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Implementation of Cryptographic Algorithms for Elliptic Curves

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Geometry, Algebra and Applications: From Mechanics to Cryptography

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 161))

Abstract

The development of side-channel and fault injection attacks against the implementation of algorithms used in elliptic curve cryptography (ECC), has pointed out that it is not enough to implement efficient algorithms that are secure from a theoretical point of view. In this sense, it is necessary to design algorithms that do not leak information which could allow an attacker to obtain the used keys, thus making the physical implementations of those algorithms resistent to this kind of attacks. In this work, some of the options to implement the scalar multiplication for elliptic curves are described.

To our colleague and friend Jaime Muñoz Masqué, so that he had something to “think about” and give us ideas for the future, on the occasion of his 65th birthday

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References

  1. Brier, E., Joye, M.: Weierstrass elliptic curves and side-channel attacks. Lect. Notes Comput. Sci. 2274, 335–345 (2002)

    Google Scholar 

  2. Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates. Lect. Notes Comput. Sci. 1514, 51–65 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Coron, J.S.: Resistance against differential power analysis for elliptic curve cryptosystems. Lect. Notes Comput. Sci. 1717, 292–302 (1999)

    Article  MATH  Google Scholar 

  4. Fischer, W., Giraud, C., Knudsen, E.W., Seifert, J.P.: Parallel scalar multiplication on general elliptic curves over \({\mathbb{F}}_p\) hedged against non-differential side-channel attacks. Cryptology ePrint Archive, Report 2002/007

    Google Scholar 

  5. Fúster Sabater, A., Hernández Encinas, L., Martín Muñoz, A., Montoya Vitini, F., Muñoz Masqué, J.: Criptografía, protección de datos y aplicaciones. RA-MA, Madrid (2012)

    Google Scholar 

  6. Gordon, D.M.: A survey of fast exponentiation methods. J. Algorithms 27(1), 129–146 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goundar, R., Joye, M., Miyaji, A.: Co-Z addition formula and binary ladders on elliptic curves. Lect. Notes Comput. Sci. 6225, 65–79 (2010)

    Article  MATH  Google Scholar 

  8. Goundar, R., Joye, M., Miyaji, A., Rivain, A., Venelli, A.: Scalar multiplication on Weierstrass elliptic curves from co-Z arithmetic. J. Cryptogr. Eng. 1(2), 161–176 (2011)

    Article  Google Scholar 

  9. Hankerson, D., Menezes, A.J., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, New York (2004)

    MATH  Google Scholar 

  10. Izu, T., Takagi, T.: A fast parallel elliptic curve multiplication resistant against side channel attacks. Lect. Notes Comput. Sci. 2274, 280–296 (2002)

    Article  MATH  Google Scholar 

  11. Joye, M.: Highly regular right-to-left algorithms for scalar multiplication. Lect. Notes Comput. Sci. 4727, 135–147 (2007)

    Article  MATH  Google Scholar 

  12. Joye, M.: Fast point multiplication on elliptic curves without precomputation. In: Proceedings of WAIFI 2008

    Google Scholar 

  13. Joye, M.: Highly regular \(m\)-ary powering ladders. Lect. Notes Comput. Sci. 5867, 350–363 (2009)

    Article  MATH  Google Scholar 

  14. Koyama, K., Tsuruoka, Y.: Speeding up elliptic cryptosystems by using a signed binary window method. Lect. Notes Comput. Sci. 740, 345–357 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Meloni, N.: New point addition formulae for ECC applications. Lect. Notes Comput. Sci. 4547, 189–201 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Möller, B.: Improved techniques for fast exponentiation. Lect. Notes Comput. Sci. 2587, 298–312 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Montgomery, P.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comput. 48, 243–264 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. NSA, NSA Suite B cryptography. National Security Agency (2005)

    Google Scholar 

  19. Rivain, M.: Fast and regular algorithms for scalar multiplication over elliptic curves. Cryptology ePrint Archive, Report 2011/338

    Google Scholar 

  20. Venelli, A., Dassance, F.: Faster side-channel resistant elliptic curve scalar multiplication. Contemp. Math. 521, 29–40 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Verneuil, V.: Elliptic curve cryptography and security of embedded devices. Ph.D. Thesis. Ècole Doctorale de Mathèmatiques et Informatique, Universitè de Bordeaux (France) (2010)

    Google Scholar 

  22. Yao, A.C.C.: On the evaluation of powers. SIAM J. Comput. 5(1), 100–103 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yen, S.M., Joye, M.: Checking before output may not be enough against fault-based cryptanalysis. IEEE Trans. Comput. 49(9), 967–970 (2000)

    Article  MATH  Google Scholar 

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Acknowledgments

This work has been partially supported by project TIN2014-55325-C2-1-R (ProCriCiS), funded by Ministerio de Economía y Competitividad, Spain.

We thank Jaime Muñoz Masqué for his continuous and challenging ideas to address new research objectives, as well as for all his useful suggestions to overcome so many obstacles.

Author information

Authors and Affiliations

  1. Institute of Physical and Information Technologies, Spanish National Research Council, C/ Serrano 144, 28006, Madrid, Spain

    Víctor Gayoso Martínez, Luis Hernández Encinas & Agustín Martín Muñoz

Authors
  1. Víctor Gayoso Martínez
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  2. Luis Hernández Encinas
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  3. Agustín Martín Muñoz
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Corresponding author

Correspondence to Luis Hernández Encinas .

Editor information

Editors and Affiliations

  1. Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid, Spain

    Marco Castrillón López

  2. Physical and Information Technology, Spanish National Research Council (CSIC), Madrid, Spain

    Luis Hernández Encinas

  3. Institute of Fundamental Physics, Spanish National Research Council (CSIC), Madrid, Spain

    Pedro Martínez Gadea

  4. Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, Madrid, Spain

    Mª Eugenia Rosado María

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Gayoso Martínez, V., Hernández Encinas, L., Martín Muñoz, A. (2016). Implementation of Cryptographic Algorithms for Elliptic Curves. In: Castrillón López, M., Hernández Encinas, L., Martínez Gadea, P., Rosado María, M. (eds) Geometry, Algebra and Applications: From Mechanics to Cryptography. Springer Proceedings in Mathematics & Statistics, vol 161. Springer, Cham. https://doi.org/10.1007/978-3-319-32085-4_11

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