Lightweighted Secure Searching Over Public-Key Ciphertexts for Edge-Cloud-Assisted Industrial IoT Devices

For the industrial Internet of Things (IIoT), public-key encryption with keyword search (PEKS) is a type of applicable and promising encryption technique to maintain the data stored in clouds secure and searchable. To further improve the efficiency of searching, it is popular to introduce edge computing near the IIoT as the substitute for a cloud. However, the straightforward collaboration of the two techniques performs negatively in latency-sensitive applications since the sluggish encryption of PEKS by IIoT devices negates the instant reaction of edges. To meet this challenge, in this article, we exploit the capability of the edge-cloud architecture and propose a lightweight-designed scheme called edge-aided searchable public-key encryption (ESPE). It allows IIoT devices to delegate their costly cryptographic operations to the nearby edge for fast computing and guarantees that all outsourced ciphertexts are semantically secure. Consequently, ESPE accelerates the corresponding ciphertext procedures on edges and saves over 70% encryption cost of an IIoT device.


I. INTRODUCTION
P LENTY of works [5] show that the edge-cloud design acts as a successful substitute for the original only-cloud platforms and compromise between powerful computation and extremely facilitated reactions in emerging applications of industrial Internet of things (IIoT) [2]- [4]. Edge computing, a popular and promising method to optimize the cloud computing systems, takes cloud services from the center of clouds to the edges of Internet, which are physically and logically closer to the IoT world. This method can provide on-demand services, such as instant caching and online analyses with low latency to meet the demands of IIoT in Industry 4.0 [1].
The other phase, the IIoT, has appeared since IoT has been widely used in the industrial field to guarantee the reliability and security of industrial processes. Cloud computing, big data analyses, and the IoTs are the fundamentals of reliability in Industry 4.0. In particular, according to big data analyses, IIoT can avoid potential risks and redundancies in producing processes. However, the goals of Industry 4.0 also include maintaining the information and the process secure [1], which indicates that data in IIoT must be securely utilized only by authorized consumers.
We implement the edge-cloud architecture in IIoT in a classic scenario, as illustrated in Fig. 1. Traditionally, IIoT devices (e.g., sensors, cameras) collect the data flow and synchronize with the cloud platforms due to limited storage. However, data managers are authorized to use the data, and they retrieve them from cloud platforms. Commonly, the communication latency during a complete data service would be considerably high since cloud platforms are physically and logically far from both IIoT devices and data managers. With the assistance of edge computing, each IIoT device stores its data flow to the closest  The application consists of four objects: the IIoT device (IIoT), the edge device (Edge), the cloud, and the data manager. Let PEKS, PKE, and SE denote the encryption algorithms of schemes PEKS, PKE, and SE, respectively. Let K be a random symmetric key. Let F be the collected data. edge platform. Similar to the Cloud, edge platforms can retrieve these data according to the authorization of data managers, such as sorting weather records of a certain week or traffic data under a street name and a time log. Such applications reduce the transmission redundancies and communication latency of the data flow during online services compared with those of the traditional method. Edge platforms supply a win-win strategy to save communication costs by restraining the unnecessary data from uploading to cloud platforms, and they avoid slow data retrieval by IIoT devices or data managers that have low searching efficiency.
However, both cloud and edge platforms are usually assumed to be honest but curious [6]. Although traditional encryption techniques can guarantee the confidentiality of collected data, they make edge platforms hardly able to retrieve data on demand. A major challenge that exists is the dilemma of keeping both data privacy and searchability on edge platforms. Fortunately, we have a technique called public-key encryption with keyword search (PEKS) [7]. PEKS is a potential idea to address the above problem. However, the existing PEKS schemes are still inappropriate for IIoT devices due to their expensive encryption cost. Fig. 2 shows the application of normal PEKS to solve the above problem. In Fig. 2, each IIoT device extracts some attributes of its collected data, takes these attributes as keywords to generate searchable ciphertexts by PEKS, encrypts the collected data by the traditional public-key encryption (PKE) and symmetric-key encryption (SE) schemes (such as the PKE scheme RSA or ElGamal and the SE scheme AES), and sends all generated ciphertexts to a nearby edge platform. In order to retrieve the collected data with the expected attributes, a data manager generates search trapdoors for the chosen attributes and requires a cloud platform to forward the generated trapdoors to all edge platforms. Then, the following steps successively happen. Each edge platform finds the matching ciphertexts in its local storage and uploads these ciphertexts to the cloud platform; then, the cloud platform forwards the uploaded ciphertexts to the manager;and finally, the manager can read the expected data by decrypting the received ciphertexts. Clearly, PEKS allows edge platforms to retrieve IIoT data on demand even if data are encrypted, and a secure PEKS scheme can guarantee data confidentiality under the assumption that both edge and cloud platforms are honest but curious.
In the phase of efficiency, each IIoT device fails at accelerating ciphertext generation in the public-key setting. Most of the processing time is wasted on generating PEKS ciphertexts, since the PEKS algorithm must be run several times, and each time of running takes much more time than that of algorithms PKE and SE. Hence, designing a PEKS scheme with lightweight encryption is urgently needed. Referring to previous works [8], [9], traditional protection mechanisms such as cryptographic algorithms and security protocols have proven inefficient as long as devices have several nonnegligible constrains on resources. It becomes more critical when meeting the real-time requirements of IIoT applications such as instant monitoring due to the costly operations in traditional encryption. Moreover, to the best of our knowledge regarding previous work, only [10] has introduced a lightweight searchable PKE (LSPE) to accelerate search performance, of which the encryption is still costly in the worst case. Hence, we are motivated to design a PEKS scheme with both lightweight encryption and fast search to meet the edge-cloud architecture.
We encounter two questions regarding why and how to build an accelerated searchable PKE system for an edge-cloud architectured IIoT. Our previous review on searchable encryption schemes first shows the necessity of PEKS in such a system, as well as the inspiration for our main ideas and contributions.
Our inspiration to design an edge-aided searchable PKE (ESPE) scheme is motivated by the aim to provide an excellent service experience while sharply lowering the burden on computation and communication. In this article, we exploit the capability of edge computing as well as lightweight-designed encryption and fast search. Two phases are processed simultaneously for the same goal. In the first phase, as a contrast to a searchable public-key ciphertexts with hidden structures (SPCHS) scheme [10], which is the first work that retains both semantic security and fast search, ESPE achieves not only lower encryption complexity but also dramatically reduced encryption latency by delegating the costly computation of bilinear mapping to an edge platform. In the second phase, to accelerate search performance, ESPE employs the idea of the hidden structure establishment among searchable ciphertexts in SPCHS. In addition to these high-efficiency expectations, ESPE guarantees that the honest-but-curious edge and cloud platforms cannot learn any information about the encrypted keyword in the sense of standard semantic security.
However, the aforementioned idea also introduces new challenges for the sake of securely delegating bilinear mapping. In SPCHS, generating different ciphertexts may compute bilinear mapping an unequal number of times, i.e., once or twice. Consequently, the honest-but-curious edge platform can break the security since it is easy to distinguish two independent ciphertexts with unequal times of delegated bilinear mappings if we directly transform SPCHS to ESPE. To address this problem, ESPE is elegantly designed to guarantee that every ciphertext just needs to delegate bilinear mapping only once.
Our contributions are as follows. We first define ESPE and its security. The ESPE model defines five main algorithms including the specified edge-aided encryption algorithm. The ESPE security definition extends the original security definition of SPCHS by additionally allowing any adversary to know all transferred information when delegating bilinear mapping. We construct an ESPE instance and prove its security based on a weak decisional bilinear Diffie-Hellman (DBDH) assumption [13]. The ESPE instance achieves the uniform delegation procedures to generate each ciphertext, and meanwhile, constructs hidden structures among ciphertexts to accelerate search performance. Finally, we experimentally compare ESPE with other works in terms of encryption cost and test its performance in a pollution monitoring application under the edge-cloud architecture. The results show that ESPE can save approximately 70% of the processing time for an IIoT device to generate one ciphertext, thereby saving the communication and decryption costs significantly in that application.
Section III models ESPE and its security. An ESPE instance is constructed in Section IV, and its security is proven in Section V. Section VI experimentally shows the practicality of ESPE. Section VII concludes this article.

II. RELATED WORKS
The costly encryption performance of PEKS is reminiscent of the high encryption performance of searchable symmetric-key encryption (SSE) [11]. However, when applying SSE under the edge-cloud architecture, all IIoT devices must share different symmetric keys with their data managers. Otherwise, a compromised IIoT device will leak the collected data of other IIoT devices. When there are large numbers of IIoT devices, there is a dramatically increased burden to any data manager to dynamically generate and revoke symmetric keys under SSE. Unfortunately, SSE multiplies the number of search trapdoors needed to meet multiple independent IIoT devices when PEKS needs only one for the same case. Consequently, the communication cost of the whole system to transfer those trapdoors increases as well. In contrast, IIoT devices never share secrets with their managers under PEKS, not to mention symmetric keys. Hence, PEKS cannot be substituted with SSE under the edge-cloud architecture. Moreover, all PEKS trapdoors are independent of IIoT devices.
Our main challenge in this article is to design a PEKS scheme with lightweight encryption. Hence, some related works on designing lightweight encryption will be introduced in the following. To our best knowledge, there are three kinds of ideas to design lightweight encryption. These ideas are heuristics in term of our article. However, only one of them is appropriate to be introduced in our article.
The first idea is to replace the costly cryptographic operations by some lightweight operations. This idea is frequently used in designing lightweight SE [14]. In contrast, it is much harder to be used in designing lightweight PEKS. Most of the PEKS schemes are constructed based on bilinear mapping. This operation has been widely recognized as the most costly cryptographic operation [15]. To date, no lightweight operation can replace it while keeping the same algebraic features. In addition, very few PEKS schemes are constructed without bilinear mapping, such as the PEKS schemes based on quadratic residuosity or lattice algebra [16]- [19]. However, all of them cannot achieve fast searching performance over ciphertexts, i.e., the sublinear search complexity, and many random numbers are chosen in their encryption algorithms. Our experiment shows that choosing numerous random numbers introduces equal or even high complexity compared with that of bilinear mapping.
The second idea is to divide a complete encryption into two parts that are, respectively, run offline and online. The offline part can be precomputed without plaintexts. Therefore, as the needed data arrive, the online part takes the results of the offline part as the input to finish the remainder of the encryption processes. Such kinds of lightweight encryption are called online/offline encryption [20]. Generally, the processor of the offline part might possibly be data producers, a trusted third party, or two noncollusive and honest-but-curious third parties. With the first processor that is also the data producer, the total time spent on encryption is not shortened. Further, the energy consumption is not reduced as well if the producer is an IIoT device. The first processor shows advantages only in employing the idle intervals of data producers to run the offline parts. The remaining two processors with semantic security need some stronger assumptions that are not easily realized.
The third processor, as a third-party idea, will be delegated with the costly cryptographic operations as long as there is encryption of certain required data. This idea is concerned with the assumptions on the delegated third party. In previous works [23], the third party is assumed either honest but curious or untrusted. The untrusted assumption on the third party means that it can return wrong or malicious results to delegators, which will be useful to design an encryption scheme with strong security. However, the satisfied schemes will be impractical under the edge-cloud architecture. In other words, the total delay caused by delegating a costly cryptographic operation to an untrusted outsourced platform, including the time cost to verify the results, is usually longer than that of running the operation locally. In such cases, devices need expensive precomputation or verification processes. Thus, there is no advantage for involving outsourced platforms. In contrast, involving the honest-but-curious assumption on the third party is more practical to balance security and practicality. Hence, our article prefers the idea to delegate the costly cryptographic operations to an honest-but-curious third party. The proposed secure data storage and searching framework in [24] gives us intuitive insight regarding our target under the edge-cloud architecture. It is also reported that in the related security analysis, no provable security was declared to support the availability of the framework yet.

III. MODELING ESPE AND ITS APPLICATION
This section models ESPE with its main algorithms, shows the application of ESPE under the edge-cloud architecture, analyzes the suffered attacks of ESPE, and finally, defines the security of ESPE.
Preliminarily, we introduce the main algorithms of ESPE. Most notions involved are listed in Table I.
Definition 1 (ESPE): An ESPE scheme consists of the following five algorithms. 1) Algorithm SystemSetup(1 k , W) takes a security parameter and a keyword space W as inputs and probabilistically generates a pair of master public-and-private keys (PK, SK). 2) Algorithm StructureSetup(PK) takes the master public key PK as input and probabilistically initializes a hidden structure by outputting a pair of public-and-private parts (PUB, PRI) of the structure. 3) Algorithm Encryption(PK, W, PRI) takes the master public key PK, a keyword W , and the private part PRI of a hidden structure as inputs, delegates the expected cryptographic operations to an edge platform, and utilizes the responses of the edge platform to generate a keyword-searchable ciphertext C. 4) Algorithm Trapdoor(SK, W ) takes the master private key SK and a keyword W as inputs and generates a keyword-search trapdoor T W . 5) Algorithm Search(PK, C, T W , PUB) takes the master public key PK, the set C of all keyword-searchable ciphertexts, a keyword-search trapdoor T W of keyword W , and the public part PUB of a hidden structure as inputs and finds all matching ciphertexts of keyword W with the hidden structure's public part PUB from C. In addition, an ESPE scheme must be consistent in the sense that given the keyword-search trapdoor T W of any keyword W and the public part PUB of any hidden structure, Algorithm Search(PK, C, T W , PUB) always finds all matching ciphertexts of W with PUB, except with a negligible probability.
Generally, ESPE has a similar structure to that of the previous work SPCHS except for the employed encryption algorithms in SPCHS. SPCHS seems to be a heuristic framework of ESPE with inseparable encryption algorithms, which is also the biggest challenge before ESPE. In particular, ESPE defines an interactive encryption algorithm, which can delegate the computeintensive cryptographic operations to an edge platform and then, use the corresponding response to generate a ciphertext. In contrast, SPCHS defines its encryption algorithm as a noninteractive one, which indicates the failure of SPCHS in our scenarios under the edge-cloud architecture.
As illustrated by Fig. 3, the tasks for the four objects in an IIoT application are changed comparing to that of PEKS. In the communication phase, when an IIoT device collects data and wishes to update data to the cloud, they first cache data to a nearby edge device. Then, the edge device relays historical data to the cloud. A data manager, who is authorized to retrieve the data, will send requests to the cloud; then, the cloud returns the data according to the requests. In the security phase, the outsourced data are searchable encrypted locally by the IIoT devices, when the searchablity is completed by the edge device. The cloud and the edge device response to the searching requests, respectively, on historical data and instant data. The corresponding ciphertexts finally will be delivered to the data manager by the cloud and decrypted out. All data flows are illuminated in Fig. 3.
The application performs in the order from the setup phase to the search phase. The operations on the vertical timelines below the IIoT devices, the edge devices, the cloud, and the data manager are implemented by sequence. The horizontal arrow lines demonstrate the information transmission. In the application, suppose that all IIoT devices know their managers' public keys for running Algorithm PKE. The application consists of three phases.
1) Setup phase: A manager generates a pair of master public-and-private keys (PK, SK) by running Al-  the matching searchable ciphertexts in its local storage and sends the PKE and SE ciphertexts that are both appended to these matching ciphertexts to the manager via the cloud platform; finally, the manager decrypts the received ciphertexts. In the abovementioned application, ESPE resists attacks from the honest-but-curious edge and cloud platforms, the compromised IIoT devices, and eavesdroppers. The previous work SPCHS also suffers the same attacks as ESPE except the attacks from the honest-but-curious edge platform. Each edge platform is mainly used to achieve the delegated cryptographic operations from IIoT devices, and then, IIoT devices take the responses from their corresponding edge platforms as auxiliary values to generate keyword-searchable ciphertexts. Hence, edge platforms know not only the generated ciphertexts but also some important values that are used to generate these ciphertexts. In contrast, cloud platforms only know the generated ciphertext. Hence, a malicious edge platform can launch more serious attacks than a malicious cloud platform can.
ESPE resists the same attacks as does SPCHS, implying that ESPE has the same security definition as SPCHS to a certain degree. Furthermore, their major differences in terms of the security definition concern how to generate keyword-searchable ciphertexts, such as the place to generate the challenge keyword-searchable ciphertext. Specifically, the security of ESPE is defined as the indistinguishability under chosen keyword-and-structure attacks (IND-CKSA). It allows a probabilistic polynomial-time (PPT) adversary to query the keyword-search trapdoors of the expected keywords and the private parts of the expected hidden structures as defined by SPCHS. When the adversary queries any keyword-searchable ciphertext (including the challenge one) with SPCHS, the difference in semantic security between SPCHS and ESPE shows that with ESPE, the IND-CKSA security allows the adversary to know not only the expected ciphertext but also the corresponding state information. Generally, the state information of a keyword-searchable ciphertext includes all observable information of the whole delegation process during the generation of the ciphertext, which means that ESPE faces more critical challenges. The IND-CKSA definition of ESPE is as follows. In practice, an IND-CKSA secure ESPE scheme means that: 1) all ESPE ciphertexts can keep the privacy of their keywords as well as hidden structures for someone who does not know any keyword-search trapdoor or any hidden structure's private part and 2) given a keyword-search trapdoor or a hidden structure's private part, one can only infer which ciphertexts have the same keyword with the given trapdoor or which ciphertexts are related to the given private part. In addition, the IND-CKSA definition does not consider keyword guessing attacks, since the previous work [23] proposed a general solution for PEKS including ESPE against such attacks. Guessing attack is not considered in this work although how to construct a lightweight design based on the naive combination is still a challenging work. However, we will consider the correlated works in the future.

IV. INSTANTIATING ESPE
This section constructs an ESPE instance in which the encryption algorithm can delegate bilinear mapping to an edge platform and take the response from the edge platform as input to generate a ciphertext. Moreover, the generated ciphertext is semantically secure under the assumption that the edge platform is honest but curious. Let G 1 , G 2 , and G T be three multiplicative groups, all with the prime order q. Let g 1 and g 2 be the generators of groups G 1 and G 2 , respectively. Letê : G 1 × G 2 → G T be an efficient bilinear mapping. Equationê(g a 1 , g b 2 ) =ê(g 1 , g 2 ) ab holds for any a ∈ Z * q and b ∈ Z * q , andê(g 1 , g 2 ) is a generator of group G T . Let W = {0, 1} * be the keyword space. Our ESPE scheme is constructed as follows.
1) Algorithm SystemSetup(1 k , W) takes a security parameter 1 k (k ∈ N) and the keyword space W as inputs and performs the following step. a) Initialize a bilinear mappingê : G 1 × G 2 → G T and choose a generator g 2 ∈ G 2 , where the binary size of prime order q equals k. b) Randomly pick s ∈ Z * q and set p = g s 2 . c) Pick a cryptographic hash function H : W → G 1 . d) Return a pair of master public-and-private keys (PK, SK), where PK = (q, G 1 , G 2 , G T ,ê, g 2 , p, H) and SK = s. 2) Algorithm StructureSetup(PK) takes the master public key PK as input and performs the following steps. a) Randomly pick u and r 1 both from Z * q and compute g u 2 and g r 1 2 .
b) Initialize a pair of public-and-private parts (PUB, PRI), where PUB = g u 2 and PRI = (u, r 1 , g r 1 2 ). Note that PRI is a variable list formed as (u, r 1 , g r 1 2 , [u, w] denotes an element of group G T . c) Algorithm Encryption(PK, W, PRI) takes the master public key PK, a keyword W , and the private part PRI of a hidden structure as inputs, parses PRI as (u, r 1 , g r 1 2 ) and some records such as (w, P t[u, w]) ∈ W × G T , and performs the following steps. a) Randomly pick r 2 ∈ Z * q and compute g r 1 ·r 2 2 . b) Compute and send H(W ) r 2 to an edge platform. c) Receive valueê(H(W ) r 2 , p) from the edge platform. d) Seek record (W, P t[u, W ]) from PRI according to both W and u. e) If existing, generate the keyword-searchable ciphertext C = (P t, g r 1 ·r 2 2 ) and update record ) and add (W, P t[u, W ] =ê(H(W ) r 2 , p) r 1 ) into PRI. 4) Algorithm Trapdoor(SK, W ) takes the master private key SK and a keyword W as inputs and generates a keyword-search trapdoor T W = H(W ) s . 5) Algorithm Search(PK, C, T W , PUB) takes the master public key PK, the set C of all keyword-searchable ciphertexts, a keyword-search trapdoor T W , and the public part PUB of a hidden structure as inputs, initializes an empty set C = ∅ ⊂ C, and performs the following steps. a) Compute P t =ê(T W , PUB). b) Seek a keyword-searchable ciphertext C = (C 1 , C 2 ) ∈ C having C 1 = P t. c) If existing, add the found ciphertext C into C , set P t =ê(T W , C 2 ), and go to step 2. d) Otherwise, return the set C . The Correctness Proof of ESPE. Suppose that: 1) PRI is initialized as (u, r 1 , g r 1 2 ) by Algorithm StructureSetup(PK) and 2) there are, in total, two keyword-searchable ciphertexts (C 1 , C 2 ), which are sequentially generated by running Algorithm Encryption(PK, W , PRI) two times. Hence, the ciphertexts C 1 and C 2 can be parsed as , respectively, where r 2 and r 3 are the two random numbers picked from the space Z * q by the double implementations of Algorithm Encryption(PK, W, PRI).
To prove the correctness of ESPE, it is without loss of generality to prove that Algorithm Search(PK, C, T W , PUB) can just find the above-mentioned two ciphertexts C 1 and C 2 . According to step 1 of Algorithm Search(PK, C, T W , PUB), we have P t =ê(T W , PUB) =ê(H(W ) s , g u 2 ) =ê(H(W ), p) u . The ciphertext C 1 can be found since equation C 1 1 = P t holds in step 2 of Algorithm Search. Furthermore, we have P t = e(T W , C 1 2 ) =ê(H(W ) s , g r 1 ·r 2 2 ) =ê(H(W ), p) r 1 ·r 2 according to step 3 of Algorithm Search. It is clear that the ciphertext C 2 can be also found by step 2 of Algorithm Search since equation C 2 1 = P t holds.
In addition, we need to prove that no more ciphertext can be found by Algorithm Search(PK, C, T W , PUB) except the matched ones. Without loss of generality, suppose that: 1) this algorithm finds an incorrect keyword-searchable ciphertext C i ; 2) the ciphertext C i has keyword W and the public part PUB = g u 2 ; 3) either W = W or PUB = PUB holds; 4) the corresponding private part of PUB is initialized as PRI = (u , r 1 , g r 1 2 ). We now prove that this assumption is false by the following two cases.
Case 1: The ciphertext C i is the first ciphertext of both keyword W and the public part PUB . In this case, we have that the ciphertext C i can be parsed as Case 2: The ciphertext C i is not the first ciphertext of both keyword W and the public part PUB . This case implies that the ciphertext C i can be parsed as where r 2 ∈ Z * q is randomly chosen, and one of the following three equations holds if Algorithm Search(PK, C, T W , PUB) can find the ciphertext C i . The three equations areê(H(W ), p) r 1 ·r 2 =ê(H(W ), p) u , e(H(W ), p) r 1 ·r 2 =ê(H(W ), p) r 1 ·r 2 , andê(H(W ), p) r 1 ·r 2 = e(H(W ), p) r 1 ·r 3 . It is also clear that none of the above equations hold in practice except with a negligible probability.
In the above statements, we simplify the proof by supposing that there are only two matching ciphertexts C 1 and C 2 . Indeed, Algorithm Search(PK, C, T W , PUB) is correct for any given number of matching ciphertexts, since all the other matching ciphertexts can be found using the same method to find C 2 . Moreover, it is easy to extend the above-mentioned Case 2 to prove that no incorrect ciphertext can be found even if there are more than two matching ciphertexts. In summary, Algorithm Search(PK, C, T W , PUB) can only find all matching keyword-searchable ciphertexts of keyword W with the public part PUB except with a negligible probability in practice.

V. PROVING THE IND-CKSA SECURITY OF ESPE
Generally, bilinear mappingê in ESPE can be instantiated as per three different types in practice. The corresponding security proofs have the same essence with only minor differences in detail. Hence, we will only prove that the ESPE instance based on the type-3 bilinear mapping [13] is IND-CKSA secure. Before the proof, it is conventional to introduce a mathematical hardness assumption.
Definition 3 (The Weak DBDH Assumption in Type 3 [13]): Given the public parameters (q, G 1 , G 2 , G T ,ê, g 1 , g 2 ) of a type-3 bilinear mapping, the weak DBDH problem in type-3 is defined as the advantage of any PPT Algorithm B to distinguish between tuples (g a 1 , g a 2 , g b 2 , g c 1 ,ê(g 1 , g 2 ) abc ) and ]| be the advantage of Algorithm B to solve the above-mentioned DBDH problem, where k equals the binary length of q. We say that the weak DBDH assumption in type-3 holds if Adv DBDH

B
(1 k ) is negligible in the parameter k. Theorem 1 proves that the IND-CKSA security of ESPE is based on the weak DBDH assumption in type-3 (due to page limitations, please see Supplemental Materials for the proof details). This theorem implies that since the weak DBDH assumption in type-3 holds in practice, no PPT adversary can break the IND-CKSA security of ESPE with the nonnegligible advantage.
Theorem 1: Let the hash function H of ESPE be modeled as a random oracle Q H . Suppose that: 1) there are at most N ∈ N hidden structures; 2) there is a PPT adversary A who can break the IND-CKSA security of ESPE with the advantage Adv IND-CKSA ESPE,A ; and 3) A issues the trapdoor query Q Trap at most q t times and the private query Q Pri at most q p times. Then, there is a PPT Algorithm B that solves the weak DBDH problem in type-3 with the advantage of approximately Adv DBDH where e is the base of the natural logarithms.

VI. TESTING ESPE
This section experimentally shows that ESPE has significant advantages in saving the encryption cost of IIoT devices and keeping the advantages of the edge-cloud architecture even in the ciphertext setting.

A. Reduction of Approximately 70% Encryption Cost 1) Experimental Environment:
We choose a bilinearmapping-friendly elliptic curve [25] from plenty of candidates. This curve has faster exponentiation operations of both groups G 1 and G 2 than those of other curves, causing ESPE to have the same security level as that of the widely used AES-128. We use the pairing-based cryptography (PBC) library to code the chosen elliptic curve and ESPE, while a Raspberry Pi device (a single-board computer with wireless LAN and Bluetooth connectivity) is used as an IIoT device and a laptop as an edge platform. The IIoT device is directly connected to the laptop via WiFi. More notions are shown in Table II. 2) Experimental Results: To show the advantage of ESPE via the encryption cost saving, we compare ESPE with two other schemes. One is the scheme of ESPE without delegation. It means that the bilinear mapping operation is not delegated to the edge platform in ESPE. The other is the original scheme SPCHS, which does not ever take lightweight encryption into account.  In it, each edge platform serves 20 IIoT sensors via the WiFi network. Each sensor detects five types of pollutants, which are ozone, particulate matter, carbon monoxide, sulphur dioxide, and nitrogen dioxide, and the detection period is five minutes. All pollution records are introduced by the CityPulse project [28]. With the pollution records, the sensors take the detection time, device location, and pollution levels of the aforementioned five pollutants as keywords to generate ESPE ciphertexts, where the pollution levels refer to the standards of the U.K. [27]. The pollution records are encrypted by an elliptic-curve-based ElGamal with a 256-bit group order and AES-128. IIoT devices upload all ciphertexts to the corresponding edge platforms. To query the pollution records of the expected keywords, such as keyword High_O3, an environment manager generates the corresponding search trapdoor by ESPE and sends it to all edge platforms via the cloud platform. Each edge platform finds the matching ciphertexts in its local storage by ESPE and sends them to the manager also via the cloud platform. Finally, the manager decrypts the received ciphertexts and obtains the queried records. Each edge platform just stores the ciphertexts generated in the last week and provides the related search services since the historical data are infrequently queried. In practice, the historical data can be uploaded to the cloud platform by the edge platform, and the search services on the historical data can be queried via the cloud platform.
2) Numerical Results: In the resulting system, each edge platform stores approximately 40 320 encrypted records and 282 240 ESPE ciphertexts. Suppose that each keyword has the same distribution in all edge platforms. Referring to Table III, Fig. 4(a) shows the search performance of an edge platform. For example, suppose that the environment protector wants to query the pollution records of keyword High_O3; each edge platform takes approximately 170 s to find all matching ciphertexts. In practice, the search performance can be significantly improved with more CPU cores on the edge platforms since ESPE supports parallel search.
To show the advantages of ESPE under the edge-cloud architecture, we test the performance of the pollution monitoring system with ESPE and without ESPE, respectively. In the system without ESPE, all edge platforms will send the entire ciphertext database to the environment protector, and the protector will decrypt out all plaintexts and subsequently, find the expected records. Fig. 4(b) shows that ESPE saves approximately 81.5%-96.4% of the communication cost of one edge platform when replying ciphertexts to the protector. Moreover, following the increasing of the edge amount, ESPE achieves a linearly enhanced advantage on decryption and communication, as shown in Fig. 4(c) and (d). Fig. 4(b) is the average cost of each record and Fig. 4(d) is the total cost of all records. Apparently, the advantages are not noticeable for single records; but for the big data records, the advantages of ESPE are tremendous on communication.

VII. CONCLUSION
In this article, a dilemma that happened often in the modern industry that IIoT devices need both lowered-cost and highsecurity requirements when retrieving private data under the edge-cloud architecture was discussed. The lightweight designed ESPE employed outsourcing PEKS and hidden structures in its models, which are promising to accelerate encryption and searching according to our analyses. It also showed the possibility that IIoT devices are free to delegate partial encryption, especially the most costly operations to the curious third party so as to save their energy and storage, which is critical. We also consider it as a novel method to build security guarantee among the trust-lacked objects in the IIoT. Summarily, ESPE under the edge-cloud architecture is over and above the given presentation as a milestone of exploring the highest capability of outsourcing secure data in the IIoT.