3.1 Verifiable delay function (VDF)

3.1.1 VDF definition.

A verifiable delay function is a function f: X → Y that requires a specified wall clock time to compute even on a parallel processor and outputs a unique result that is effectively verified [26]. Even when evaluated on a massively parallel processor, the function still has to be evaluated in a series of consecutive steps [27]. Most importantly, given input x and output y, anyone can quickly verify that the output result y = f(x) is correct, where f: = X→Y satisfies the following requirements: (1)

The design of verifiable delay functions takes into account two key elements: delay and verifiability [28]. Latency refers to the time required to complete the computation, while verifiability refers to the verifiability of the output result. By designing functions that satisfy these two elements, verifiable delay functions are used in distributed applications in scenarios where time delays need to be added [29].

A verifiable delay function is usually defined as a tuple (N, x, T), where N is the input domain of the function, x is the input value, and T is the number of iteration steps. For each input tuple, the function outputs a unique result y, which can be verified as the correct output of the function f(x) under publicly verifiable conditions, as shown in Eq (2): (2)

Where N is the RSA modulus, which consists of the product of two large prime numbers p and q, i.e., N = p * q, x is a random seed belonging to , denoting a random value located in the range of modulus N. T is the time delay parameter, which is a non-negative integer. In addition, φ(N) = (p-1)(q-1) is an Eulerian function to denote the order of modulus N.

In addition, the triple algorithm for the verifiable delay function consists of three steps of Setup, Eval and Verify as shown in Fig 1 Algorithmic flow of the verifiable delay function.

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Fig 1. Algorithmic flow of the verifiable delay function.

https://doi.org/10.1371/journal.pone.0302325.g001

Setup(λ, T) → pp: = (ek, vk) is a stochastic algorithm with inputs the delay parameter T and the security parameter λ, outputs the public parameter pp consisting of the evaluation key parameter ek and the authentication key parameter vk. The algorithm is under default assumptions required to guarantee a certain security strength.

For avoiding the impact on security, the running time of the algorithm should be limited to an acceptable range. Thus, a suitable running time based on the security parameter λ has to be chosen [30]. Also, choosing random numbers is an important issue, which should be secret and sufficiently random to ensure security.

Eval(ek, x) → (y, π) is a slow cryptographic algorithm that accepts as input the evaluation key parameter ek and a random seed x and outputs the result y along with a possibly null proof π. In order to ensure sequentiality, the Eval algorithm has to complete in a specified amount of time and needs to satisfy the requirements of a polynomial logarithmic parallel processor.

Verify(vk, x, y, π) → {accept, reject} is a deterministic cryptographic algorithm that accepts as input a verification key parameter vk, a random seed x, an output value y, with a possibly null proof π. The Verify algorithm is a simple, simple, and easy to use algorithm. It outputs accept if y is equal to f(x) and reject otherwise. The Verify algorithm runs much faster than the Eval algorithm, which takes a total time polynomial in log(T) and λ to complete its execution.

3.1.2 VDF in blockchain.

Existing centralized timestamping service methods suffer from a single point of failure in the blockchain structure, where users need to assume that the timestamping server is secure and trustworthy [31]. In contrast, timestamping data on the chain can be overly burdensome for users. In addition, the existing distributed methods must provide timestamps through a set of voluntary issuers. Users require real-time interactions with all issuers to guarantee the security of outsourced data. This not only increases the communication burden of users but also increases the cost overhead of all issuers [32].

To solve the above problems, a blockchain-distributed timestamping model architecture with a continuously verifiable delay function can be used, as shown in Fig 2 Distributed timestamping model for blockchain based on continuously verifiable delay function, which does not rely on any third-party trusted entity and uses the blockchain-distributed network for storage.

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Fig 2. Distributed timestamping model for blockchain based on continuously verifiable delay function.

https://doi.org/10.1371/journal.pone.0302325.g002

The model is divided into three main layers: hash calculation service layer, distributed timestamp service layer, and blockchain layer.

Hash calculation service layer: The main function of this layer is to provide hash calculation for the input data and return its hash value. In this layer, existing hash functions or hash algorithms can be used to complete the calculation without writing their hash functions or implementing hash algorithms [33]. Hash computing services have a very important role in data security, encryption, and integrity verification, which helps users achieve data security and integrity assurance [34].

Distributed Timestamp Service Layer: This layer adopts a precise time source and a high-intensity and high-standard security mechanism, which is used to confirm the existence of the data processed by the system and the relative time sequence of the relevant operations, providing a basic service for the prevention of time repudiation and tampering in the information system [35]. The timestamp server is a timestamp authority system based on public key infrastructure technology, providing accurate and trustworthy timestamp service for all users. In the past, the scheme of providing timestamp service to data through a single timestamp server easily led to collusion between users and timestamp service providers, failing to ensure the security of data adequately [36]. Therefore, the distributed timestamping service layer adopts the blockchain distributed timestamping scheme based on a continuous verifiable delay function to provide timestamping service, and maintains the timestamps of the data through multiple timestamping servers together. In this process, the main timestamp server is responsible for providing a time synchronization service for the whole system.

Blockchain Layer: This layer is the core of the whole model, which utilizes the characteristics of blockchain to store and protect timestamp information. The decentralized feature of blockchain can prevent a single point of failure and ensure the availability and stability of timestamp service [37]. In addition, the transparency and non-comparability of the blockchain can also ensure the trustworthiness and security of the timestamp [38].

3.2 Smart contract

3.3 Quantum computers and anti-quantum cryptography

4.1 Design objectives and requirements

Traditional public key cryptosystems have taken a huge hit with the rapid development of quantum computers, as large integer factorization and discrete logarithm problems become solvable on quantum computers [65]. This also means that the elliptic curve-based digital signature regimes used in blockchain are no longer secure against quantum computing attacks [66]. In addition, researchers have pointed out that the user’s identity information in blockchain transactions is "pseudo-anonymity", which is a problem that plagues the development of blockchain technology. Therefore, this paper proposes a fast certificate-less signature algorithm based on lattice to solve the problem of user identity privacy leakage in blockchain transactions [67].

4.2 Infrastructure framework

The overall construction of user identity privacy and security against quantum computing attacks security question in the post-quantum blockchain anonymous transaction scheme constructed using the The Lattice Cryptography-based Certificate-Less Rapid Signature (LWE-CLRS) algorithm is shown in Fig 3. Architecture of fast signature algorithms for quantum-resistant blockchains.

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Fig 3. Architecture of fast signature algorithms for quantum-resistant blockchains.

https://doi.org/10.1371/journal.pone.0302325.g003

As shown in the figure, the user needs to send a transaction request to the supplier who provides the resources for this transaction to make a transaction with him. To ensure the security and privacy of the transaction, the LC-LRS algorithm is utilized for ring signing. First, the user requests the public parameters and a portion of his/her public-private key pair from the key generation centre (KG). The user has to determine the appropriate number of ring membership, the more number, the higher the privacy of the transaction, but the computational and verification complexity of the transaction will also increase. The user utilizes the LC-LRS algorithm to sign the transaction quickly and ensure the transaction’s validity. Therefore, the user sends this transaction request to the supplier. Eventually, the supplier receives the transaction request, performs confirmation to verify whether the transaction is legitimate and writes the transaction record in the blockchain, thus completing the entire transaction process.

4.3 Detailed steps of the algorithm

This subsection details a fast certificate-less signature algorithm based on lattice cryptography(LWE-CLRS).

System Initialization: The input parameters are n, a prime number q greater than 3, and m = 0 (), L > 0 (), 0 ≤ σ < . KG (Key Generation Center) generates the system parameters P and the master key MSK in the manner described in the System Initialization section. The system parameters P consist of the public and key parameters, and the master key MSK is used to generate the user’s private key. The master key MSK is used to generate the user’s private key. Execute the trapdoor generation algorithm so as to output the matrix A and lattice cipher T_A as shown in Eq 3.

(3)

Next, three secure hash functions are selected,as shown in Eqs (4)–(6), and the final output is the public parameter P and the system master key MSK = T_A.

(4)

(5)

(6)

Key Generation: Key generation needs to be performed in the KG, which generates the private key sk and public key pk for the user based on the public parameter P and the master key MSK.

KG uses the matrix original image sampling algorithm when generating partial private keys for users with identity IDs. This algorithm is used to generate matrices that satisfy specific constraints to ensure that the generated private key meets the requirements of security and certificate-free linkability, as shown in Eq (7). (7) where , , and satisfies . then KG generates the user partial private key , thus KG along with the selection matrix ,and sends the and A′ to the user.

The secret matrix is randomly selected and saved by the user himself or generated and sent to the user by the KG.

Since m > n, if the user wants to generate a public-private key pair, randomly select the n rows of the matrix and transpose to get the matrix,thus calculating A″, A_ID and S_ID, as shown in Eqs (8)–(10).

(8)

(9)

(10)

Where , , , I is the unit matrix. The final computation yields ,which outputs the user’s public key A_ID, and private key S_ID.

Fast Signature: Set the user of fast signature as U = {ID1, ID2, ID3,…, IDl}, public key as pk = {A_ID1, A_ID2, A_ID3,…, A_IDl}, l as the maximum number of members. μ is the surrogate signature message and IDi expresses the signer identification already, therefore the formula for fast signature is shown in Eq (11).

(11)

Verification: Input membership information U and pk are verified, if all i∈[l], have and then the condition holds to pass the verification, otherwise the fast signature is invalid.

4.4 Security analysis of algorithms

The LWE-CLRS algorithm is a state-of-the-art cryptographic signature algorithm whose security is based on the LWE difficulty problem, which is the difficulty of approximating the solution of a linear problem in a system of polynomial equations. The difficulty of the LWE problem has been studied and verified many times and is considered relatively secure.

The LWE-CLRS algorithm provides aggregation security and chaining security whereby the signatures of multiple signers are aggregated and guaranteed to be equivalently secure. This aggregation security is achieved by a rational scheme design and is related to the chain length of the signatures. Various secure aggregation schemes have been shown to be feasible, allowing the algorithm to provide high-strength security in practical applications.

The chain structure is one of the core features of the LWE-CLRS algorithm, which allows chain linking of ring signatures, thus enabling fine-grained authorization management and security guarantees. The security of the chain structure requires maintaining the relevance of the signatures and needs to have the aggregation security of the chain structure. Such a chain structure can provide more flexible signing and verification methods.

In addition, the LWE-CLRS algorithm supports revocability, which allows the signer to revoke a previously published signature without revealing the private key. This feature is realized through the support of a revocation key, which needs to be kept private and unbreakable. Such a mechanism allows the algorithm to provide a secure solution in cases where a specific signature needs to be revoked.

5.1 Smart contract platform selection

Ether is a great platform to consider when choosing a smart contract platform. Ethernet is one of the most popular and widely used smart contract platforms, with an extensive developer community and a mature ecosystem of tools. Writing smart contracts using the Solidity programming language leverages Ethernet’s vast ecosystem for complex distributed applications.

Second, Ethernet is one of the leading scalable smart contract platforms, offering multiple solutions to address network scaling. Layer-2 solutions, such as Rollups and Sidechains, can significantly improve network throughput and performance. In addition, EtherNet is also working on Ethereum 2.0, a new, more scalable blockchain platform that will dramatically increase EtherNet’s performance and throughput.

Beyond that, Ether has many advantages. For example, it provides a wide range of development tools, debuggers and libraries, which can significantly improve the efficiency and quality of smart contract development. Ethernet also supports token issuance and management based on standards such as ERC-20, ERC-721, and ERC-1155, which means that assets can be transferred and exchanged more easily.

Additionally, Ether is committed to developing a community ecosystem that enables Ether smart contracts to interact with other Ether-compatible blockchains through EVM (Ether Virtual Machine) compatibility. This interoperability enables more application scenarios, such as leveraging the DeFi protocol, non-homogenized token (NFT) markets, etc.

5.2 Smart contract deployment

The identity management aspect can be done by the smart contract through the agent layer. When an identity (an identity on the chain or a user’s identity) is registered, the smart contract can index it and perform identity logout or update operations. However, as the number of identities increases, the identity information table maintained by the agent layer becomes enormous, which makes the management of identity lookup more and more inconvenient.

For improving the efficiency of identity lookup, the Bloom filter algorithm can be considered. Bloom filter is a fast and efficient data structure for checking whether an element exists in a set. Using an array of bits along with several hash functions, it determines whether an identity exists or not by hashing the identity to several locations in the bit array and checking whether these locations have been set, as shown in Algorithm 1.

Algorithm 1: Trusted Identity Search

Input:BF = {0}ⁿ, identifier[m]→ Array of query identity collections. id → Identity to be queried

 i = 0

 // Loop over the m identity sets identifier to be queried

 While(i<m){

  j = 0;

  // For each identity identifier[i], compute the hash value by h_j(identifier[i]) and set the corresponding BF array element to 1 to indicate that the identity may exist.

  While(i< = k){

  BF[ℎ_j(identifier[i])] = 1;

  j = j+1;}

  i = i+1;}

 //Find id

 For i in range(0, k+1):

  // Iterate through the BF array, if there exists any BF array element of 0, it means the corresponding identity does not exist, return False

  IF (h_i(identifier))! = 1:

   Return False

  //If all the BF array elements are 1, it means all the identities to be queried exist, return True.

  Return True

5.3 Integration of smart contracts with signature algorithms

In the case of organizing nodes on multiple heterogeneous chains into an identity agent layer P2P network, a trusted identity management and message authentication mechanism is established between different chains using on-chain transactions. This is achieved through smart contracts, which are used to manage the identity certificates of the nodes on the identity agent layer as well as the blockchains they belong to, as show in Fig 4 Architecture diagram for the integration of smart contracts and signature algorithms.

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Fig 4. Architecture diagram for the integration of smart contracts and signature algorithms.

https://doi.org/10.1371/journal.pone.0302325.g004

First, a smart contract is created on each chain for identity management. This smart contract maintains the identity information of each node, including the identity book and the corresponding public key. When a node needs to join the identity agent layer network, it registers its identity through on-chain transactions. It stores its identity book and public key in the corresponding smart contract.

When inter-chain communication is performed, the agent layer nodes of the participating nodes will first perform inter-chain identity authentication. The authentication is performed through the identity information in the smart contract. For example, when user A needs to communicate with user B, the agent layer nodes of their respective chains query each other’s identity information, including identity certificates and public keys, through the smart contract. The node then signs the message using the appropriate signature algorithm and performs message verification to ensure that the other user is from a legitimate blockchain network.

To enhance security, smart contracts are integrated with signature algorithms. In the smart contract, the relevant verification logic and signature algorithm are defined to ensure the authenticity and integrity of the message. In this way, when a node makes a transaction on the chain, the smart contract verifies and determines whether the transaction is legitimate or not. Meanwhile, the signature algorithm is used to sign the transaction and the message to ensure the identity of the sender of the message and the integrity of the message.

6.1 Experimental settings

The effectiveness of the scheme proposed in this paper is tested through simulation experiments to verify whether the scheme proposed in this paper is capable of meeting the resistance to quantum computation in cross-chain communication scenarios. The experiments in this paper are run on Ubuntu 21.10 operating system. During the experiments, the system is utilized to achieve the fusion of blockchain smart contracts that are resistant to quantum computing attacks, with the application configuration shown in Table 1.

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Table 1. Experimental environments.

https://doi.org/10.1371/journal.pone.0302325.t001

In this experiment, the following configurations and tools were used: an AMD Ryzen 9 5900HX processor was used for the CPU, and the RAM was 32 GB. For the operating system, Ubuntu 21.10 was chosen as the base platform. For the programming of smart contracts, the Solidity platform was used, while the writing of consensus algorithms was done on the Python platform. As the blockchain platform, HyperLedger Fabric was chosen. The above combination of configurations and tools will provide a stable and efficient operating environment for the experiments in order to facilitate the related simulation work.

6.2 Speed of each stage

In the simulation system in this paper, transactions are performed in a "serial" fashion. Whenever a transaction is created, a signature and a verification result are required. For each simulation, 1000 different experimental tests were conducted to obtain the durations of signatures, verifications, and transactions. This experimental design provides a more comprehensive understanding of the performance of the system for evaluation, as shown in Table 2.

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Table 2. Time for each phase of a digitally signed blockchain transaction.

https://doi.org/10.1371/journal.pone.0302325.t002

According to the results in Table 2, the durations of each phase are all at the microsecond level, which means that the performance of the whole blockchain system system is less affected by the fact that the transactions are done in a "serial" way. However, the serial approach clearly shows the specific flow of each transaction, including the signature and verification process, which is easy to understand and debug. Each transaction is executed in a precise sequence without the complexity and competitive conditions associated with concurrent operations.

The result indicates that the algorithm is extremely efficient in processing transactions with fast response capability. Since the duration of each stage is very short, the entire transaction process can be completed quickly, thereby not creating a significant delay to the processing speed of other transactions. Thus, the current blockchain system has excellent efficiency in processing transactions.

6.3 Security assessment

Simultaneously, in this paper, malicious nodes that become KG nodes are tested. These KG nodes have a reputation value of 50 in the initial stage, which increases significantly as they continue to work honestly. However, if the KG nodes start to perform malicious behaviors, Fig 5 Results of the 50 rounds of voting on the evil KG node shows that their reputation value will drop dramatically. When the reputation value falls below a predetermined threshold, the KG node will be stripped of its identity and the system will initiate the process of re-voting for election.

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Fig 5. Results of the 50 rounds of voting on the evil KG node.

https://doi.org/10.1371/journal.pone.0302325.g005

The mechanism imposes severe penalties on nodes with high scores but evil behaviors, while rewarding nodes with well-intentioned behaviors handsomely. The entire blockchain encourages nodes to cooperate with each other for overall stable operation, while taking a firm stance against malicious nodes to create a safe operating environment.

6.4 Experimental comparison

In this section, the efficiency of the cross-chain communication system is tested in simulation comparison. In cross-chain communication, various entities interact with information, which mainly involves smart contracts and identity fast digital signatures on the blockchain. Owing to the redundancy of the blockchain system, each agent layer node backs up the identity data. However, an increase in the number of user node authentication and verification requests, or an increase in the number of blockchain networks, leads to more hashing operations by the agent layer nodes.

As shown in the Fig 6 Number of cross-chain communication process operations, the BSCTQCAT algorithm proposed in this paper outperforms the other two algorithms in terms of system efficiency through multiple runs of the test for fixed-size files. Therefore, the scheme in this paper can ensure the timeliness of the operation and the throughput of the system, and side by side, it highlights the stability.

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Fig 6. Number of cross-chain communication process operations.

https://doi.org/10.1371/journal.pone.0302325.g006

7.1 Application scenarios

A transaction is defined as an exchange of value between parties or multiple parties using currency as a medium. Transactions are realized by transferring digital assets representing monetary value from the sender’s ledger to the receiver’s ledger. As with traditional banking transactions, each transaction in the blockchain must clearly contain information such as the address where the transaction was generated, where the transaction was sent out, and the amount of the transaction. Through the transaction, a digital currency is created and the digital currency is transferred and tracked, which is one of the core features of the blockchain.

Each transaction in the blockchain requires a clear record of the sender, the receiver, the transaction amount and the associated transaction-generating address and transaction-output address information. This information makes up the transaction record, through which the transaction is monitored and traced, thus guaranteeing the security of the transaction. During the transaction process, the amount of digital currency will change with the transaction, and sometimes, it is necessary to carry out a change operation. The quantity and amount of the transaction, as well as the way to make change, need to be described and recorded in detail in the transaction record.

7.2 Analysis of case studies

7.2.2 Implementation strategies.

In a blockchain system, a wallet can be considered as an account which is composed of a signature mechanism, a public key, public key address mechanism. The digital signature mechanism is used to uniquely authenticate the transaction initiated by the account to ensure the authenticity and integrity of the transaction. The public key, on the other hand, is used to verify the validity and authenticity of that transaction during the transaction process. Meanwhile, the public key address is the address that must be used by other account nodes to transfer money to this account node.

In a wallet, the wallet holder creates his/her own public and private keys and associates them with his/her account using a digital signature. Other nodes in the network can then use this public key to validate transactions initiated by the account and confirm their authenticity. The public key address is a unique address derived from the data processing of the public key, which is used by other nodes to transfer money to the account, as shown in Fig 7. The wallet structure of the blockchain.

In addition to verifying the authenticity and uniqueness of the transaction, the signature mechanism and the public key address mechanism also guarantee the security of the transaction. The transaction is encrypted while the initiator of the transaction signs the transaction using a digital signature, thus guaranteeing the security of the transaction process.

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Fig 7. The wallet structure of the blockchain.

https://doi.org/10.1371/journal.pone.0302325.g007

8.1 Major findings and conclusions

In this paper, a blockchain smart contract technology based on resistance to quantum computing attacks is proposed to address the cross-chain communication, data sharing and security challenges faced by current blockchain technology. The technique first introduces a digital signature scheme of lattice cyphers to improve the ability to resist quantum search algorithm attacks. Second, it uses a smart contract authentication scheme to organize nodes on multiple heterogeneous chains to construct an identity agent layer P2P network. Data sharing and establishing a trusted identity management and message authentication mechanism between different chains are realized through transactions, effectively facilitating cross-chain communication.

8.2 Limitations and perspectives

With the development of quantum computing technology, blockchain technology faces more and more severe security challenges. Therefore, the proposal in this paper is in line with the current technology trend. In future research, the standardization of cross-chain communication and data sharing also has to be deeply explored and promoted for promoting interoperability among different blockchains, achieving a larger range of data exchange to enhance the network effect, and further realizing the application of blockchain technology.