A Message Transmission Scheme for IoT Inspired by The Shamir Scheme and Based on The Hidden Number Problem (HNP)

Authors

Mohamed El Ouafi
Department of Mathematics, Pluridisciplinary Faculty, Mohammed First University, Laboratory of Applied Mathematics and Information Systems, Selouane, Nador, Morocco
Abderrahim Aslimani
Department of Mathematics, Pluridisciplinary Faculty, Mohammed First University, Laboratory of Applied Mathematics of the Oriental, BP 300-62700, Selouane, Nador,Morocco
Kaoutar Lamrini Uahabi
Department of Mathematics, Pluridisciplinary Faculty, Mohammed First University, Laboratory of Applied Mathematics and Information Systems, Selouane, Nador, Morocco
Abderrahim Zannou
ERCI2A, FSTH, Abdelmalek Essaadi University, Tetouan, Morocco
DOI: https://doi.org/10.52549/ijeei.v13i4.7366
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Published: 18-11-2025 • Issue: Vol. 13 No. 4: December 2025 • Section: Regular paper

Abstract

The Internet of Things (IoT) requires cryptographic mechanisms that are both lightweight and resistant to emerging attacks. Classical public-key protocols such as RSA or ElGamal, as well as many post-quantum lattice-based schemes, consume too many resources for devices with limited memory, processing power, and energy. In this study, we propose a three-pass message transmission protocol that avoids any prior key exchange. Inspired by Shamir’s keyless scheme and relying on the hardness of the Hidden Number Problem (HNP) and the Decisional Diffie–Hellman (DDH) assumption, the protocol operates in a finite field with safe primes and refreshes random masks at each execution, providing strong resistance to brute-force and small-subgroup attacks. We formally prove IND-CPA security and implement the HNP 3-pass scheme, showing that each pass executes in 1.4--1.8 ms on a workstation, with 132-byte public keys and 192--256-byte secret keys. Estimated energy consumption per iteration is 101.562 mJ. Comparative simulations on a workstation and embedded platforms (Arduino Uno and Raspberry Pi) against RSA-512, ECC (secp256r1/secp521r1), and post-quantum Kyber-512 show that our scheme achieves execution times comparable to ECC and Shamir’s 3-pass protocol, is significantly faster than RSA, and consumes less energy than Kyber-512. This combination of low latency, moderate key sizes, and energy efficiency highlights the practicality of the HNP 3-pass protocol for resource-constrained IoT environments
Keywords: lightweight cryptography; Internet of Things (IoT); Hidden Number Problem; secure transmission; keyless protocol

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Author Biographies

Mohamed El Ouafi
Mohamed El Ouafi is a researcher in the Laboratory of Applied Mathematics and Information Systems, Multidisciplinary Faculty of Nador, Mohammed First University, Morocco. His research interests include cryptography, Internet of Things (IoT) security, and blockchain technologies.
Abderrahim Aslimani
is a Professor in the Department of Mathematics, Multidisciplinary Faculty of Nador, Mohammed First University , Morocco. His research interests include stochastic analysis, probability theory, applied mathematics, partial differential equations, functional analysis, and measure theory.
Kaoutar Lamrini Uahabi
is a Professor in the \textit{Multidisciplinary Faculty of Nador}, \textit{Mohammed First University}, Morocco. She is a member of the \textit{Laboratory of Applied Mathematics and Information Systems,}. Her research interests include nonlinear analysis, fractal geometry, and their applications in medical imaging and complex systems. She is also interested in chaotic dynamical systems and the mathematical modeling of random phenomena.
Abderrahim Zannou
is a Professor at the Faculty of Sciences and Techniques of Al Hoceima, Abdelmalek Essaadi University, Morocco. His research interests include the Internet of Things (IoT), and blockchain technologies.

How to Cite

[1]
M. El Ouafi, A. Aslimani, K. Lamrini Uahabi, and A. Zannou, “A Message Transmission Scheme for IoT Inspired by The Shamir Scheme and Based on The Hidden Number Problem (HNP)”, IJEEI, vol. 13, no. 4, pp. 873–887, Nov. 2025, doi: 10.52549/ijeei.v13i4.7366.