Securing RSA against Fault Analysis by Double Addition Chain Exponentiation

  • Matthieu Rivain
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5473)

Abstract

Fault Analysis is a powerful cryptanalytic technique that enables to break cryptographic implementations embedded in portable devices more efficiently than any other technique. For an RSA implemented with the Chinese Remainder Theorem method, one faulty execution suffices to factorize the public modulus and fully recover the private key. It is therefore mandatory to protect embedded implementations of RSA against fault analysis.

This paper provides a new countermeasure against fault analysis for exponentiation and RSA. It consists in a self-secure exponentiation algorithm, namely an exponentiation algorithm that provides a direct way to check the result coherence. An RSA implemented with our solution hence avoids the use of an extended modulus (which slows down the computation) as in several other countermeasures. Moreover, our exponentiation algorithm involves 1.65 multiplications per bit of the exponent which is significantly less than the 2 required by other self-secure exponentiations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amiel, F., Feix, B., Villegas, K.: Power analysis for secret recovering and reverse engineering of public key algorithms. In: Adams, C., Miri, A., Wiener, M. (eds.) SAC 2007. LNCS, vol. 4876, pp. 110–125. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Aumüller, C., Bier, P., Fischer, W., Hofreiter, P., Seifert, J.-P.: Fault attacks on RSA with CRT: Concrete results and practical countermeasures. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 260–275. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Bao, F., Deng, R., Han, Y., Jeng, A., Narasimhalu, A.D., Ngair, T.-H.: Breaking Public Key Cryptosystems an Tamper Resistance Devices in the Presence of Transient Fault. In: Christianson, B., Lomas, M. (eds.) Security Protocols 1997. LNCS, vol. 1361, pp. 115–124. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Berzati, A., Canovas, C., Goubin, L.: (In)security Against Fault Injection Attacks for CRT-RSA Implementations. In: Breveglieri, L., Gueron, S., Koren, I., Naccache, D., Seifert, J.-P. (eds.) FDTC 2008, pp. 101–107. IEEE Computer Society, Los Alamitos (2008)Google Scholar
  5. 5.
    Berzati, A., Canovas, C., Goubin, L.: Perturbating RSA public keys: An improved attack. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 380–395. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Biham, E., Shamir, A.: Differential fault analysis of secret key cryptosystems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 513–525. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  7. 7.
    Blömer, J., Otto, M., Seifert, J.-P.: A New RSA-CRT Algorithm Secure against Bellcore Attacks. In: Jajodia, S., Atluri, V., Jaeger, T. (eds.) CCS 2003, pp. 311–320. ACM Press, New York (2003)Google Scholar
  8. 8.
    Boneh, D., DeMillo, R.A., Lipton, R.J.: On the importance of checking cryptographic protocols for faults. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 37–51. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  9. 9.
    Boreale, M.: Attacking right-to-left modular exponentiation with timely random faults. In: Breveglieri, L., Koren, I., Naccache, D., Seifert, J.-P. (eds.) FDTC 2006. LNCS, vol. 4236, pp. 24–35. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Boscher, A., Naciri, R., Prouff, E.: CRT RSA algorithm protected against fault attacks. In: Sauveron, D., Markantonakis, K., Bilas, A., Quisquater, J.-J. (eds.) WISTP 2007. LNCS, vol. 4462, pp. 229–243. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Bos, J., Coster, M.: Addition chain heuristics. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 400–407. Springer, Heidelberg (1990)Google Scholar
  12. 12.
    Brier, É., Chevallier-Mames, B., Ciet, M., Clavier, C.: Why one should also secure RSA public key elements. In: Goubin, L., Matsui, M. (eds.) CHES 2006. LNCS, vol. 4249, pp. 324–338. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Chevallier-Mames, B., Ciet, M., Joye, M.: Low-cost Solutions for Preventing Simple Side-Channel Analysis: Side-Channel Atomicity. IEEE Transactions on Computers 53(6), 760–768 (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Ciet, M., Joye, M.: Practical Fault Countermeasures for Chinese Remaindering Based RSA. In: Breveglieri, L., Koren, I. (eds.) FDTC 2005, pp. 124–132 (2005)Google Scholar
  15. 15.
    Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  16. 16.
    Dottax, E., Giraud, C., Rivain, M., Sierra, Y.: On Second-Order Fault Analysis Resistance for CRT-RSA Implementations. Cryptology ePrint Archive, Report 2009/24 (2009), http://eprint.iacr.org/2009/024
  17. 17.
    Fouque, P.-A., Valette, F.: The Doubling Attack: Why Upwards is Better than Downwards. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 269–280. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Giraud, C.: An RSA Implementation Resistant to Fault Attacks and to Simple Power Analysis. IEEE Transactions on Computers 55(9), 1116–1120 (2006)CrossRefGoogle Scholar
  19. 19.
    Gordon, D.M.: A Survey of Fast Exponentiation Methods. J. Algorithms 27(1), 129–146 (1998)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Itoh, K., Izu, T., Takenak, M.: Address-bit Differential Power Analysis of Cryptographic Schemes OK-ECDH and OK-ECDSA. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 129–143. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  21. 21.
    Itoh, K., Izu, T., Takenaka, M.: A Practical Countermeasure against Address-Bit Differential Power Analysis. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 382–396. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Joye, M., Lenstra, A., Quisquater, J.-J.: Chinese Remaindering Based Cryptosystems in the Presence of Faults. Journal of Cryptology 12(4), 241–245 (1999)CrossRefMATHGoogle Scholar
  23. 23.
    Kim, C.H., Quisquater, J.-J.: Fault Attacks for CRT Based RSA: New Attacks, New Results, and New Countermeasures. In: Sauveron, D., Markantonakis, K., Bilas, A., Quisquater, J.-J. (eds.) WISTP 2007. LNCS, vol. 4462, pp. 215–228. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Kim, C.H., Shin, J.H., Quisquater, J.-J., Lee, P.J.: Safe-error attack on SPA-FA resistant exponentiations using a HW modular multiplier. In: Nam, K.-H., Rhee, G. (eds.) ICISC 2007. LNCS, vol. 4817, pp. 273–281. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Knuth, D.: The Art of Computer Programming, 3rd edn. Addison-Wesley, Reading (1988)Google Scholar
  26. 26.
    Kocher, P., Jaffe, J., Jun, B.: Differential Power Analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  27. 27.
    Kocher, P.: 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
  28. 28.
    Koç, Ç.: Analysis of the Sliding Window Techniques for Exponentiation. Computer & Mathematics with applications 30(10), 17–24 (1995)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)MATHGoogle Scholar
  30. 30.
    Messerges, T., Dabbish, E., Sloan, R.: Power Analysis Attacks of Modular Exponentiation in Smartcard. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 144–157. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  31. 31.
    Mitrinovic, D.S., Sándor, J., Crstici, B.: Handbook of Number Theory. Springer, Heidelberg (1995)MATHGoogle Scholar
  32. 32.
    Möller, B.: Algorithms for multi-exponentiation. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 165–180. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  33. 33.
    Rivest, R., Shamir, A., Adleman, L.: A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM 21(2), 120–126 (1978)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Schmidt, J., Herbst, C.: A Practical Fault Attack on Square and Multiply. In: Breveglieri, L., Gueron, S., Koren, I., Naccache, D., Seifert, J.-P. (eds.) FDTC 2008, pp. 53–58. IEEE Computer Society, Los Alamitos (2008)Google Scholar
  35. 35.
    Seifert, J.-P.: On Authenticated Computing and RSA-based Authentication. In: Atluri, V., Meadows, C., Juels, A. (eds.) ACM CCS 2005, pp. 122–127. ACM Press, New York (2005)Google Scholar
  36. 36.
    Shamir, A.: Improved Method and Apparatus for Protecting Public Key Schemes from Timing and Fault Attacks. Publication number: WO9852319 (November 1998)Google Scholar
  37. 37.
    Sun Microsystems. Application Programming Interface – Java CardTM Plateform, Version 2.2.2 (March 2006), http://java.sun.com/products/javacard/specs.html
  38. 38.
    Vigilant, D.: RSA with CRT: A new cost-effective solution to thwart fault attacks. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 130–145. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  39. 39.
    Wagner, D.: Cryptanalysis of a Provable Secure CRT-RSA Algorithm. In: Pfitzmann, B., Liu, P. (eds.) CCS 2004, pp. 82–91. ACM Press, New York (2004)Google Scholar
  40. 40.
    Yen, S.-M., Joye, M.: Checking Before Output Not Be Enough Against Fault-Based Cryptanalysis. IEEE Transactions on Computers 49(9), 967–970 (2000)CrossRefMATHGoogle Scholar
  41. 41.
    Yen, S.-M., Kim, S.-J., Lim, S.-G., Moon, S.-J.: A countermeasure against one physical cryptanalysis may benefit another attack. In: Kim, K.-c. (ed.) ICISC 2001. LNCS, vol. 2288, pp. 414–427. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  42. 42.
    Yen, S.-M., Kim, S.-J., Lim, S.-G., Moon, S.-J.: RSA Speedup with Residue Number System Immune against Hardware Fault Cryptanalysis. IEEE Transactions on Computers 52(4), 461–472 (2003)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Matthieu Rivain
    • 1
  1. 1.Oberthur Technologies & University of LuxembourgLuxembourg

Personalised recommendations