Effects of Optimizations for Software Implementations of Small Binary Field Arithmetic

  • Roberto Avanzi
  • Nicolas Thériault
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)


We describe an implementation of binary field arithmetic written in the C programming language. Even though the implementation targets 32-bit CPUs, the results can be applied also to CPUs with different granularity.

We begin with separate routines for each operand size in words to minimize performance penalties that have a bigger relative impact for shorter operands – such as those used to implement modern curve based cryptography. We then proceed to use techniques specific to operand size in bits for several field sizes.

This results in an implementation of field arithmetic where the curve representing field multiplication performance closely resembles the theoretical quadratic bit-complexity that can be expected for small inputs.

This has important practical consequences: For instance, it will allow us to compare the performance of the arithmetic on curves of different genera and defined over fields of different sizes without worrying about penalties introduced by field arithmetic and concentrating on the curve arithmetic itself. Moreover, the cost of field inversion is very low, making the use of affine coordinates in curve arithmetic more interesting. These applications will be mentioned.


Binary fields efficient implementation curve-based cryptography 


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  1. Avanzi, R.: Aspects of hyperelliptic curves over large prime fields in software implementations. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 148–162. Springer, Heidelberg (2004)Google Scholar
  2. Avanzi, R., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: The Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, Boca Raton (2005)Google Scholar
  3. Fan, X., Wollinger, T., Wang, Y.: Efficient Doubling on Genus 3 Curves over Binary Fields. IACR ePrint 2005/228Google Scholar
  4. Fong, K., Hankerson, D., López, J., Menezes, A.: Field Inversion and Point Halving Revisited. IEEE Trans. Computers 53(8), 1047–1059 (2004)CrossRefGoogle Scholar
  5. Guyot, C., Kaveh, K., Patankar, V.M.: Explicit algorithm for the arithmetic on the hyperelliptic Jacobians of genus 3. J. Ramanujan Math. Soc. 19(2), 75–115 (2004)zbMATHMathSciNetGoogle Scholar
  6. Hankerson, D., López-Hernandez, J., Menezes, A.: Software Implementation of Elliptic Curve Cryprography over Binary Fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. Karatsuba, A., Ofman, Y.: Multiplication of Multidigit Numbers on Automata. Soviet Physics - Doklady 7, 595–596 (1963)Google Scholar
  8. King, B.: An Improved Implementation of Elliptic Curves over GF(2n) when Using Projective Point Arithmetic. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 134–150. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. Knuth, D.: The Art of Computer Programming, 3rd edn. Seminumerical Algorithms, vol. 2. Addison Wesley Longman, Redwood City (1998)Google Scholar
  10. Lange, T.: Efficient Arithmetic on Genus 2 Hyperelliptic Curves over Finite Fields via Explicit Formulae. Cryptology ePrint Archive, Report 2002/121Google Scholar
  11. Lange, T., Stevens, M.: Efficient doubling for genus two curves over binary fields. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 170–181. Springer, Heidelberg (2005)Google Scholar
  12. Lim, C., Lee, P.: More flexible exponentiation with precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)Google Scholar
  13. López, J., Dahab, R.: High-speed software multiplication in \(F_{2^m}\). In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 203–212. Springer, Heidelberg (2000)Google Scholar
  14. Pelzl, J., Wollinger, T., Guajardo, J., Paar, C.: Hyperelliptic curve cryptosystems: closing the perfomance gap to elliptic curves (Update). IACR ePrint 2003/026Google Scholar
  15. Pelzl, J., Wollinger, T., Paar, C.: Low cost Security: Explicit Formulae for Genus 4 Hyperelliptic Curves. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 1–16. Springer, Heidelberg (2004)Google Scholar
  16. Pippenger, N.: On the evaluation of powers and related problems (preliminary version). 17th Annual Symp. on Foundations of Comp. Sci, pp. 258–263. IEEE Computer Society, Los Alamitos (1976)Google Scholar
  17. Schönhage, A., Grotefeld, A.F.W., Vetter, E.: Fast Algorithms–A Multitape Turing Machine Implementation. BI Wissenschafts-Verlag, Mannheim (1994)zbMATHGoogle Scholar
  18. Schroeppel, R., Orman, H., O’Malley, S., Spatscheck, O.: Fast key exchange with elliptic curve systems. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 43–56. Springer, Heidelberg (1995)Google Scholar
  19. Shoup, V.: NTL: A Library for doing number theory. URL:
  20. Sun Corporation’s Elliptic Curve Cryptography contributions to OpenSSL. Available at
  21. Weimerskirch, A., Stebila, D., Shantz, S.C.: Generic GF(2m) Arithmetic in Software and its Application to ECC. In: Safavi-Naini, R., Seberry, J. (eds.) ACISP 2003. LNCS, vol. 2727, pp. 79–92. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. Wollinger, T.: Software and Hardware Implementation of Hyperelliptic Curve Cryptosystems. Ph.D. Thesis, Ruhr-Universität Bochum, Germany (2004)Google Scholar
  23. Wollinger, T., Pelzl, J., Paar, C.: Cantor versus Harley: Optimization and Analysis of Explicit Formulae for Hyperelliptic Curve Cryptosystems. IEEE Transactions on Computers (To appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Roberto Avanzi
    • 1
  • Nicolas Thériault
    • 2
  1. 1.Fakultät für Mathematik, Ruhr-Universität Bochum and the Horst Görtz Institut für IT-SicherheitGermany
  2. 2.Instituto de Matemática y Física, Universidad de TalcaChile

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