Affine Pairings on ARM

  • Tolga Acar
  • Kristin Lauter
  • Michael Naehrig
  • Daniel Shumow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7708)


We report on relative performance numbers for affine and projective pairings on a dual-core Cortex A9 ARM processor. Using a fast inversion in the base field and doing inversion in extension fields by using the norm map to reduce to inversions in smaller fields, we find a very low ratio of inversion-to-multiplication costs. In our implementation, this favors using affine coordinates, even for the current 128-bit minimum security level specified by NIST. We use Barreto-Naehrig (BN) curves and report on the performance of an optimal ate pairing for curves covering security levels between 128 and 192 bits. We compare with other reported performance numbers for pairing computation on ARM CPUs.


Optimal ate pairing BN curves ARM architecture 


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  1. 1.
    Akinyele, J.A., Lehmanny, C.U., Green, M.D., Pagano, M.W., Peterson, Z.N.J., Rubin, A.D.: Self-protecting electronic medical records using attribute-based encryption. Cryptology ePrint Archive, Report 2010/565 (2010),
  2. 2.
    Aranha, D.F., Karabina, K., Longa, P., Gebotys, C.H., López, J.: Faster Explicit Formulas for Computing Pairings over Ordinary Curves. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 48–68. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Barreto, P.S.L.M., Naehrig, M.: Pairing-Friendly Elliptic Curves of Prime Order. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319–331. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Beuchat, J.-L., González-Díaz, J.E., Mitsunari, S., Okamoto, E., Rodríguez-Henríquez, F., Teruya, T.: High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 21–39. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Ciet, M., Joye, M., Lauter, K., Montgomery, P.L.: Trading inversions for multiplications in elliptic curve cryptography. Des. Codes Cryptography 39(2), 189–206 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Costello, C., Lange, T., Naehrig, M.: Faster Pairing Computations on Curves with High-Degree Twists. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 224–242. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Granger, R., Scott, M.: Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 209–223. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Koç, Ç.K., Acar, T.: Analyzing and comparing Montgomery multiplication algorithms. IEEE Micro 16, 26–33 (1996)CrossRefGoogle Scholar
  9. 9.
    Lauter, K., Montgomery, P.L., Naehrig, M.: An Analysis of Affine Coordinates for Pairing Computation. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 1–20. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Lynn, B.: The Pairing-Based Cryptography Library (PBC),
  11. 11.
    Montgomery, P.L.: Modular multiplication without trial division. Mathematics of Computation 44(170), 519–521 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Naehrig, M., Niederhagen, R., Schwabe, P.: New Software Speed Records for Cryptographic Pairings. In: Abdalla, M., Barreto, P.S.L.M. (eds.) LATINCRYPT 2010. LNCS, vol. 6212, pp. 109–123. Springer, Heidelberg (2010), Corrected version CrossRefGoogle Scholar
  13. 13.
    Oliveira, L.B., Aranha, D.F., Gouvêa, C.P.L., Scott, M., Câmara, D.F., López, J., Dahab, R.: TinyPBC: Pairings for Authenticated Identity-Based Non-Interactive Key Distribution in Sensor Networks. Computer Communications 34(3), 485–493 (2011)CrossRefGoogle Scholar
  14. 14.
    Pereira, G.C.C.F., Simplício, Jr., M.A., Naehrig, M., Barreto, P.S.L.M.: A family of implementation-friendly BN elliptic curves. Journal of Systems and Software (2011) (to appear), doi:10.1016/j.jss.2011.03.083Google Scholar
  15. 15.
    Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L.J., Kachisa, E.J.: On the Final Exponentiation for Calculating Pairings on Ordinary Elliptic Curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 78–88. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Yoshitomi, M., Takagi, T., Kiyomoto, S., Tanaka, T.: Efficient Implementation of the Pairing on Mobilephones Using BREW. In: Kim, S., Yung, M., Lee, H.-W. (eds.) WISA 2007. LNCS, vol. 4867, pp. 203–214. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tolga Acar
    • 1
  • Kristin Lauter
    • 1
  • Michael Naehrig
    • 1
    • 2
  • Daniel Shumow
    • 1
  1. 1.Microsoft ResearchRedmondUSA
  2. 2.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenNetherlands

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