International Journal of Modern Science and Technology · ISSN 2456-0235

Review · Computer & Comm. Tech. · Volume 1, Issue 8 · Nov 2016 · Pages 269–275

Elliptic Curve and Associate Cryptosystem

S. Revathi, A. R. Rishivarman

  • S. Revathi: Department of Mathematics, Theivanai Ammal College for Women (Autonomous), Villupuram - 605 401. Tamilnadu, India.
  • A. R. Rishivarman: Department of Mathematics, Theivanai Ammal College for Women (Autonomous), Villupuram - 605 401. Tamilnadu, India.

Abstract

Elliptic curve is a study of points on two-variable polynomials of degree three. With curve defined over a finite field, this set of points are acted by an addition operation forms a finite group structure. Encryption and decryption transform a point into another point in the same set. Besides providing conceptual understanding, discussions are targeting the issues of security and efficiency of elliptic curve cryptosystem. Cryptography is an evolving field that research into discreet mathematical equation that is representable by computer algorithm for providing message confidentiality. The scheme has been widely used by nation-states, corporate and individual who seek privacy for data in storage and during transmission. This paper provides a ground up survey on elliptic curve cryptography. The present paper serves as a basis to understand the fundamental concept behind this cryptosystem. Moreover, we also highlight subareas of research within the scope of elliptic curve cryptosystem.

Keywords

Elliptic Curve Cryptography; Finite field; Addition; Multiplication

References

  1. Mohamed MA. A Survey on Elliptic Curves Cryptography. Applied Mathematical Science . 2014; 8(154):7665- 7691.
  2. Kumar A, Lala K, Kumar A. VHDL Implementation using Ell iptic Curve Point Multiplication. International Journal of Advanced Research in Computer and Communication Engineering . 2012;1(6)392-398.
  3. Faltings G, Taylor R, Wiles A. The Proof of Fermat’s Last Theorem . Notices of the American Mathematical Society . 1995;42(7):743-746.
  4. Lavanya M, Natarajan V. Improved Elliptic Curve Arithmetic Over GF(P) Using Different Projective Coordinate System. Applied Mathematics Science . 2015;9(45):2235-2243.
  5. Koblitz. Elliptic Curve Cryptosys tems. Mathematics of Computatio n. 1987;48: 2003-2009.
  6. Miller V. Use of Elliptic Curves in Cryptography. Advances in Cryptology. 1986;218(483):417-426.
  7. ANSI X9.62. Public Key Cryptography for Financial Services Industry: The Cur ve Digital Signature Algorithm, 1999.
  8. ANSI X9.62, Public Key Cryptog raphy for The Financial Services Industry: The Elliptic curve key Agreement and Key Transport Protocols. 2000.
  9. Hemnath Ravindra, Jalaja S . ASIC Architectures for Implementing ECC Arithmetic over Finite Fields. International Journals of Science and Researc h. 2015;4(6):2319-7064.
  10. ISO IEC 14888, Information Technology Securit y Techniques -Digital Signatures. Appendix part 3. Certificate based Mechanism. 1998.
  11. ISO IEC15946, Information Security Technology-Cryptography Techniques based on Elliptic Curve. 1999.
  12. Digital Signat ure standard. FIPS Publication. 2000:186-192.
  13. Mishal N, Singh RK. A Study on Finite Field Multiplication Over GF(2m) and its Application on Elliptic Curve Cryptography. International Journal of Science and Engineering Research . 2014;5(5):1672-1684.
  14. Bailey D , Christof Paar C. Arithmetic in Finite Field Extensions with Application in Elliptic Curve Crypt ography. Journal of Cryptology. 2001;14(3):153-176.
  15. Bednara M, Daldrup J, Shokrollahi.et al Tradeoff analysis o f pga Based Elliptic Curve Crypto graphy. Proc. Of the IEEE International Symposium on Circuits and Systems, Scottsdale, Arizona, USA, 2002.
  16. King B. Mapping an Arbitary Message to an Elliptic Curve When Defined over GF (2n). International Journal of Network Security. 2009;8(2):169-17.
  17. Koblitz N. A family of Jacobian Suitable for Discrete Log Cryptosy stem. Advances in Cryptology. 1988;88:94-99.
  18. Uwe Krieger Signature, Diplomaebeit C. University Essen Faahbereich . Mathematics and Informatics. 1997.
  19. Sakai Y, Sakurai K. On The Pra ctical Performance of Hyperelliptic Curve Cryptosystem in Software Implementation . IEICE Transaction on Fundamental of Electronics, Communication and Computer Sciences. 2000;E83-A(4):692-703.
  20. Wollinger T. Computer Architecture for Cryptosystems Based on Hyperelliptic Curves. Thesis: Worc ester Polytechnique Institute, 2001.
  21. Boston N, Clancy T, Liow Y, Webster J. Genus Two Hyperelliptic Curve Coprocessor. Workshop on Cryptographic Hardware and Embeded System, 2002.
  22. Smart NP. The Hessian form of An Elliptic C urve. Berlin Heidelberg : Springer-Verlag; 2001.
  23. Edwards H M. A Normal form for Elliptic Curves. Bulletin of the American Mathematical Society, 2007;44(3):393-422.
  24. Bernstein D, Lange T. Faster Addition and Doubling on Elliptic Curves, 2007.
  25. Gouvea F, Mazur B. The Squa re Free Sieve and The Rank of Elliptic Curve . Journal of American Mathematical Society. 1991;4(1):1-23.
  26. Twisted Hessian Curves Retrieved . Auto-Twisted Hessian.html, 2010.
  27. Daniel J. Bernstein, Marc Joye, Tania Lange et al. Twisted Edward curves, 2008.
  28. Olivier Billet, Marc Joye. The Jacobi Model of An Elliptic Curv e and The Side - channel Analysis . Berlin Heidelberg: Springer-Verlag; 2003.
  29. Montgomery PL. Speeding the Pollard and Elliptic Curve Methods of Factorization. American Mathematical Society. 1987;48:243-264.
  30. Silverman, Joseph H. The Arithmetic of Elliptic Curves. Gradute Texts in Mathematics. New York. Springer Verlag: 1994.
  31. Standards for Efficient Cryptography , Sec 2. Elliptic Curve Cryptography . Certicom Research. 2000.
  32. Standards for Efficient Cryptogr aphy, Sec 2. Recommended Ellip tic Curve Domain parameters, Certicom Research. 2010.
  33. Sravana Kumar D, Suneetha CH, Chandrasekhar A. Encryption of Data Using Elliptic Curve Over Finite Fields . International Jo urnals and Parallel System. 2012;3(1):301-308.
  34. Gajbhiye S, Sharma M, Dashputre S. A Survey Report on Elliptic Curve Cryptography. International Journal of Electrical and Computer Engineering , 2011;1(2):195-201.
  35. Gayoso Martinez V, Hernandez Encinas L, Avila S. A Survey of the Elliptic Curve Integrated Enc ryption Scheme . Journal of Computer Science and Engineering . 2010;2(2):7-13.
  36. Marisa W, Paryasto, Kuspriyante, Sarwan et al. Issues in Elliptic Curve Crypyography Implementation. International Indonesia Journal, 2009;1(1):29-33.
  37. Morales-Sandoval M, An Interoperable and Reconfigurable Hardware Architecture for Elliptic Curve Cryptography Thesis: Mexico. National Institute for Astrophysics. Optics and Electronics. 2006.
  38. Mathew Estes, Phillip Hin es, Efficient Implementation of an Elliptic Curve Cryptosystems Over Binary Galios Field in Normal and Polynomial Ba ses. George Massion University. 2006.