International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 10, No. 1, 8 Feb. 2024
Cover page and Table of Contents: PDF (size: 478KB)
Author(s)
Index Terms
Trapdoor function, cryptography, integer factorization, RSA, ElGamal, Pollard Rho
Abstract
At the bedrock of cryptosystems lie trapdoor functions, serving as the fundamental building blocks that determine the security and efficacy of encryption mechanisms. These functions operate as one-way transformations, demonstrating an inherent asymmetry: they are designed to be easily computable in one direction, while proving computationally challenging, if not infeasible, in the opposite direction. This paper contributes to the evolving landscape of cryptographic research by introducing a novel trapdoor function, offering a fresh perspective on the intricate balance between computational efficiency and security in cryptographic protocols.
The primary objective of this paper is to present and scrutinize the proposed trapdoor function, delving into a comprehensive analysis that unveils both its strengths and weaknesses. By subjecting the function to rigorous examination, we aim to shed light on its robustness as well as potential vulnerabilities, contributing valuable insights to the broader cryptographic community. Understanding the intricacies of this new trapdoor function is essential for assessing its viability in practical applications, particularly in securing sensitive information in real-world scenarios.
Moreover, this paper does not shy away from addressing the pragmatic challenges associated with deploying the proposed trapdoor function at scale. A thorough discussion unfolds, highlighting the potential hurdles and limitations when attempting to integrate this function into large-scale environments. Considering the practicality and scalability of cryptographic solutions is pivotal, and our analysis strives to provide a clear understanding of the circumstances under which the proposed trapdoor function may encounter obstacles in widespread implementation.
In essence, this paper contributes to the ongoing discourse surrounding trapdoor functions by introducing a new entrant into the cryptographic arena. By meticulously exploring its attributes, strengths, and limitations, we aim to foster a deeper understanding of the intricate interplay between cryptographic theory and real-world applicability.
Cite This Paper
Awnon Bhowmik, "An Unorthodox Trapdoor Function", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.10, No.1, pp. 31-38, 2024. DOI: 10.5815/ijmsc.2024.01.04
Reference
[1]Policy, vol. 20, no. 4, pp. 231-242, 2007.
[2]A. C. Yao, "Theory and application of trapdoor functions," in 23rd Annual Symposium on Foundations of Computer Science (SFCS 1982), IEEE, 1982, pp. 80-91.
[3]E. Ochoa-Jim'enez, L. Rivera-Zamarripa, N. Cruz-Cort'es and F. Rodriguez-Henr'iquez, "Implementation of RSA signatures on GPU and CPU architectures," IEEE Access, vol. 8, pp. 9928-9941, 2020.
[4]L. Qiu, Z. Liu, G. C. CF Pereira and H. Seo, "Implementing RSA for sensor nodes in smart cities," Personal and Ubiquitous Computing, vol. 21, pp. 807-813, 2017.
[5]R. Granger, T. Kleinjung and J. Zumbr"agel, "On the discrete logarithm problem in finite fields of fixed characteristic," Transactions of the American Mathematical Society, vol. 370, no. 5, pp. 3129-3145, 2018.
[6]A. M. Odlyzko, "Discrete logarithms in finite fields and their cryptographic significance," in Workshop on the Theory and Application of of Cryptographic Techniques, 1984.
[7]H. I. Hussein and W. M. Abduallah, "An efficient ElGamal cryptosystem scheme," International Journal of Computers and Applications, vol. 43, no. 10, pp. 1088-1094, 2021.
[8]S. Y. Yan, "Primality testing and integer factorization in public-key cryptography," Advances In Information Security, 2009.
[9]K. Rabah, "Review of Methods for Integer Factorization Applied to Cryptography," Journal of applied Sciences, vol. 6, no. 1, pp. 458-481, 2006.
[10]J. Hoffstein, "Integer Factorization and RSA," in An Introduction to Mathematical Cryptography, Springer, 2008, pp. 1--75.
[11]E. Bach, "Toward a theory of Pollard's rho method," Information and Computation, vol. 90, no. 2, pp. 139-155, 1991.
[12]O. Danvy, "The Tortoise and the Hare Algorithm for Finite Lists, Compositionally," ACM Transactions on Programming Languages and Systems, vol. 45, no. 1, pp. 1-35, 2023.
[13]R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey and D. E. Knuth, "On the LambertW function," Advances in Computational mathematics, vol. 5, no. 1, pp. 329-359, 1996.
[14]J. Boyd, "Global approximations to the principal real-valued branch of the Lambert W-function," Applied Mathematics Letters, vol. 11, no. 6, pp. 27-31, 1998.
[15]F. N. Fritsch, R. Shafer and W. Crowley, "Solution of the transcendental equation wew= x," Communications of the ACM, vol. 16, no. 2, pp. 123-124, 1973.
[16]D. Barry, S. Barry and P. Culligan-Hensley, "Algorithm 743: WAPR--a Fortran routine for calculating real values of the W-function," ACM Transactions on Mathematical Software (TOMS), vol. 21, no. 2, pp. 172-181, 1995.
[17]D. Barry, P. Culligan-Hensley and S. Barry, "Real values of the W-function," ACM Transactions on Mathematical Software (TOMS), vol. 21, no. 2, pp. 161-171, 1995.
[18]N. N. Schraudolph, "A fast, compact approximation of the exponential function," Neural Computation, vol. 11, no. 4, pp. 853-862, 1999.
[19]H. Corrigan-Gibbs and D. Kogan, "The discrete-logarithm problem with preprocessing," in Advances in Cryptology--EUROCRYPT 2018: 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29-May 3, 2018 Proceedings, Part II 37, 2018.
[20]C. Diem, "On the discrete logarithm problem in elliptic curves," Compositio Mathematica, vol. 147, no. 1, pp. 75-104, 2011.