, the value can be according to the Lucas coded OAM entanglement state formula | phi>And Charlie measurements, using formula L_k＝F_k+1+F_k-1K is not less than 2, k is an integer, and a lucas value for generating key information is obtained and recorded as L_k(ii) a Then, Alice and Bob can simultaneously obtain F according to the Fibonacci sequence and the Lucas sequence_kAnd L_kGo forward toStep (b) to obtain a key seed F_2k＝F_k×L_k(ii) a The process does not need to exchange classical information and does not need to carry out bit flipping;

step 4, Alice and Bob repeat steps 1-3 until a sufficiently long key seed is obtained;

step 5, according to the result of the Charlie detector, Alice and Bob obtain a key seed F of the Fibonacci key matrix_2k(ii) a Then, they generate Fibonacci key matrices with the same rank using a random number generator;

and 6, enabling Alice and Bob to pass through a determinant det (D) corresponding to the Fibonacci key matrix_2k) Verifying whether the key matrix is correct or not as 1; if the rank of the determinant is not equal to 1, communication is aborted; otherwise, they can encrypt the digital message by matrix multiplication using the Fibonacci diagonal key matrix.