Quantum walks-based simple authenticated quantum cryptography protocols for secure wireless sensor networks

Wireless sensor networks (WSNs) play a crucial role in various applications, ranging from environmental monitoring to industrial automation that require high levels of security. With the development of quantum technologies, many security mechanisms may be hacked due to the promising capabilities of quantum computation. To address this challenge, quantum protocols have emerged as a promising solution for enhancing the security of wireless sensor communications. One of the common types of quantum protocols is quantum key distribution (QKD) protocols, which are investigated to allow two participants with fully quantum capabilities to share a random secret key, while semi-quantum key distribution (SQKD) protocols are designed to perform the same task using fewer quantum resources to make quantum communications more realizable and practical. Quantum walk (QW) plays an essential role in quantum computing, which is a universal quantum computational paradigm. In this work, we utilize the advantages of QW to design three authenticated quantum cryptographic protocols to establish secure channels for data transmission between sensor nodes: the first one is authenticated quantum key distribution (AQKD), the second one is authenticated semi-quantum key distribution (ASQKD) with one of the two participants having limited quantum capabilities, and the last one is ASQKD but both legitimate users possess limited quantum resources. The advantages of the proposed protocols are that the partners can exchange several different keys with the same exchanged qubits, and the presented protocols depend on a one-way quantum communication channel. In contrast, all previously designed SQKD protocols rely on two-way quantum communication. Security analyses prove that the presented protocols are secure against various well-known attacks and highly efficient. The utilization of the presented protocols in wireless sensor communications opens up new avenues for secure and trustworthy data transmission, enabling the deployment of resilient WSNs in critical applications. This work also paves the way for future exploration of quantum-based security protocols and their integration into WSNs for enhanced data protection.


Introduction
Wireless sensor networks (WSNs) have become a ubiquitous technology in modern society, offering versatile communication platforms for various applications.These networks play a crucial role in fields such as monitoring, logistics, surveillance, smart homes, and healthcare [1], where ensuring high levels of security is paramount [2,3].However, with the advancements in quantum technologies, many traditional security Meslouhi et al pointed out that the protocol presented in [31] suffered from man-in-the-middle attack.In [29], Yuan et al designed a quantum authentication protocol based on ping-pong method.Yuan et al's protocol can verify the identity of legitimate participants and update the initial authentication key for reuse.In [30], Guan et al presented a three-party AQKD protocol to share a random secret key between two parties with the help of a trusted center.However, Luo et al [33] pointed out that the protocol presented in [30] suffers from information leakage and intercept-measure attack.In 2016, Huang et al [27] presented two AQKD protocols based on single photons and the idea of collective detection.The first protocol is a two-party AQKD, and the other protocol is a multiparty AQKD with star network topology.In both protocols, the legitimate participants pre-share an m-bit secret key.In [34], Li et al presented two ASQKD protocols without classical channels.
Hash functions play a critical role in various cryptographic tasks and are an essential component of modern cryptographic applications [35].They are widely used in both classical and quantum cryptography to ensure data integrity, authentication, and confidentiality.In the context of quantum authentication protocols, classical hash functions are extensively employed to enhance the security of the established quantum channel [27,28,34].
The development of quantum computers and quantum algorithms may endangered some classical cryptographic methods [36].Simultaneously, several novel proposals using quantum systems to strengthen classical cryptographic protocols have been developed, among them the use of quantum walks (QWs), a universal quantum computation model [37,38], which can be utilized to create hash functions [36,[39][40][41][42][43] because of its nonlinear chaotic dynamical behavior.Furthermore, QWs have high sensitivity to initial states, non-periodicity, stability and can be used as a tool to produce very large keyspaces capable of withstanding different attacks [44][45][46][47].
In quantum cryptographic protocols, privacy amplification is a crucial process that ensures the security of the generated secret key.The purpose of privacy amplification is to remove any potential correlations or knowledge that the eavesdropper may have acquired during the key generation process.The quantum hash function (QHF) that has been constructed using QWs can be used perfectly for privacy amplification [39].In order to enhance the security of quantum cryptographic protocols, we investigate three authenticated quantum cryptography protocols based on QHF that have been constructed using quantum walks.The first protocol is AQKD and the other two protocols are ASQKD.
Any SQKD protocol relies on a two-way quantum communication (qubits are allowed to travel from the sender to the receiver, then back again to the sender), in which the quantum user (Alice) sends a quantum state to the classical user (Bob) [48].The classical user is limited to performing the following operations: (1) to measure the received particle in computational basis (Z-basis), (2) to prepare fresh qubits in the computational basis, (3) to reflect qubits, and (4) to reorder qubits.
To reduce the amount of consumed quantum resources to establish a random secret key between two parties, Krawec [49] and Liu and Hwang [50] presented a SQKD protocol to establish a random secret key in which two classical participants utilize a mediated server with quantum capabilities.In all above-mentioned SQKD and ASQKD protocols, users are actually semi-classical users as they require some quantum resources for preparing qubits in computational basis or quantum memory to reorder qubits [20].In this paper, we present two ASQKD protocols that reduce the consumed quantum resources for establishing a random secret key with participants sensor nodes.In the first protocol, the receiver node (Bob) has access to those quantum resources only necessary for running QWs and measuring qubits in the computational basis.In the second protocol, both legitimate participants nodes are semi-classical.The sender node (Alice) has access only to those quantum resources required to run QWs, as well as to prepare and send qubits in the computational basis {|0⟩,|1⟩}, while the receiver node has access to the quantum resources needed to measure qubits in the computational basis as well as to run QWs.For distinguishing between the proposed two ASQKD protocols, we call the first as ASQKD1 (has one semi-classical user) and the other as ASQKD2 (both users are semi-classical).We can summarize the role of QWs in designing the proposed protocols as follows: (1) act as a QHF for the privacy amplification technique with higher security; (2) decide the encoding basis for the transmitted qubits; (3) decide the measurement basis for the received qubits and its positions; and (4) perform the authentication process between legitimate parties for the proposed protocols.
Some key properties of our proposed quantum cryptography protocols are: (i) The pre-shared control parameters are numerical values and can be reused several times due to both the pre-shared key and some publicly announced parameters being used as control parameters for the QW.(ii) The authenticated members can establish different secret keys with the same established qubits because the privacy amplification process is based on both the output of established qubits and some publicly announced parameters.By announcing different parameters, the legitimate parties obtain different shared keys without any potential correlations between them.(iii) The number of bits for the shared secret key may be greater than the number of shared qubits several times due to performing the privacy amplification process.(iv) In order to reduce the cost of the consumed quantum resources in practical implementations, the proposed ASQKD protocols rely on one-way quantum communication channel while the previous presented ASQKD and SQKD protocols based on two-way quantum communication.
The layout of this paper is set as follows: the presented scenario for wireless sensors communications in quantum scenario is provided in section 2, while the preliminary knowledge for QWs is given in sections 3 and 4 is devoted to the proposed authenticated quantum cryptography protocols.The analysis of the suggested protocols is given in section 5 and the conclusion drowns in section 6.

WSNs scenario
WSNs have become a fundamental technological component for intelligent communities because of their potential benefits.The applications of WSNs extend from body area networks to local and home area networks and further to a wide variety of services in smart cities [51].However, applications of WSNs lack stringent privacy and security protection due to the involvement of human life.
The architecture of WSNs and the restricted nature of the node's resources make it vulnerable and open to numerous attacks.In addition, an adversary can intercept and hereafter fabricate sensitive data to be transmitted as the original data.Moreover, an attacker can pass incorrect data and even reveal it as a sensor itself, and modify the collected data.If the communication of sensor nodes is not secure, then a malicious entity can extract the transmitted data and the code associated with that node, which leads to many security threats and challenges.
Also, with the development of quantum technologies, many cryptographic mechanisms may be hacked due to the promising capabilities of quantum computers.Therefore, security and privacy are the foremost challenges for WSNs, which need new cryptographic mechanisms to have the ability to withstand the promising attacks from digital computers besides quantum computers.For these reasons, the goal of this paper is to design new authentication protocols for secure wireless communication based on quantum technologies and open the door for integrating quantum technologies with WSNs and various Internet of Things devices to achieve high security and efficiency.The suggested scenario for wireless sensor communications in the quantum scenario is presented in figure 1.

QWs based QHF
A coined discrete QW is composed of the following elements: a walker, a coin, evolution operators for both coin and walker and a set of observables.A walker is a quantum system |ψ⟩ p exists in a Hilbert space H p of dimension d where d = N for a QW run on an N-node circle.The coin is typically a quantum system existing in a two-dimensional Hilbert space |ψ⟩ c ∈ H c .In each step t of acting QWs, a Unitary operator Û applied on the entire quantum state |φ ⟩ where Ĉ is a Unitary operator to be applied on the coin state, Î is the identity operator, and Ŝ is the Shift operator (that is, the operator that diffuses the quantum particle over the topology over which the QW is run).If the QW is run over an N circle, then the shift operator Ŝ can be expressed as in equation ( 2) The coin operator Ĉ can be written in matrix for as in equation ( 3) The final state |φ ⟩ final of the quantum state after t steps is provided by equation ( 4) The probability P(x, t) of locating the walker at location x after t steps can be represented as in equation ( 5) In 2013, Li et al [36] proposed the first QHF based on 1-D two-walker QWs on a circle controlled by a bit string.After that, numerous QHFs based on QWs have been constructed [39,41,42].Furthermore, please note that the probability P(x, t) have nonzero in any location x if the number of steps t is greater than or equal to the number of vertices N [43,52].All constructed QHFs based on QWs [43] survive from this defect.To avoid this defect we will modify the QHF presented in [42] by appending another coin operator.
In [42], Yang et al built a QHF for a binary m-bit based on one-walker QWs on a circle with N nodes.Unitary operators Û0 and Û1 are applied when the tth bit of m is '0' and '1', respectively.In the modified QHF, we use three coins Ĉ0 , Ĉ1 and Ĉ2 to construct the three evolution operators Û0 , Û1 and Û2 , respectively.Where the evolution operator Û0 ( Û1 ) is performed when the tth bit of m is '0' ('1'), and the evolution operator Û2 is applied when the tth step exceeds to the size of m and does not reach N. As an example, if m is '101' and N = 25, the final state can be given as in equation ( 6) According to QHF presented in [42], the final state can be written as in equation ( 7) where probability P is equal to zero in some locations.For more illustration, the hash values constructed by the two QHFs (modified QHF and Yang et al's QHF [42]) using the same parameters for message m = '101' and N = 25 are given in figure 2 for binary format and in the hexadecimal format as follows: • Modified QHF: 9581 D3E3 0E6A 2956 2FB6 E3F7 5298 7609 8C31 A6E3 3B8F B2F4 53 • Yang et al's QHF [42]: 0000 0000 0000 0000 003F 003F 003F 003F 0000 0000 0000 0000 00 The modified QHF is outlined in the following steps.
(i) Select initial parameters (N, m, ω, θ 0 , θ 1 , θ 2 ) for operating one-particle QWs on a circle of N vertices governed by m-bit to generate a probability distribution P of size N.Here θ 0 , θ 1 and θ 2 are parameters for the coin operators Ĉ0 , Ĉ1 and Ĉ2 , respectively.The coin initial state is prepared as Construct the hash value for m string by transforming P to a binary values as in equation ( 8): where 8×N is the length of binary hash value.

The proposed quantum authentication protocols
There are two types of participants in any quantum cryptography protocol: quantum participants and semi-quantum (classical) participants.The quantum participant has full quantum capabilities and can prepare, measure, and manipulate complex quantum states using advanced quantum techniques.The classical participant has access to limited quantum resources to carry out the following procedures: (1) receive and resend quantum states via the quantum channel; (2) perform measurements using the computational basis; (3) prepare quantum states on the computational basis; (4) reflect quantum states; and (5) reorder quantum states (that need quantum memory) [20].This section presents three variants of quantum authentication protocols designed to establish random secret keys based on quantum walks.The first protocol is AQKD and the others are ASQKD, which the proposed protocols required pre-share master key parameters (ω, θ 0 , θ 1 , θ 2 ) for operating one-particle QWs on a circle with odd N nodes (for instance, this can be prepared in advance once in a closed environment).The pre-shared master key parameters are used to establish a common secret key between two or more entities.This process typically involves securely exchanging the pre-shared key parameters through a trusted channel, or it can be prepared in advance in a closed environment.Once the entities have the same pre-shared master key parameters, they can be used as a basis for authentication operations and further cryptographic operations.
In the proposed quantum authentication protocols, the used quantum communication environment is based on single photons, and the quantum channel is assumed to be lossless and noiseless.

The AQKD protocol
The procedure of AQKD protocol is illustrated in figure 3 and given in the following steps:  (vii) Eventually, Alice informs Bob an odd number N key , to operates QWs (N key , R, ω, θ 0 , θ 1 , θ 2 ) for constructing the hash value K ∈ {0, 1} 8N key as a secret key of length 8×N key bits.
From the declared data via the classical channel around the hash value, no one can get any information regarding the master key (ω, θ 0 , θ 1 , θ 2 ) or the final secret key K. Consequently, the master key can be reused several times later due to both the pre-shared key and some publicly announced parameters being used as control parameters for operating QW and performing the privacy amplification process.Also, there are no potential correlations between the shared final keys and the measurement results R or any information regarding the master key.For more illustration, see the given example in figure 4, in which both authorized partners can establish various secret keys utilizing the same transmitted qubits by replicating step (vii) various times with publishing another N key in each time.

The ASQKD1 protocol
In any SQKD protocol, Alice with fully quantum capabilities communicates with the classical receiver (Bob) to establish a random secret key via two-way quantum communication channel.The main contribution of presenting this class of protocols is to reduce the quantum capabilities to make quantum communications more realizable and practical.Therefore, the proposed ASQKD1 protocol relies on one-way quantum communication and Bob has limited quantum capabilities to execute the following operations: (1) receives and measures qubits with computational basis, and (2) runs one-walker quantum walks on a circle.
The procedures of ASQKD1 protocol are given in figure 5 and described in the following steps:  S i in computational basis as |0⟩ or |1⟩, and records the corresponding classical bit value of ith qubit as R ∈ {0, 1} x .Otherwise, Alice randomly prepares the qubit S i with diagonal basis as + ⟩ or | − ⟩. (iii) Perform step (iii) as in our AQKD protocol (section 4.1).(iv) As Bob receives a photon stream S, he performs quantum measurements in computational basis on each incoming qubit and stores the corresponding measurement results of ith qubit when ith bit of B is '1' to get the classical bit string R ∈ {0, 1} x .(v) Performing steps (v)-(vii) in the proposed AQKD protocol, which presented in section 4.1.

The ASQKD2 protocol
In our ASQKD2 protocol, both legitimate participants have access to both classical and quantum resources: Alice has access only to those quantum resources required to send and prepare qubits in the computational basis, and running quantum walks, while Bob has access to those quantum resources needed to receive and measure qubits with the computational basis as well as for running quantum walks.Our ASQKD2 protocol relies on one-way quantum communication, not rely on two-way quantum communication as stated in the previous presented SQKD protocols.
The procedure of ASQKD2 protocol is illustrated in figure 7 and presented as follows: (i) Perform step (i) as in our AQKD protocol (section 4.1) (ii) Alice produces a random sequence of single photons S ∈ {|0⟩, |1⟩} 8N using the computational basis and records the corresponding classical values of ith qubit when ith bit of B is '1'.(iii) Performing step (iii) in the proposed AQKD protocol.(iv) As Bob receives a photon stream S, he measures all received qubits using the computational basis and stores the measurement results of ith qubit when ith bit of B is '1', to get the classical bit string R ∈ {0, 1} x .(v) Perform steps (v)-(vii) in the proposed AQKD protocol, as presented in section 4.1.
The main contribution of proposing ASQKD2 is to share a random secret key between two semi-quantum participants without using a mediated quantum server, to reduce the amount of consumed quantum resources and to make quantum communications more realizable and practical.For more illustration see the example presented in figure 8.

Analysis of the presented protocols
In this section, we prove that the proposed authenticated quantum cryptography protocols are high efficient and secure against several well-known attacks such as intercept-and-resend attack, impersonation attack, and collective attack.

Efficiency analysis
One of the essential tools to measure the efficiency of quantum protocols is the qubit efficiency, which presented by Cabello [53] and can be denoted as follows: where K is the total number of bits for the established secret key, B represents the number of whole generated qubits, and C refers to the number of whole exchanged classical bits over the classical channel except those used for eavesdropping check [54].One of the main advantages of QHF based on quantum walks is the length of the hash value variant with the number of nodes in the circle.In the proposed protocols, Alice informs Bob of a value for N, to run QWs (N, ω, θ 2 ) to produce a hash value B ∈ {0, 1} 8N with length 8×N (step (i)), and announces another odd integer number N key , to run QWs (N key , R, ω, θ 0 , θ 1 , θ 2 ) to generate a hash value K ∈ {0, 1} 8N key with length 8×N key as a secret key (step (vii)).Therefore, the qubit efficiency of the proposed authenticated quantum cryptography protocols is dependent only on the number of nodes in There are also other measures for efficiency, pre-shared key efficiency, which can be stated as in equation ( 10) where K is the total number of bits for the established secret key, M represents the total number of bits for the pre-shared master key.The participants of the presented protocols require a pre-share master key parameters (ω, θ 0 , θ 1 , θ 2 ) for acting one-walker quantum walks on a circle with odd N vertices, which are numerical values that typically have a compact representation, requiring less storage space and reducing transmission overhead.This efficiency is particularly beneficial when dealing with large master keys or when transmitting them over limited-bandwidth networks.In the illustrated examples (see figures 4, 6 and 8), ω = 0 (1-bit), θ 0 = 60 (6-bit), θ 1 = 45 (6-bit), and θ 2 = 36 (6-bit).So, the total number of bits for the pre-shared master key is 19-bit.Therefore, the pre-shared key efficiency for the illustrated examples only is 1 demonstrates the efficiency of the presented protocols, which gives the qubit efficiency and the pre-shared key efficiency for the proposed authenticated quantum cryptography protocols and its related ASQKD protocols [31,34].Subsequently, the proposed protocols are highly efficient.

Security analysis
The main goal of Eve is to get any information about the established key from the transferred qubits.Hence, security analysis is an essential task for any quantum protocol.The security of our proposed quantum cryptography protocols is guaranteed by both standard protocols like the quantum no-cloning theorem and Figure 7.The proposed ASQKD2 protocol, which limited to prepare qubits in computational basis and running quantum walks, while Bob limited to measure the received qubits with computational basis besides running quantum walks.quantum uncertainty postulate to prevent unconditional attacks as well as on quantum walks and their key parameters.In the case of both participants are restricted to quantum capabilities, the key established between them cannot achieve conditional security but its security is based on mathematical computation [21].However, the security of the two proposed ASQKD protocols is based on quantum walks, not mathematical computation.In this regard, we exhibit the security analysis for the presented protocols in a detailed way and confirm that the presented protocols are effective in exposing any active attack.

Impersonation analysis
In these types of attack, Eve plays Bob's role in communicating with Alice, and in other words, plays Alice's role in communicating with Bob to obtain the fully established secret key or part of it.
In the presented AQKD protocol, we assumed that Eve impersonates the Alice task to contact Bob.In step (i), Eve announces with Bob the value of N, to produce a bit string B ′ ∈ {0, 1} 8N by running QWs (N, ω, θ ).
Nevertheless, it is extremely difficult to produce the correct hash value B, because Eve does not have the full key parameters (ω, θ 2 ).Furthermore, it is very difficult for Eve to prepare a sequence of single photons S ∈ {|0⟩, |1⟩, | + ⟩, | − ⟩} 8N with correct bases according to B. Eventually Eve transfers to Bob a fake stream of photons S according to its own bit string B ′ .In step (iv), Bob measures the received qubits according to the hash value B. Then Eve informs Bob about a value for N check , for producing a bit string K check ∈ {0, 1} 8N check by running QWs (N check , R, ω, θ 0 , θ 1 , θ 2 ).The bit values of R recorded by Eve (step (ii)) is distinct from the measurement results R stored by Bob (step (iv)) due to the generated B stream for both partners is distinct from each other and Eve do not possessing the correct key ω, θ 0 , θ 1 , and θ 2 (step (i)).Therefore, Bob reveals the presence of Eve in step (vi), once Eve publishes the first 4×N check -bit of K check , thus the protocol terminated by Bob.On the other hand, if Eve wants to play the role of Bob to contact Alice.Eve revealed by 8.An illustrated paradigm for the presented ASQKD2 protocol, in the case of pre-established master key is ω = 0, θ0 = 60, θ1 = 45, θ2 = 36.
Table 1.Qubit efficiency and pre-shared key efficiency for the proposed authenticated cryptography protocols and its related ASQKD [31,34].

Protocol
The Alice in step (vi), when Alice informs Eve with the first 4×N check -bit of K check sequence and waits Eve to response with the remaining bits of K check .By checking the remaining bits announced of K check , Alice reveals the presence of Eve and terminates the protocol.
As for the security of the proposed ASQKD1 and ASQKD2 protocols, the same analyses can be performed and detect the existence of Eve in step (vi) of each protocol.Therefore, the proposed authenticated quantum cryptography protocols are secure against active attacks.Moreover, Eve does not obtain any information about the master key (ω, θ 0 , θ 1 , θ 2 ) therefore, the key parameters can be used later several times.

Intercept-resend analysis
In this type of attack, Eve tries to obtain any secret information by intercepting the sequence of photons S sent by Alice, and then resends it to Bob.
In the proposed AQKD protocol, Eve tries to measure the photons in the sequence S transferred by Alice.Eve does not possess any information regarding the keys (ω, θ 2 ) to measure the transferred qubits in correct bases according to the correct hash value B. Ultimately, Eve performs quantum measurement in random bases and then sends it to Bob with a new sequence photons according to her used measurement bases and corresponding measurement results.Then, Bob measures the received according to the hash value B to obtain a measurement results R (step (iv)) that different from classical bit values R recorded by Alice (step (ii)).That leads to Bob generate a checking hash value K check by running QWs (N check , R, ω, θ 0 , θ 1 , θ 2 ) different from Alice's K check (step (v)) due to the bit string R is different for both participants.Consequently, both participants detect the presence of Eve and terminate the protocol (step (vi)).Therefore, the proposed AQKD protocol is secure against intercept-resend attack.
In the proposed ASQKD1 protocol, Bob performs quantum measurement in computational basis on all received qubits.Eve knows this fact and the protocol is one-way communication, therefore she measures all qubits in computational basis over the sequence S sent by Alice, then prepares a new sequence of photons in computational basis according to her measurement results and sends it to Bob.In step (iv) of the protocol, Bob performs quantum measurement with computational basis on all qubits and stores only the measurement results of ith qubit when ith bit of B is '1' to get the classical bit string R. Eve does not have any information regarding to the key parameters (ω, θ 2 ) to store the measurement results when ith bit of B is '1', so Eve randomly extract a fake bit string R ′ .In step (vi) of the protocol, Alice and Bob check K check that there is no detection for the existence of Eve (the quantum channel assumed to be ideal) due to Eve sending to Bob a sequence of photons with computational basis according to her measurement results, so the protocol continues for sharing the random secret key (step (vii)).Eve may be successful to obtain R partly, however, Eve fails to obtain any information regarding the established secret key due to Eve not possessing the used master key parameters for generating the secret key QWs (N key , R, ω, θ 0 , θ 1 , θ 2 ).Therefore, the proposed ASQKD1 protocol is secure against intercept-resend attack.
Regarding the analysis of the proposed ASQKD2 protocol, the same analysis of the proposed ASQKD1 protocol can be performed.

Collective attack
Collective attacks are a type of security threat that can compromise the integrity of the quantum communication channel.Unlike individual attacks in which Eve interacts with each signal coming from Alice (Bob) separately, in collective attacks, Eve attaches ancilla state to each state transmitted independently between the legitimate parties, Alice and Bob.Eve uses unitary operators to extract the information from the target states to the ancilla states.She then retains her quantum state and retransmits the quantum state originally transmitted between Alice and Bob.Once Alice and Bob complete their state measurements and announce the measurement bases, Eve can store the ancilla states, wait for the participants of the protocol to publish some useful information, and then measure the ancilla states to obtain the information.This method allows Eve to gather information without introducing detectable disturbances and poses a significant threat to the security of the quantum communication system.
In the proposed AQKD protocol, Eve attaches the ancilla state to each photon in the sequence S to Bob.Once Alice and Bob complete their state measurements and announce the hash value K check to authenticate each other, Eve can measure the ancilla states, trying to obtain some information about the measurement results R.But the announced hash value K check does not help Eve measure the ancilla states on the correct basis, due to the announced hash value K check being the hash code for the measurement results R. Hence, it is hard for Eve to get the correct measurement results R. Therefore, the proposed AQKD protocol is secure against collective attack.
In the proposed ASQKD1 protocol, Bob performs quantum measurement on computational basis on all received qubits.Eve knows this fact, and the protocol is one-way communication; therefore, she measures all ancilla states on computational basis.Eve does not have any information regarding the key parameters (ω, θ 2 ) to store the measurement results when ith bit of B is '1', so Eve may be successful in obtaining R partly.However, Eve fails to obtain any information regarding the established secret key due to Eve not possessing the used master key parameters for generating the secret key QWs (N key , R, ω, θ 0 , θ 1 , θ 2 ), and tiny changes in the sequence R lead to a massive change in the established secret key.Therefore, the proposed ASQKD1 protocol is secure against collective attack.
Regarding the analysis of the proposed ASQKD2 protocol, the same analysis of the proposed ASQKD1 protocol can be performed.

Conclusions
In this paper, we used the benefits of quantum walks characteristics to propose three authenticated quantum cryptography protocols based on quantum walks for secure wireless sensor communications: the first one is AQKD, the second is ASQKD with one of the two participants having limited quantum capabilities, and the last protocol is ASQKD with both legitimate users having limited quantum capabilities.The advantages of the proposed quantum cryptography protocols are: (a) the preshared master key parameters can be reused several times; (b) the authenticated partners can establish various secret keys with the same transferred qubits; (c) the number of bits for the shared secret key may be greater than the number of shared qubits several times according to the used N and N key ; and (d) the protocols rely on a one-way quantum communication channel, while all previous proposed SQKD protocols rely on two-way quantum communication.Security analyses showed that the proposed protocols are highly efficient and secure against several well-known attacks, such as intercept-and-resend attack, impersonation attack, and collective attack.The main goal of the presented work is to open the door for integrating quantum technologies with WSNs and various Internet of Things devices to achieve high security and efficiency.

Figure 1 .
Figure 1.The suggested scenario for wireless sensor communications in quantum scenario.

Figure 2 .
Figure 2. Plots of 200-bit hash value constructed by the two QHFs (modified QHF and Yang et al's QHF [42]) using the same parameters for message '101' and N is 25.
(i) The sender (Alice) informs the receiver (Bob) publicly an odd number N, for performing QHF (N, ω, θ 2 ) to produce a hash value B ∈ {0, 1} 8N of length 8×N.(ii) Alice prepares a stream of single photons S ∈ {|0⟩, |1⟩, | + ⟩, | − ⟩} 8N according to B sequence as follows and records its corresponding classical values R ∈ {0, 1} 8N .B decides the encoding bases.If the ith bit of B is '1' then Alice randomly prepares the qubit S i in computational basis (Z-basis) as |0⟩ or |1⟩.Moreover, if the ith bit of B is '0' then Alice randomly prepares the qubit S i in Hadamard basis (X-basis) as | + ⟩ or | − ⟩. (iii) Alice transmits the photon stream S to Bob through an ideal quantum channel.(iv)As Bob receives the photon sequence S, he measures the qubits with correct bases according to the bit string B. If the ith bit of B is '1' then Bob measures the qubit S i in the computational basis.Furthermore, if the ith bit of B is '0' then Bob measures qubit S i in Hadamard basis.Thereby, Bob obtains the measurement results R ∈ {0, 1} 8N .(v) Alice publicly agrees with Bob an odd number N check , to perform QHF (N check , R, ω, θ 0 , θ 1 , θ 2 ) for creating a bit string K check ∈ {0, 1} 8N check with length 8×N check .(vi) For detecting an eavesdropper, Alice informs Bob the first 4×N check -bit of K check sequence.If the comparing outcomes are identical, then Bob declares the remaining bits of K check to be reviewed by Alice.Thereby, both partners authenticate each other.If there is any mistake, both members end the protocol.

Figure 3 .
Figure3.The proposed AQKD protocol, which both the legitimate users have fully quantum capabilities and the size of the exchanged final key is dependent on the stated N key only and not on the number of shared qubits.
(i) Perform step (i) as in our AQKD protocol (section 4.1).(ii) Alice prepares a stream of single photons S ∈ {|0⟩, |1⟩, | + ⟩, | − ⟩} 8N according to B sequence as follows: Let B decides the encoding bases.If the ith bit of B is '1', then Alice randomly prepares the qubit

Figure 5 .
Figure 5.The proposed ASQKD1 protocol, which Alice has fully quantum capabilities and Bob limited to measures qubits with computational basis, besides of running quantum walks.