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Implicit Lightweight Proxy Based Key Agreement for the Internet of Things (ILPKA)

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Abstract

Due to the heterogeneity of devices available in the Internet of Things and the limitations of the resources connected to it, simplifying and lightening the algorithms used in the Internet of Things is an interesting area of study for researchers. One of the critical areas in the Internet of Things is the secret key establishment in communicating parties to make secure communication. The importance of communication security, on the one hand, and the need to create lightweight algorithms in the Internet of Things, on the other hand, have created an additional incentive to establish a lightweight key agreement protocol. Therefore, we present a lightweight method for key agreement in this article, while specifying the requirements of an appropriate key agreement protocol. To reduce the communication overhead, we implicitly perform operations of establishing the agreed key by sending a message. The results of analyzing the proposed method show a relative decrease compared to other methods concerning the number of main operations. Furthermore, the evaluation of the proposed method using formal tools indicates the success of establishing the agreed key, authentication of the communication parties, and also resistance to the relevant attacks.

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The authors have no relevant financial or non-financial interests to disclose.

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Author notes

  1. Rasoul Roustaei and Midia Reshadi have contributed equally to this work.

Authors and Affiliations

  1. Department of Computer Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Rasoul Roustaei & Midia Reshadi

  2. Department of Computer Engineering, Shahed University, Tehran, Iran

    Hamid Haj Seyyed Javadi

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  1. Rasoul Roustaei
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Correspondence to Hamid Haj Seyyed Javadi.

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Appendix A: The Implementation of the Proposed and Basic Algorithms

Appendix A: The Implementation of the Proposed and Basic Algorithms

In this section we present implementation of proposed and basic algorithms. this appendix contains supplementary information that may be helpful in providing a more comprehensive understanding of the research problem.these implementations formally verify the correctness of our proposed scheme by formally analyse the security goals of the scheme.

1.1 A.1: Basic Algorithm(ALPKA)

The ALPKA protocol [6] is a lightweight key agreement approach proposed in 2019 by Braeken et al. A formalized protocol consists of three components:

  1. 1.

    Initial assumptions (conditions that hold before the protocol starts);

  2. 2.

    Protocol steps (the messages exchanged between the principals);

  3. 3.

    Protocol goals (conditions that are expected to hold if the protocol terminates successfully).

1.1.1 A.1.1: Protocol Assumptions

$$\begin{aligned}{} & {} Express TTP possessions at time t0 \\{} & {} \quad Nx,Ny,Nz \end{aligned}$$
  1. A1:

    TTP possess at [0] Nx;

  2. A2:

    TTP know at [0] NOT (Zero possess at [0] Nx);

  3. A3:

    TTP possess at [0] Ny;

  4. A4:

    TTP know at [0] NOT (Zero possess at [0] Ny);

  5. A5:

    TTP possess at [0] Nz;

  6. A6:

    TTP know at [0] NOT (Zero possess at [0] Nz);

    $$\begin{aligned} Ai = H(Ui,Ny), Bi = H(Nz,Ny) XOR H(Ui,Ny), H(Nx) \end{aligned}$$
  7. A7:

    TTP possess at [0] H(Ui,Ny);

  8. A8:

    TTP know at [0] Ui possess at [0] H(H(Ui,Ny));

  9. A9:

    TTP possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  10. A10:

    TTP know at [0] Ui possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

    $$\begin{aligned} si = H(H(Ui,Ny),Nz) \end{aligned}$$
  11. A11:

    TTP possess at [0] H(H(Ui,Ny),Nz);

  12. A12:

    TTP know at [0] Ui possess at [0] H(H(Ui,Ny),Nz);

    $$\begin{aligned} Aj = H(Uj,Ny), Bj = H(Nz,Ny) XOR H(Uj,Ny),H(Nx) \end{aligned}$$
  13. A13:

    TTP possess at [0] H(Uj,Ny);

  14. A14:

    TTP know at [0] Uj possess at [0] H(H(Uj,Ny));

  15. A15:

    TTP possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  16. A16:

    TTP know at [0] Uj possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

    $$\begin{aligned} si = H(H(Uj,Ny),Nz) \end{aligned}$$
  17. A17:

    TTP possess at [0] H(H(Uj,Ny),Nz);

  18. A18:

    TTP know at [0] Uj possess at [0] H(H(Uj,Ny),Nz);

    $$\begin{aligned}{} & {} Express\; Ui\; possessions \;at \;time\; t0 \\{} & {} \quad Bi= XOR(H(Nz,Ny), \;H(Ui,Ny)), \;H(Ai)= H(H(Ui,Ny)),\; si= \\{} & {} \quad H(H(Ui,Ny),Nz); \end{aligned}$$
  19. A19:

    Ui possess at [0] Uj;

  20. A20:

    Ui possess at [0] Sk;

  21. A21:

    Ui possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  22. A22:

    Ui possess at [0] H(H(Ui,Ny));

  23. A23:

    Ui possess at [0] H(H(Ui,Ny),Nz);

  24. A24:

    Ui possess at [0] H(Nx);

  25. A25:

    Ui possess at [0] Nt;

  26. A26:

    Ui know at [0] NOT (ZERO possess at [0] Nt);

    $$\begin{aligned}{} & {} Express \;Uj \;possessions \;at\; time \;t0 \\{} & {} \quad Bj= XOR(H(Nz,Ny), H(Uj,Ny)), \; H(Aj)= H(H(Uj,Ny)),\; sj= H(H(Uj,Ny),Ny); \end{aligned}$$
  27. A27:

    Uj possess at [0] Ui;

  28. A28:

    Uj possess at [0] Sk;

  29. A29:

    Uj possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  30. A30:

    Uj possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  31. A31:

    Uj possess at [0] H(H(Uj,Ny));

  32. A32:

    Uj possess at [0] H(H(Uj,Ny),Nz);

  33. A33:

    Uj possess at [0] H(Nx);

  34. A34:

    Uj possess at [0] H(H(H(Uj,Ny),Ny), XOR(H(Nz,Ny), H(Ui,Ny)));

    $$\begin{aligned} Express \;Sk \;possessions \;at \;time \;t0 \end{aligned}$$
  35. A35:

    Sk possess at [0] H(Nz,Ny);

  36. A36:

    Sk possess at [0] H(Sk,H(Nx));

  37. A37:

    Sk know at[0] ( Sk receive at [1] XOR(XOR(H(Nz,Ny), H(Ui,Ny)), H(H(Sk,H(Nx)), H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny)))))) IMPLY Ui send at [1] XOR(XOR(H(Nz,Ny), H(Ui,Ny)), H(H(Sk,H(Nx)), H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny)))))));

  38. A38:

    Uj know at [0] (Uj receive at[2] XOR(H(XOR(H(Nz,Ny), H(Uj,Ny)), NtH(H(H(Uj,Ny),Ny),XOR(H(Nz,Ny), H(Ui,Ny)))), XOR(H(Nz,Ny), H(Ui,Ny))) IMPLY Sk send at[2] XOR(H(XOR(H(Nz,Ny), H(Uj,Ny)), NtH(H(H(Uj,Ny),Ny),XOR(H(Nz,Ny), H(Ui,Ny)))), XOR(H(Nz,Ny), H(Ui,Ny))));

  39. A39:

    Uj know at[0] ( Uj receive at [2] mH(Nt) IMPLY Ui send at [1] mH(Nt));

1.1.2 A.1.2: Protocol Steps

$$\begin{aligned} Step 1: C1, C2, C3, CIDi, C \end{aligned}$$
  1. S1:

    Sk receive at [1] XOR(H(Sk,H(Nx)),H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny)))));

  2. S1:

    Sk receive at [1] XOR(H(H(H(Ui,Ny)),XOR(H(Sk,H(Nx)),H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny)))))),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny))));

  3. S1:

    Sk receive at [1] XOR(H(XOR(H(Sk,H(Nx)),H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny))))), H(H(Ui,Ny))),XOR(H(Nz,Ny), H(Uj,Ny)));

  4. S1:

    Sk receive at [1] XOR(XOR(H(Nz,Ny), H(Ui,Ny)), H(H(Sk,H(Nx)), H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny))))));

  5. S1:

    Sk receive at [1] mH(Nt);

    $$\begin{aligned} Step 2: C4, C5, C \end{aligned}$$
  6. S2:

    Uj receive at [2] XOR(H(H(H(Uj,Ny)), mH(Nt)),NtH(H(H(Uj,Ny),Ny),XOR(H(Nz,Ny), H(Ui,Ny))));

  7. S2:

    Uj receive at [2] XOR(H(XOR(H(Nz,Ny), H(Uj,Ny)), NtH(H(H(Uj,Ny),Ny),XOR(H(Nz,Ny), H(Ui,Ny)))), XOR(H(Nz,Ny), H(Ui,Ny)));

  8. S2:

    Uj receive at [2] mH(Nt);

1.1.3 Protocol Goals

$$\begin{aligned}{} & {} Verify \;if \;the \;establishment \;of\; the \;session\; keys\; is \;done \\{} & {} \quad correctly\; (i.e. \;secrecy\; of\; the \;keys \;components \;and \;their\; possessions \\{} & {} \quad by\; the \;legitimate\; users \end{aligned}$$
  1. G1:

    Sk possess at [1] H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny))));

  2. G2:

    Sk possess at [1] XOR(H(Nz,Ny), H(Ui,Ny));

  3. G3:

    Sk possess at [1] H(Ui,Ny);

  4. G4:

    Sk possess at [1] H(H(H(Ui,Ny)),XOR(H(Sk,H(Nx)),H(XOR(H(Nz,Ny),H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny))))));

  5. G5:

    Sk possess at [1] NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny)));

  6. G6:

    Uj possess at [2] H(H(H(Uj,Ny)), mH(Nt));

  7. G7:

    Uj possess at [2] NtH(H(H(Uj,Ny),Nz),XOR(H(Nz,Ny), H(Ui,Ny)));

  8. G8:

    Uj possess at [2] H(XOR(H(Nz,Ny), H(Uj,Ny)), NtH(H(H(Uj,Ny),Nz),XOR(H(Nz,Ny), H(Ui,Ny))));

  9. G9:

    Uj possess at [2] H(H(H(Uj,Ny),Nz),XOR(H(Nz,Ny), H(Ui,Ny)));

  10. G10:

    Uj possess at [2] Nt;

  11. G11:

    Uj possess at [2] m;

    $$\begin{aligned}{} & {} authentication \;of \;users \\{} & {} \quad Sk \;authenticate\; Ui \end{aligned}$$
  12. G12:

    Sk know at [1] Ui send at [1] XOR(XOR(H(Nz,Ny), H(Ui,Ny)), H(H(Sk,H(Nx)), H(XOR(H(Nz,Ny), H(Ui,Ny)),NtH(H(H(Ui,Ny),Nz),XOR(H(Nz,Ny), H(Uj,Ny))))));

  13. G13:

    Uj know at [2] Sk send at [2] XOR(H(XOR(H(Nz,Ny), H(Uj,Ny)), NtH(H(H(Uj,Ny),Nz),XOR(H(Nz,Ny), H(Ui,Ny)))), XOR(H(Nz,Ny), H(Ui,Ny))); Uj authenticate Ui

  14. G14:

    Uj know at [2] Ui send at [1] mH(Nt);

  15. G15:

    AttackDetection;

1.2 A.2: The Implementation of Proposed Algorithm

The ILPKA protocol is a implicit lightweight proxy based key agreement for the Internet of Things. A formalized protocol consists of three components:

  1. 1.

    Initial assumptions (conditions that hold before the protocol starts);

  2. 2.

    Protocol steps (the messages exchanged between the principals);

  3. 3.

    Protocol goals (conditions that are expected to hold if the protocol terminates successfully).

1.2.1 A.2.1: Protocol Assumptions

$$\begin{aligned}{} & {} Express TTP(KDC) possessions at time t0 \\{} & {} \quad (Nx,Ny,Nz,rij,Ai,Bi,si,Aj,Bj,sj) \\ \end{aligned}$$
  1. A1:

    TTP possess at [0] Nx;

  2. A2:

    TTP know at [0] NOT (Zero possess at [0] Nx);

  3. A3:

    TTP possess at [0] Ny;

  4. A4:

    TTP know at [0] NOT (Zero possess at [0] Ny);

  5. A5:

    TTP possess at [0] Nz;

  6. A6:

    TTP know at [0] NOT (Zero possess at [0] Nz);

  7. A7:

    TTP possess at [0] XOR(H(H(H(Ui,Ny),Nz), XOR(H(Nz,Ny), H(Uj,Ny))),H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny))));

  8. A8:

    TTP know at [0] Sk possess at [1] XOR(H(H(H(Ui,Ny),Nz), XOR(H(Nz,Ny), H(Uj,Ny))),H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny))));

    $$\begin{aligned}{} & {} Components Produced by KDC for each Entity \\{} & {} \quad Ai = H(Ui,Ny), Bi = H(Nz,Ny) XOR H(Ui,Ny),H(Nx),si = H(H(U,Ny),Nz) \end{aligned}$$
  9. A9:

    TTP possess at [0] H(Ui,Ny);

  10. A10:

    TTP know at [0] Ui possess at [0] H(H(Ui,Ny));

  11. A11:

    TTP possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  12. A12:

    TTP know at [0] Ui possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  13. A13:

    TTP possess at [0] H(Nx);

  14. A14:

    TTP know at [0] Ui possess at [0] H(Nx);

  15. A15:

    TTP possess at [0] H(H(Ui,Ny),Nz);

  16. A16:

    TTP know at [0] Ui possess at [0] H(H(Ui,Ny),Nz);

    $$\begin{aligned} Aj = H(Uj,Ny), Bj = H(Nz,Ny) XOR H(Uj,Ny),H(Nx),sj = H(H(Uj,Ny),Nz) \end{aligned}$$
  17. A17:

    TTP possess at [0] H(Uj,Ny);

  18. A18:

    TTP know at [0] Uj possess at [0] H(H(Uj,Ny));

  19. A19:

    TTP possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  20. A20:

    TTP know at [0] Uj possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  21. A21:

    TTP know at [0] Uj possess at [0] H(Nx);

  22. A22:

    TTP possess at [0] H(H(Uj,Ny),Nz);

  23. A23:

    TTP know at [0] Uj possess at [0] H(H(Uj,Ny),Nz);

    $$\begin{aligned}{} & {} Express Ui possessions at time t0 \\{} & {} \quad Bi= XOR(H(Nz,Ny),H(Ui,Ny)), Bj,Sk H(Ai)= H(H(Ui,Ny)), si= H(H(Ui,Ny),Nz) \end{aligned}$$
  24. A24:

    Ui possess at [0] Sk;

  25. A25:

    Ui possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  26. A26:

    Ui possess at [0] H(H(Ui,Ny));

  27. A27:

    Ui possess at [0] H(H(Ui,Ny),Nz);

  28. A28:

    Ui possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  29. A29:

    Ui possess at [0] H(Nx);

  30. A30:

    Ui know at [0] TTP possess at [0] H(Nx);

  31. A31:

    Ui possess at [0] Nt;

  32. A32:

    Ui know at [0] NOT (ZERO possess at [0] Nt);

  33. A33:

    Ui know at [0] Uj possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  34. A34:

    Ui know at [0] Uj possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  35. A35:

    Ui know at [0] Uj possess at [0] Sk;

  36. A36:

    Ui know at [0] Sk possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  37. A37:

    Ui know at [0] Sk possess at [0] H(Sk,H(Nx));

  38. A38:

    Ui know at [0] Sk possess at [0] H(H(Ui,Ny));

    $$\begin{aligned}{} & {} Express Uj possessions at time t0 \\{} & {} \quad Bj= XOR(H(Nz,Ny), H(Uj,Ny)), H(Aj)= H(H(Uj,Ny)), sj= H(H(Uj,Ny),Nz); \end{aligned}$$
  39. A39:

    Uj possess at [0] Sk;

  40. A40:

    Uj possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  41. A41:

    Uj possess at [0] H(H(Uj,Ny));

  42. A42:

    Uj possess at [0] H(H(Uj,Ny),Nz);

  43. A43:

    Uj possess at [0] H(Nx);

  44. A44:

    Uj know at [0] TTP possess at [0] H(Nx);

  45. A45:

    Uj know at [0] Ui possess at [0] XOR(H(Nz,Ny), H(Ui,Ny));

  46. A46:

    Uj know at [0] Ui possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  47. A47:

    Uj know at [0] Ui possess at [0] Sk;

  48. A48:

    Uj know at [0] Sk possess at [0] XOR(H(Nz,Ny), H(Uj,Ny));

  49. A49:

    Uj know at [0] Sk possess at [0] H(Sk,H(Nx));

  50. A50:

    Uj know at [0] Sk possess at [0] H(H(Uj,Ny));

    $$\begin{aligned} Express Sk(Proxy) possessions at time t0 \end{aligned}$$
  51. A51:

    Sk possess at [0] H(Sk,H(Nx));

  52. A52:

    Sk possess at [0] H(Nz,Ny);

  53. A53:

    Sk possess at [1] XOR(H(H(H(Ui,Ny),Nz), XOR(H(Nz,Ny), H(Uj,Ny))),H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny))));

  54. A54:

    Sk know at [0] Ui possess at [0] H(Sk,H(Nx));

  55. A55:

    Sk know at [0] Uj possess at [0] H(Sk,H(Nx));

  56. A56:

    Sk know at [0] Ui possess at [0] H(Ui,Ny);

  57. A57:

    Sk know at [0] Uj possess at [0] H(Uj,Ny);

  58. A58:

    Sk know at [0] Ui possess at [0] H(H(Ui,Ny));

  59. A59:

    Sk know at [0] Uj possess at [0] H(H(Uj,Ny));

1.2.2 A.2.2 Protocol Steps

$$\begin{aligned} Step 1:Ui SEND (M1, M2, M3, M4, M5) to Proxy \end{aligned}$$
  1. S1:

    Sk receivefrom Ui at [1] XOR(XOR(H(Nz,Ny), H(Ui,Ny)), H(Sk,H(Nx)));

  2. S1:

    Sk receivefrom Ui at [1] XOR(XOR(Nt, H(H(H(Ui,Ny),Nz), XOR(H(Nz,Ny), H(Uj,Ny)))), H(H(Sk,H(Nx)),H(H(Ui,Ny))));

  3. S1:

    Sk receivefrom Ui at [1] H(XOR(H(Nz,Ny), H(Ui,Ny)),H(H(Ui,Ny)));

  4. S1:

    Sk receivefrom Ui at [1] XOR(XOR(H(Nz,Ny), H(Uj,Ny)), H(XOR(Nt, H(H(H(Ui,Ny),Nz), XOR(H(Nz,Ny), H(Uj,Ny)))),H(H(Ui,Ny))));

  5. S1:

    Sk receivefrom Ui at [1]mH(Nt);

    $$\begin{aligned} Step 2:Proxy SEND(M'1, M'2, M'3) to Uj \end{aligned}$$
  6. S2:

    Uj receivefrom Sk at [2] XOR(H(H(Uj,Ny)), XOR(Nt,H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny)))));

  7. S2:

    Uj receivefrom Sk at [2] XOR(H(H(H(Uj,Ny)), XOR(Nt,H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny))))), XOR(H(Nz,Ny), H(Ui,Ny)));

  8. S2:

    Uj receivefrom Sk at [2] mH(Nt);

1.2.3 A.2.3: Protocol Goals

$$\begin{aligned}{} & {} Verify if the establishment of the session keys is done \\{} & {} \quad correctly (i.e. secrecy of the keys components and their possessions \\{} & {} \quad by the legitimate users \end{aligned}$$
  1. G1:

    Sk possess at [1] H(H(Sk,H(Nx)), H(H(Ui,Ny)));

  2. G2:

    Sk possess at [1] XOR(H(Nz,Ny),H(Ui,Ny));

  3. G3:

    Sk possess at [1] H(H(Ui,Ny));

  4. G4:

    Sk possess at [1] XOR(Nt, H(H(H(Ui,Ny),Nz), XOR(H(Nz,Ny),H(Uj,Ny))));

  5. G5:

    Sk possess at [1] XOR(H(Nz,Ny),H(Uj,Ny));

  6. G6:

    Sk possess at [1] H(Uj,Ny);

  7. G7:

    Uj possess at [2] XOR(H(Nz,Ny), H(Ui,Ny));

  8. G8:

    Uj possess at [2] XOR(Nt,H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny))));

  9. G9:

    Uj possess at [2] H(H(Uj,Ny),Nz);

  10. G10:

    Uj possess at [2] (H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny)));

  11. G11:

    Uj possess at [2] H(H(H(Uj,Ny),Nz), XOR(H(Nz,Ny), H(Ui,Ny)));

  12. G12:

    Uj possess at [2] Nt;

  13. G13:

    Uj possess at [2] m;

  14. G14:

    Uj know at [2] Uj receivefrom Sk at [2] mH(Nt);

  15. G15:

    Sk know at [1] Sk receivefrom Ui at [1] mH(Nt);

  16. G16:

    AttackDetection;

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Roustaei, R., Javadi, H.H.S. & Reshadi, M. Implicit Lightweight Proxy Based Key Agreement for the Internet of Things (ILPKA). Wireless Pers Commun 130, 1833–1860 (2023). https://doi.org/10.1007/s11277-023-10360-0

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