Journal of Cryptographic Engineering

, Volume 4, Issue 2, pp 91–106 | Cite as

Efficient binary polynomial multiplication based on optimized Karatsuba reconstruction

  • Chistophe Negre
Regular Paper


At Crypto 2009, Bernstein (LNCS, vol 5677. Springer, Berlin, pp 317–336, 2009) proposed two optimized Karatsuba formulas for binary polynomial multiplication. Bernstein obtained these optimizations by re-expressing the reconstruction of one or two recursions of the Karatsuba formula. In this paper we present a generalization of these optimizations. Specifically, we optimize the reconstruction of \(s\) recursions of the Karatsuba formula for \(s \ge 1\). To reach this goal, we express the recursive reconstruction through a tree and reorganize this tree to derive an optimized recursive reconstruction of depth \(s\). When we apply this approach to a recursion of depth \(s=\log _2(n)-2\) we obtain a parallel multiplier with a space complexity of \(3.75 n^{\log _2(3)}+O(n)\) XOR gates and \(1.78 n^{\log _2(3)}\) AND gates and with a delay of \((2\log _2(n)-1) D_\oplus +D_\otimes \) where \(D_\oplus \) represents the delay of an XOR gate and \(D_\otimes \) the delay of an AND gate.


Binary polynomial multiplication Karatsuba formula  Optimized recursive reconstruction Parallel multiplier 



This work was supported by PAVOIS ANR 12 BS02 002 02.


  1. 1.
    Berlekamp, E.R.: Bit-serial Reed–Solomon encoder. In: IEEE Transactions on Information Theory, IT-28 (1982)Google Scholar
  2. 2.
    Bernstein, D.J.: Batch binary Edwards. In: Proceedings of Advances in Cryptology - CRYPTO 2009. LNCS, vol. 5677, pp. 317–336. Springer, Berlin (2009)Google Scholar
  3. 3.
    Boneh, D., Franklin, M.K.: Identity-based encryption from the Weil pairing. SIAM J. Comput. 32(3), 586–615 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil Pairing. J. Cryptol. 17(4), 297–319 (2004)Google Scholar
  5. 5.
    Cenk, M., Hasan, M.A., Negre, C.: Efficient subquadratic space complexity binary polynomial multipliers based on block recombination. IEEE Trans. Comp. (2014, to appear)Google Scholar
  6. 6.
    Fan, H., Hasan, M.A.: A new approach to sub-quadratic space complexity parallel multipliers for extended binary fields. IEEE Trans. Comput. 56(2), 224–233 (2007)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fan, H., Sun, J., Gu, M., Lam, K.-Y.: Overlap-free Karatsuba–Ofman polynomial multiplication algorithm. IET Inf. Secur. 4, 8–14 (March 2010)Google Scholar
  8. 8.
    Hasan, M.A., Méloni, N., Namin, A.H., Negre, C.: Block recombination approach for subquadratic space complexity binary field multiplication based on Toeplitz matrix-vector product. IEEE Trans. Comp. (2014, to appear) Google Scholar
  9. 9.
    Karatsuba, A., Ofman, Y.: Multiplication of multidigit numbers on automata. Sov. Phys. Dokl. (Engl. Transl.) 7(7), 595–596 (1963)Google Scholar
  10. 10.
    Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48, 203–209 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Leone, M.: A new low complexity parallel multiplier for a class of finite fields. In: Proceedings of the Third International Workshop on Cryptographic Hardware and Embedded Systems (CHES ’01), pp. 160–170, Springer, London (2001)Google Scholar
  12. 12.
    Mastrovito, E.D.: VLSI Architectures for Computation in Galois Fields. PhD thesis, Linkoping University, Department of Electrical Engineering, Linkoping, Sweden (1991)Google Scholar
  13. 13.
    Miller, V.: Use of elliptic curves in cryptography. In: Advances in Cryptology, Proceedings of CRYPTO’85. LNCS, vol. 218, pp. 417–426. Springer, Berlin (1986)Google Scholar
  14. 14.
    Paar, C.: A new architecture for a parallel finite field multiplier with low complexity based on composite fields. IEEE Trans. Comput. 45(7), 856–861 (1996)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Team DALIUniversité de PerpignanPerpignanFrance
  2. 2.LIRMM, UMR 5506Université Montpellier 2 and CNRSMontpellierFrance

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