Novel Way of Generating Random Numbers Using Lucas Sequence and Associated in ATM/Ecommerce for Secured Online Transactions

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-305
Published: Nov 28, 2025
Cite as:
R. Elumalai, G.S.G.N. Anjaneyulu, Novel Way of Generating Random Numbers Using Lucas Sequence and Associated in ATM/Ecommerce for Secured Online Transactions, Int. J. Anal. Appl., 23 (2025), 305.

Main Article Content

R. Elumalai, G.S.G.N. Anjaneyulu

Abstract

This paper proposes novel pseudorandom number generators (PRNGs) based on the Lucas sequence and the SHA3-512 hashing algorithm over the finite field Fp. We introduce one primary algorithm capable of generating random numbers up to 32 digits (256 bits) in length, suitable for highly confidential applications such as international communications, defense activities, and large monetary transactions. Additionally, three associated sub-algorithms produce fixed-length random numbers of 4, 6, and 8 digits (32, 48, and 64 bits, respectively), optimized for ATM and e-commerce transactions. Unlike existing PRNGs that rely on a single seed, our approach utilizes two seeds: a userprovided, context-specific seed, and a server-generated seed derived from the Lucas sequence over the Pell curve. The server-generated seed remains entirely outside user control, enhancing the security of the generated random numbers. The PRNG process involves hashing the sum of solutions to the Pell curve, converting the resulting hexadecimal hash output into binary, and extracting the first half of the 512 bits, which is then mapped to an integer over the field Fq to produce the random number. Statistical analysis using the Kolmogorov-Smirnov test confirms that the generated numbers follow a uniform distribution. Security analysis demonstrates resilience against various attacks, including direct cryptanalytic, input-based, iterative guessing, backtracking, and gap-filling attacks. These results suggest that the proposed PRNGs offer improved security and efficiency for applications in ATM operations and e-commerce.

Article Details

References

  1. D.E. Knuth, The Art of Computer Programming: Seminumerical Algorithms, Addison–wesley, 2014.
  2. Free Software Foundation, The GNU C Library (glibc) Manual, https://sourceware.org/glibc/manual.
  3. S.K. Park, K.W. Miller, Random Number Generators: Good Ones Are Hard to Find, Commun. ACM 31 (1988), 1192–1201. https://doi.org/10.1145/63039.63042.
  4. D. Crawford, Technical Correspondence, Commun. ACM 36 (1993), 105–110. https://doi.org/10.1145/159544.376068.
  5. P. L'Ecuyer, R. Touzin, Fast Combined Multiple Recursive Generators With Multipliers of the Form $a = pm 2^q pm 2^r$, in: Proceedings of the 2000 Winter Simulation Conference, pp. 683–689, 2000.
  6. M.E. O'Neill, PCG: A Family of Simple Fast Space-Efficient Statistically Good Algorithms for Random Number Generation, Technical Report, Harvey Mudd College, (2014). https://www.cs.hmc.edu/tr/hmc-cs-2014-0905.pdf.
  7. P. L’Ecuyer, Maximally Equidistributed Combined Tausworthe Generators, Math. Comput. 65 (1996), 203–213. https://doi.org/10.1090/s0025-5718-96-00696-5.
  8. P. L’Ecuyer, Tables of Maximally Equidistributed Combined LFSR Generators, Math. Comput. 68 (1999), 261–269. https://doi.org/10.1090/s0025-5718-99-01039-x.
  9. F. Panneton, P. L'Ecuyer, M. Matsumoto, Improved Long-Period Generators Based on Linear Recurrences Modulo 2, ACM Trans. Math. Softw. 32 (2006), 1–16. https://doi.org/10.1145/1132973.1132974.
  10. S. Vigna, A PRNG Shootout, https://prng.di.unimi.it.
  11. M. Matsumoto, T. Nishimura, Mersenne Twister, ACM Trans. Model. Comput. Simul. 8 (1998), 3–30. https://doi.org/10.1145/272991.272995.
  12. M. Saito, M. Matsumoto, Simd-Oriented Fast Mersenne Twister: A 128-Bit Pseudorandom Number Generator, Monte Carlo Quasi-Monte Carlo Methods 2006 (2008), 607–622. https://doi.org/10.1007/978-3-540-74496-2_36.
  13. M. Saito, M. Matsumoto, A Uniform Real Random Number Generator Obeying the IEEE 754 Format Using an Affine Transition, in: 8th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC '08), 2008.
  14. M. Saito, M. Matsumoto, A PRNG Specialized in Double Precision Floating Point Numbers Using an Affine Transition, Monte Carlo Quasi-Monte Carlo Methods 2008 (2009), 589–602. https://doi.org/10.1007/978-3-642-04107-5_38.
  15. S. Wolfram, Random Sequence Generation by Cellular Automata, Adv. Appl. Math. 7 (1986), 123–169. https://doi.org/10.1016/0196-8858(86)90028-x.
  16. P. Hortensius, R. McLeod, W. Pries, D. Miller, H. Card, Cellular Automata-Based Pseudorandom Number Generators for Built-In Self-Test, IEEE Trans. Comput. Des. Integr. Circuits Syst. 8 (1989), 842–859. https://doi.org/10.1109/43.31545.
  17. S. Das, B.K. Sikdar, A Scalable Test Structure for Multicore Chip, IEEE Trans. Comput. Des. Integr. Circuits Syst. 29 (2010), 127–137. https://doi.org/10.1109/tcad.2009.2034349.
  18. K. Bhattacharjee, D. Paul, S. Das, Pseudo-Random Number Generation Using a 3-State Cellular Automaton, Int. J. Mod. Phys. C 28 (2017), 1750078. https://doi.org/10.1142/s0129183117500784.
  19. M. Blum, S. Micali, How to Generate Cryptographically Strong Sequences of Pseudo Random Bits, in: Providing Sound Foundations for Cryptography: On the Work of Shafi Goldwasser and Silvio Micali, Association for Computing Machinery, 2019: 227–240. https://doi.org/10.1145/3335741.3335751.
  20. C.J. Ballot, H.C. Williams, The Lucas Sequences: Theory and Applications, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-37238-4.
  21. M.R. Rizqi, A. Waluyo, G.M. Edgy, S.U. Sunaringtyas, Enhanced Authentication Mechanism for Automated Teller Machine (ATM) Through Implementation of Soft Two-Factor Authentication, in: Proceedings of the 3rd International Conference on Electronics, Communications and Control Engineering, ACM, New York, NY, USA, 2020, pp. 3-7. https://doi.org/10.1145/3396730.3396735.
  22. M.M. Fazili, M.F. Shah, S.F. Naz, A.P. Shah, Next Generation QCA Technology Based True Random Number Generator for Cryptographic Applications, Microelectron. J. 126 (2022), 105502. https://doi.org/10.1016/j.mejo.2022.105502.
  23. K. Bhattacharjee, S. Das, A Search for Good Pseudo-Random Number Generators: Survey and Empirical Studies, Comput. Sci. Rev. 45 (2022), 100471. https://doi.org/10.1016/j.cosrev.2022.100471.
  24. B. Ryabko, Asymptotically Most Powerful Tests for Random Number Generators, J. Stat. Plan. Inference 217 (2022), 1–7. https://doi.org/10.1016/j.jspi.2021.07.007.
  25. L. Yao, X. Wu, H. Zhang, DCDRO:A True Random Number Generator Based on Dynamically Configurable Dual-Output Ring Oscillator, Integration 93 (2023), 102053. https://doi.org/10.1016/j.vlsi.2023.102053.
  26. A.A. Pandit, A. Kumar, A. Mishra, Lwr-Based Quantum-Safe Pseudo-Random Number Generator, J. Inf. Secur. Appl. 73 (2023), 103431. https://doi.org/10.1016/j.jisa.2023.103431.
  27. J. Polo, H. López, C. Hernández, Random Number Generator Based on a Memristive Circuit, Heliyon 10 (2024), e26635. https://doi.org/10.1016/j.heliyon.2024.e26635.
  28. Q. Fan, F. Ran, L. Yan, Multi-Bit Per Cycle True Random Number Generator Based on Xor-Xnor Ring Oscillator Unit, Microelectron. J. 146 (2024), 106142. https://doi.org/10.1016/j.mejo.2024.106142.
  29. Y. Chen, H. Liang, L. Zhang, L. Yao, Y. Lu, High Throughput Dynamic Dual Entropy Source True Random Number Generator Based on FPGA, Microelectron. J. 145 (2024), 106113. https://doi.org/10.1016/j.mejo.2024.106113.
  30. E.B. Barker, J.M. Kelsey, Recommendation for Random Number Generation Using Deterministic Random Bit Generators, NIST Special Publication 800-90A, NIST, (2012). https://doi.org/10.6028/nist.sp.800-90a.