# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021009
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## Formal security proof for a scheme on a topological network

 1 Department of Information Engineering, Computer Science, and Mathematics, University of L'Aquila, Via Vetoio, 67100 L'Aquila (AQ), Italy 2 Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy

* Corresponding author

Received  November 2020 Revised  February 2021 Early access April 2021

Fund Project: The authors are members of INdAM-GNSAGA (Italy). This work was partially supported by the Centre of EXcellence on Connected, Geo-Localized and Cybersecure Vehicles (EX-Emerge), funded by Italian Government under CIPE resolution n. 70/2017 (Aug. 7, 2017)

Key assignment and key maintenance in encrypted networks of resource-limited devices may be a challenging task, due to the permanent need of replacing out-of-service devices with new ones and to the consequent need of updating the key information. Recently, Aragona et al. proposed a new cryptographic scheme, ECTAKS, which provides a solution to this design problem by means of a Diffie-Hellman-like key establishment protocol based on elliptic curves and on a prime field. Even if the authors proved some results related to the security of the scheme, the latter still lacks a formal security analysis. In this paper, we address this issue by providing a security proof for ECTAKS in the setting of computational security, assuming that no adversary can solve the underlying discrete logarithm problems with non-negligible success probability.

Citation: Roberto Civino, Riccardo Longo. Formal security proof for a scheme on a topological network. Advances in Mathematics of Communications, doi: 10.3934/amc.2021009
##### References:
 [1] R. Aragona, R. Civino, N. Gavioli and M. Pugliese, An authenticated key scheme over elliptic curves for topological networks, preprint, arXiv: 2006.02147. To appear in Journal of Discrete Mathematical Sciences & Cryptography Google Scholar [2] D. Boneh, The decision Diffie-Hellman problem, Algorithmic Number Theory (Portland, OR), Lecture Notes in Comput. Sci., 1423, Springer, (1998), 48-63. doi: 10.1007/BFb0054851.  Google Scholar [3] W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory, 22 (1976), 644-654.  doi: 10.1109/tit.1976.1055638.  Google Scholar [4] S. Marchesani, L. Pomante, M. Pugliese and F. Santucci, Definition and development of a topology-based cryptographic scheme for wireless sensor networks, in Sensor Systems and Software, Springer International Publishing, (2013), 47-64. doi: 10.1007/978-3-319-04166-7_4.  Google Scholar [5] S. Marchesani, L. Pomante, F. Santucci and M. Pugliese, A cryptographic scheme for real-world wireless sensor networks applications, in Proceedings of the ACM/IEEE 4th International Conference on Cyber-Physical Systems, Association for Computing Machinery, 2013. doi: 10.1145/2502524.2502568.  Google Scholar [6] M. Pugliese, Managing Security Issues in Advanced Applications of Wireless Sensor Networks, Ph.D thesis, Department of Electrical Engineering and Computer Science, University of L'Aquila, 2008, available at https://mpugliese.webnode.it/_files/200000061-a7608a760b/24.%20phd_thesis.pdf. Google Scholar [7] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, Graduate Texts in Mathematics, 2009. doi: 10.1007/978-0-387-09494-6.  Google Scholar

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##### References:
 [1] R. Aragona, R. Civino, N. Gavioli and M. Pugliese, An authenticated key scheme over elliptic curves for topological networks, preprint, arXiv: 2006.02147. To appear in Journal of Discrete Mathematical Sciences & Cryptography Google Scholar [2] D. Boneh, The decision Diffie-Hellman problem, Algorithmic Number Theory (Portland, OR), Lecture Notes in Comput. Sci., 1423, Springer, (1998), 48-63. doi: 10.1007/BFb0054851.  Google Scholar [3] W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory, 22 (1976), 644-654.  doi: 10.1109/tit.1976.1055638.  Google Scholar [4] S. Marchesani, L. Pomante, M. Pugliese and F. Santucci, Definition and development of a topology-based cryptographic scheme for wireless sensor networks, in Sensor Systems and Software, Springer International Publishing, (2013), 47-64. doi: 10.1007/978-3-319-04166-7_4.  Google Scholar [5] S. Marchesani, L. Pomante, F. Santucci and M. Pugliese, A cryptographic scheme for real-world wireless sensor networks applications, in Proceedings of the ACM/IEEE 4th International Conference on Cyber-Physical Systems, Association for Computing Machinery, 2013. doi: 10.1145/2502524.2502568.  Google Scholar [6] M. Pugliese, Managing Security Issues in Advanced Applications of Wireless Sensor Networks, Ph.D thesis, Department of Electrical Engineering and Computer Science, University of L'Aquila, 2008, available at https://mpugliese.webnode.it/_files/200000061-a7608a760b/24.%20phd_thesis.pdf. Google Scholar [7] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, Graduate Texts in Mathematics, 2009. doi: 10.1007/978-0-387-09494-6.  Google Scholar
An example of ${\rm{ANT}}$, where red nodes represent ${\rm{ANT}}_{{i}}$
Target ANT
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