Searching for Best Karatsuba Recurrences

  • Çağdaş Çalık
  • Morris Dworkin
  • Nathan Dykas
  • Rene PeraltaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11544)


Efficient circuits for multiplication of binary polynomials use what are known as Karatsuba recurrences. These methods divide the polynomials of size (i.e. number of terms) \(k \cdot n\) into k pieces of size n. Multiplication is performed by treating the factors as degree-\((k-1)\) polynomials, with multiplication of the pieces of size n done recursively. This yields recurrences of the form \( M(k n) \le \alpha M(n) + \beta n + \gamma ,\) where M(t) is the number of binary operations necessary and sufficient for multiplying two binary polynomials with t terms each. Efficiently determining the smallest achievable values of (in order) \(\alpha , \beta , \gamma \) is an unsolved problem. We describe a search method that yields improvements to the best known Karatsuba recurrences for k = 6, 7 and 8. This yields improvements on the size of circuits for multiplication of binary polynomials in a range of practical interest.


  1. 1.
    Barbulescu, R., Detrey, J., Estibals, N., Zimmermann, P.: Finding optimal formulae for bilinear maps. In: Özbudak, F., Rodríguez-Henríquez, F. (eds.) WAIFI 2012. LNCS, vol. 7369, pp. 168–186. Springer, Heidelberg (2012). Scholar
  2. 2.
    Bernstein, D.J.: Batch binary Edwards. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 317–336. Springer, Heidelberg (2009). Scholar
  3. 3.
    Boyar, J., Find, M.G., Peralta, R.: Small low-depth circuits for cryptographic applications. Crypt. Commun. 11(1), 109–127 (2018). Scholar
  4. 4.
    Boyar, J., Matthews, P., Peralta, R.: Logic minimization techniques with applications to cryptology. J. Cryptol. 26(2), 280–312 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brent, R.P., Gaudry, P., Thomé, E., Zimmermann, P.: Faster multiplication in GF(2)[x]. In: van der Poorten, A.J., Stein, A. (eds.) ANTS 2008. LNCS, vol. 5011, pp. 153–166. Springer, Heidelberg (2008). Scholar
  6. 6.
    Cenk, M., Hasan, M.A.: Some new results on binary polynomial multiplication. J. Cryptogr. Eng. 5, 289–303 (2015)CrossRefGoogle Scholar
  7. 7.
    De Piccoli, A., Visconti, A., Rizzo, O.G.: Polynomial multiplication over binary finite fields: new upper bounds. J. Cryptogr. Eng. 1–14, April 2019.
  8. 8.
    Fan, H., Hasan, M.A.: Comments on five, six, and seven-term Karatsuba-like formulae. IEEE Trans. Comput. 56(5), 716–717 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Find, M.G., Peralta, R.: Better circuits for binary polynomial multiplication. IEEE Trans. Comput. 68(4), 624–630 (2018). Scholar
  10. 10.
    Fuhs, C., Schneider-Kamp, P.: Optimizing the AES S-box using SAT. In: Proceedings International Workshop on Implementation of Logics (IWIL), pp. 64–70 (2010)Google Scholar
  11. 11.
    Karatsuba, A.A., Ofman, Y.: Multiplication of multidigit numbers on automata. Sov. Phys. Doklady 7, 595–596 (1963)Google Scholar
  12. 12.
    Montgomery, P.L.: Five, six, and seven-term Karatsuba-like formulae. IEEE Trans. Comput. 54(3), 362–369 (2005). Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the United States; foreign copyright protection may apply 2019

Authors and Affiliations

  • Çağdaş Çalık
    • 1
  • Morris Dworkin
    • 1
  • Nathan Dykas
    • 2
  • Rene Peralta
    • 1
    Email author
  1. 1.Computer Security DivisionNISTGaithersburgUSA
  2. 2.Mathematics DepartmentUniversity of MarylandCollege ParkUSA

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