Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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GENERALIZED RSA CIPHER AND DIFFIE-HELLMAN PROTOCOL

  • Received : 2020.06.10
  • Accepted : 2020.08.27
  • Published : 2021.01.30
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Abstract

In this paper I am considering several cryptological threads. The problem of the RSA cipher, like the Diffie-Hellman protocol, is the use of finite sets. In this paper, I generalize the RSA cipher and DH protocol for infinite sets using monoids. In monoids we can not find the inverse, which makes it difficult. In the second part of the paper I show the applications in cryptology of polynomial composites and monoid domains. These are less known structures. In this work, I show different ways of encrypting messages based on infinite sets.

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References

  1. D.D. Anderson, GCD domains, Gauss' Lemma and contents of polynomials, Non-Noetherian Commutative Ring Theory 520 (2000), 1-31.
  2. D.D. Anderson and D.F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Pauli 28 (1979), 215-221.
  3. D.D. Anderson, D.F. Anderson, and M. Zafrullah, Rings between D[X] and K[X], Houston J. of Mathematics 17 (1991), 109-129.
  4. D.F. Anderson and A. Ryckaert, The class group of D + M, J. Pure Appl. Algebra 52 (1988), 199-212. https://doi.org/10.1016/0022-4049(88)90091-6
  5. J. Brewer and E. Rutter, D + M construction with general overrings, Mich. Math. J. 23 (1976), 33-42.
  6. D. Costa, J. Mott, and M. Zafrullah, The construction D+XDS[X], J. Algebra 153 (1978), 423-439.
  7. M. Cozzens and S. J. Miller, The Mathematics of Encryption: An Elementary Introduction, American Mathematical Society, 2013.
  8. A. David, Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice, CCS '15: Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security 12 (2015), 5-17.
  9. W. Diffie and M.E. Hellman, New Directions in Cryptography, IEEE Transactions on Information Theory 22 (1976), 644-654. https://doi.org/10.1109/TIT.1976.1055638
  10. J.H. Ellis, The possibility of Non-Secret digital encryption, CESG Research Report Corpus ID: 18813089, 2014.
  11. M. Fontana and S. Kabbaj, On the Krull and valuative dimension of D+XDS[X] domains, J. Pure Appl. Algebra 63 (1990), 231-245. https://doi.org/10.1016/0022-4049(90)90069-T
  12. M.E. Hellman and E. Martin, An overview of public key cryptography, IEEE Communications Magazine 40 (2002), 42-49.
  13. P. Jeffdrzejewicz, M. Marciniak, L. Matysiak and J. Zielinski, On properties of square-free elements in commutative cancellative monoids, Semigroup Forum 100 (2020), 850-870. https://doi.org/10.1007/s00233-019-10022-3
  14. N. Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Math., Springer, ISBN 978-1-4419-8592-7.
  15. S. McAdam and D.E. Rush, Schreier Rings, Bull. London Math. Soc. 10 (1978), 77-80. https://doi.org/10.1112/blms/10.1.77
  16. L. Matysiak, On properties of composites and monoid domains, Accepted for printing in Advances and Applications in Mathematical Sciences arXiv: 2006.14948, (2020).
  17. L. Matysiak, ACCP and atomic properties of composites and monoid domains, Accepted for printing in Indian Journal of Mathematics http://lukmat.ukw.edu.pl/ACCP%20and%20atomic%20properties%20of%20composites%20and%20monoid%20domains.pdf, (2020).
  18. Merkle and C. Ralph, Secure Communications Over Insecure Channels, Communications of the ACM 21 (1978), 294-299. https://doi.org/10.1145/359460.359473
  19. G. Picavet, About GCD domains, Advances in commutative ring theory 205 (1999), 501-519.
  20. R. Rivest, A. Shamir and L. Adleman, A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, Communications of the ACM 21 (1978), 120-126. https://doi.org/10.1145/359340.359342
  21. N. Smart, Dr Clifford Cocks CB, Bristol University, (2008), Retrieved August 14, 2011.
  22. M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895-1920. https://doi.org/10.1080/00927878708823512
  23. M. Zafrullah, The D + XDS[X] construction from GCD-domains, J. Pure Appl. Algebra 50 (1988), 93-107. https://doi.org/10.1016/0022-4049(88)90006-0
  24. Dr. Clifford Cocks CB, BBC News, https://www.bbc.com/news/uk-englandgloucestershire-11475101, 2010.