Implementation of Elliptic Curves in the Polynomial Blom Key Pre-Distribution Scheme for Wireless Sensor Networks and Distributed Ledger Technology
- We propose a modified scheme that can be used for any arbitrary ;
- We discuss and prove the security of the proposed scheme against the coalition of the sensor nodes.
2. Literature Review
2.1. Application of Blom’s Scheme in WSN
2.2. Application of Blom’s Scheme in DLT
3.1. Greatest Common Divisor
3.2. Elliptic Curves
- Let H be any subgroup of G. The order of H divides the order of G;
- Let . The order of g divides the order of G.
3.3. Lagrange Interpolation Polynomial
4. Materials and Methods
4.1. Blom’s Key Pre-Distribution Scheme
- Before the deployment of the sensor nodes, an offline key distribution center will first construct a matrix C over a finite field of order q, where q is a prime power and . Matrix C is publicly known, which means any sensors and adversaries are allowed to know C. Let be the column of matrix C, where . Note that is a -tuple over . Column is assigned to node .
- Then, the key distribution center will create a random symmetric matrix D over . Matrix D must be kept secret, which means any sensors and adversaries are not allowed to know D.
- The key distribution center will compute an matrix M such that , where is the transpose of .
- Let be the i row of matrix M, where . Note that is a -tuple over . The key distribution center will then give to node over a secure channel.
- Let us say node wants to communicate with node . Both will compute the same key, as follows:
- Node will compute , which we call . Note that is a single element in .
- Node will compute , which we call . Note that is a single element in .
4.2. Scalar Multiplication on Elliptic Curves over Finite Field
4.3. Proposed Algorithm for Arbitrary
- Let p be a prime greater than 3, and p is publicly known to all. Let N be the total number of nodes. Let be the s node where . The trusted authority (TA) chooses an elliptic curve E over prime field such that
- is the set of integers modulo . TA selects an element for node , which is also made public such that for .
- Let . For , the TA chooses random points where such that and forms the polynomialThe chosen and the polynomial f above are privately known only by the TA. The polynomial f is symmetric, i.e., . The number of points to be chosen by the TA depends on the value of , where
- For each node , the TA computesThe TA then privately sends to node over a secure channel. Note that node only knows and does not know the coefficients . is privately known only by the TA and node . Note that the scalar of each point can be reduced to modulo based on Theorem 3.
- If two nodes, and , want to communicate with each other, they individually compute the common key (shared key), (the same as ), where node computesNote that .
4.4. Example for
- Let and p be publicly known to all. TA chooses an elliptic curve E over prime field such that
- Note that , which is a prime. TA selects an element for node , which is also made public such that for , as shown below.
- Let . For , the TA chooses random points where such that , as shown below.The TA then forms the secret polynomial , as shown below.
- For node , the TA computesFor node , the TA computesFor node , the TA computesThe TA then privately sends , , and to nodes , and , respectively, over a secure channel. Note that the scalar of each point can be reduced to modulo based on Theorem 3.
- If and want to communicate with each other, they individually compute the common key (shared key), (the same as ), where node computesNote that .
- is symmetric, i.e., ;
- For , it holds that ;
- has a degree of at most in both x and y, since has a degree of at most in both x and y and
6.1. The Application of the Proposed Scheme in Hyperledger Fabric DLT
6.2. The Pros and Cons
- The proposed scheme has high connectivity, which means all nodes in the network are able to compute the shared keys among each other. In other words, the probability of sharing keys between nodes is 1.
- The proposed scheme has high scalability, which means our proposed scheme can be used in networks with a huge number of nodes.
- The proposed scheme supports the mobility of a node as long as the identities or the public keys of the new neighboring nodes are already stored in the moving node.
- The information needed to establish the shared keys are stored in the nodes before deployment by an offline TA. Therefore, an adversary cannot attack the TA to obtain the secret polynomial and the information required to compute the shared keys.
- Elliptic curves are used in the proposed scheme, which increases the complexity of the calculation. Scalar multiplication and the group law of elliptic curves are implemented, instead of just adding and multiplying integers.
- This scheme can be implemented in WSNs, and also possibly in DLT technology, as discussed earlier.
- The proposed scheme does not support the flexibility requirement. In other words, if our proposed scheme is implemented in a network, then new joining nodes cannot simply join the network, since the identities or the public keys of the new nodes were not distributed in the existing nodes before deployment.
- The proposed scheme is not secure against the capture of nodes. However, Albakri et al.  mentioned that there are several security mechanisms that can be utilized to eliminate this problem, such as tamper-proof mechanisms to protect the information in the nodes from an attacker, even if the attacker captures the nodes.
6.3. The Comparison of the Proposed Scheme with Other Existing Schemes
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
|WSN||wireless sensor network|
|DLT||distributed ledger technology|
|MSP||membership service provider|
|MRD||maximum rank distance|
|MDS||maximum distance separable|
|PKI||public key infrastructure|
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|Finite field with p elements|
|N||The total number of nodes|
|A positive integer|
|gcd||The greatest common divisor of integers a and b|
|E||An elliptic curve defined over field K of form where|
|The set of points with coordinates in some field L on E, i.e.,|
|The number of points on E|
|P||A point on E|
|Point at infinity on E|
|A secret symmetric bivariate polynomial known only by the trusted authority|
|The node where|
|The public key of node used in the proposed scheme where|
|, i.e., secret information given to node where|
|, i.e., the shared key between node and node where ,|
|Resilience||Capturing devices must have a minimal impact on the network security|
|Connectivity||The probability of sharing keys between nodes must be maximum|
|Mobility||Moving devices must share keys with their new neighbors|
|Flexibility||Devices must be able to join or leave the network at any time|
|Scalability||Increasing the network size must not degrade performance|
|Scheme||Resilience||Connectivity||Mobility||Flexibility||Scalability||Use of Elliptic Curves||Value of|
|Blom ||1||Yes (within network)||No||High||No||Any positive integer|
|Lazos ||1||Yes (within network)||No||High||No||Any positive integer|
|Khan et al. ||N (total number of nodes)||1||Yes (within network)||Yes||High||No||Any positive integer|
|Wang et al. ||1||Yes (within network)||Yes||High||No||Any positive integer|
|Udin et al. ||Unknown||1||Yes (within network)||No||High||Yes||1|
|Our scheme||1||Yes (within network)||No||High||Yes||Any positive integer|
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Antony, S.N.F.M.A.; Bahari, M.F.A. Implementation of Elliptic Curves in the Polynomial Blom Key Pre-Distribution Scheme for Wireless Sensor Networks and Distributed Ledger Technology. J. Sens. Actuator Netw. 2023, 12, 15. https://doi.org/10.3390/jsan12010015
Antony SNFMA, Bahari MFA. Implementation of Elliptic Curves in the Polynomial Blom Key Pre-Distribution Scheme for Wireless Sensor Networks and Distributed Ledger Technology. Journal of Sensor and Actuator Networks. 2023; 12(1):15. https://doi.org/10.3390/jsan12010015Chicago/Turabian Style
Antony, Siti Noor Farwina Mohamad Anwar, and Muhammad Fatihin Afiq Bahari. 2023. "Implementation of Elliptic Curves in the Polynomial Blom Key Pre-Distribution Scheme for Wireless Sensor Networks and Distributed Ledger Technology" Journal of Sensor and Actuator Networks 12, no. 1: 15. https://doi.org/10.3390/jsan12010015