Timing Attack against Implementation of a Parallel Algorithm for Modular Exponentiation

  • Yasuyuki Sakai
  • Kouichi Sakurai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2846)

Abstract

We describe a parallel algorithm for modular exponentiation y ≡ xkmodn. Then we discuss timing attacks against an implementation of the proposed parallel algorithm for modular exponentiation. When we have two processors, which perform modular exponentiation, an exponent k is scattered into two partial exponents k(0) and k(1), where k(0) and k(1) are derived by bitwise AND operation from k such that \(k^{(0)}=k \wedge(0101...01)_{2}\) and \(k^{(1)}=k \wedge(1010...10)_{2}\). Two partial modular exponentiations y0 ≡ xk0modn and y1 ≡ xk1modn are performed in parallel using two processors. Then we can obtain y by computing y ≡ y0y1 modn. In general, the hamming weight of k(0) and k(1) are smaller than that of k. Thus fast computation of modular exponentiation y ≡ xkmodn can be achieved. Moreover we show a timing attack against an implementation of this algorithm. We perform a software simulation of the attack and analyze security of the parallel implementation.

Keywords

Parallel modular exponentiation Montgomery multiplication Side channel attack Timing attack RSA cryptosystems 

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References

  1. [BDF98]
    Boneh, D., Durfee, G., Frankel, Y.: An attack on RSA given a small fraction of the private key bits. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 25–34. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  2. [DKLMQW98]
    Dhem, J.F., Koeune, F., Leroux, P.A., Mestré, P., Quisquater, J.J.: A practical implementation of the timing attack. In: Schneier, B., Quisquater, J.-J. (eds.) CARDIS 1998. LNCS, vol. 1820, pp. 175–190. Springer, Heidelberg (1998)Google Scholar
  3. [GG02]
    Garcia, J.M.G., Garcia, R.M.: Parallel algorithm for multiplication on elliptic curves. Cryptology ePrint Archive, Report 2002/179 (2002), http://eprint.iacr.org
  4. [HQ00]
    Hachez, G., Quisquater, J.J.: Montgomery exponentiation with no final subtractions: Improved Results. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 293–301. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. [IT02]
    Izu, T., Takagi, T.: Fast parallel elliptic curve multiplications resistant to side channel attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 335–345. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. [IYTT02]
    Itoh, K., Yajima, J., Takenaka, M., Torii, N.: DPA countermeasures by improving the window method. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 303–317. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. [KJJ99]
    Kocher, P.C., Jaffe, J., Job, B.: Differential power analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)Google Scholar
  8. [Ko96]
    Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)Google Scholar
  9. [Mo85]
    Montgomery, P.L.: Modular multiplication without trial division. Math. Comp. 44(170), 519–521 (1885)CrossRefGoogle Scholar
  10. [OS00]
    Okeya, K., Sakurai, K.: Power analysis breaks elliptic curve cryptosystems even secure against the timing attack. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 178–190. Springer, Heidelberg (2000)Google Scholar
  11. [Sc00]
    Schindler, W.: A timing attack against RSA with the Chinese Remainder Theorem. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 109–124. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. [SQK01]
    Schindler, W., Quisquater, J.-J., Koeune, F.: Improving divide and conquer attacks against cryptosystems by better error detection correction strategies. In: Proc. of 8th IMA International Conference on Cryptography and Coding, pp. 245–267 (2001)Google Scholar
  13. [Wa99]
    Walter, C.D.: Montgomery exponentiation needs no final subtractions. Exercises in Computer Systems Analysis 35(21), 1831–1832 (1999)CrossRefGoogle Scholar
  14. [WT01]
    Walter, C.D., Thompson, S.: Distinguishing exponent digits by observing modular subtractions. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 192–207. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Yasuyuki Sakai
    • 1
  • Kouichi Sakurai
    • 2
  1. 1.Mitsubishi Electric CorporationKanagawaJapan
  2. 2.Kyushu UniversityFukuokaJapan

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