Skip to main content
SpringerLink
Log in

Partial key exposure attack on RSA using some private key blocks

  • Original Paper
  • Published:
Journal of Computer Virology and Hacking Techniques Aims and scope Submit manuscript

Abstract

RSA is a well-known cryptosystem in public-key cryptography and the strength of the cryptosystem depends on the hardness of factoring large integers. Several attacks have been proposed by using the partial information of the secret parameters, which can be obtained by side-channel attacks. Partial key exposure attacks exploit the information gained by a side-channel attack(s) and identify the potential of the RSA cryptosystem if an attacker knows that partial information. In this paper, we investigate the strength of RSA, if an attacker obtains some blocks of the secret exponent, and by guessing successfully a few most significant bits (MSBs) of any of the primes in RSA. Some blocks of the secret exponent can be extracted by cold boot attack and some MSBs of any of the primes can be guessed correctly. We apply LLL algorithm to attack the RSA and follow the Jochemsz and May approach to construct the lattice.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Rivest, R.L., Shamir, A., Adleman, M.: A Method for obtaining digital signatures and public key cryptosystems. Commun. ACM 21(2), 120–126 (1978)

    Article  MathSciNet  Google Scholar 

  2. Wiener, M.: Cryptanalysis of short RSA secret exponents. IEEE Trans. Inf. Theory 36, 553–558 (1990)

    Article  MathSciNet  Google Scholar 

  3. Boneh, D., Durfee, G.: Cryptanalysis of RSA with Private Key d Less than N^292. IEEE Trans. Inf. Theory 46(4), 1339–1349 (2000)

    Article  MathSciNet  Google Scholar 

  4. Blomer, J., May, A.: Low secret exponent RSA revisited. In: Silverman, J.H. (ed.) CaLC, Volume 2146 of Lecture Notes in Computer Science, pp. 4–19. Springer (2001)

    Google Scholar 

  5. De Weger, B.: Cryptanalysis of RSA with small prime difference. Appl. Algebra Eng. Commun. Comput. 13(1), 17–28 (2002)

    Article  MathSciNet  Google Scholar 

  6. Santosh Kumar, R., Narasimham, C., Pallamsetty, S.: Cryptanalysis of RSA with a small prime difference by using Unravelled linearization. Int. J. Comput. Appl. 61(3), 14–16 (2013)

    Google Scholar 

  7. Santosh Kumar, R., Krishna, S.R.M.: Cryptanalysis of RSA with small difference of primes and two decryption exponents: Jochemsz and May approach. Cryptologia (2022)

  8. Takayasu, A., Kuniharo, N.: Cryptanalysis of RSA with multiple secret exponents. In: Takayasu, A., Kunihiro, N. (eds.) ACISP, LNCS, vol. 8544, pp. 176–191. Springer, NSW, Australia (2014)

    Google Scholar 

  9. Sarkar, S., Maitra, S.: Cryptanalysis of RSA with more than one decryption exponent. Inf. Process. Lett. 110(8–9), 336–340 (2009)

    MathSciNet  Google Scholar 

  10. Sarkar, S., Maitra, S.: Cryptanalysis of RSA with two decryption exponents. Inf. Process. Lett. 110(5), 178–181 (2010)

    Article  MathSciNet  Google Scholar 

  11. Susilo, W., Tonien, J., Yang, G.: Divide and capture: an improved cryptanalysis of the encryption standard algorithm RSA. Comput. Stand. Interfaces 74, 103470 (2021)

    Article  Google Scholar 

  12. Nitaj, A., Ariffin, M.R.K., Adenan, N.N.H., Merenda, D.S., Ahmadian, A.: Exponential increment of RSA attack range via lattice-based cryptanalysis. Multimedia Tools Appl. 40, 1–16 (2021)

    Google Scholar 

  13. Luo, P., Zhou, H., Wang, D., Dai, Y.: Cryptanalysis of RSA for a special case with d > e. Sci. China Ser. F Inf. Sci. 52(4), 609–616 (2009)

    Article  MathSciNet  Google Scholar 

  14. Mumtaz, M., Ping, L.: An improved cryptanalysis for large RSA decryption exponent with constrained secret key’. Int. J. Inf. Comput. Secur. 14(2), 102–117 (2019)

    Google Scholar 

  15. Rivest, R.L., Shamir, A.: Efficient Factoring based on partial information. In: Pichler, F. (ed.) EUROCRYPT, Lecture Notes in Computer Science, vol. 219, pp. 31–34. Springer (1986)

    Google Scholar 

  16. Coppersmith, D.: Finding a small roots of a univariate modular equation. In: Maurer, U.M. (ed.) EUROCRYPT, Lecture Notes in Computer Science, vol. 1070, pp. 155–165. Springer (1996)

    Google Scholar 

  17. Lenstra, A.K., Lenstra, H.W., Jr., Lovasz, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)

    Article  MathSciNet  Google Scholar 

  18. Coppersmith, D.: Finding a small root of a bivariate integer equation: factoring with high bits known. In: Maurer, U.M. (ed.) EUROCRYPT, Lecture Notes in Computer Science, vol. 1070, pp. 178–189. Springer (1996)

    Google Scholar 

  19. 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.) ASIA-CRYPT, Lecture Notes in Computer Science, vol. 1514, pp. 25–34. Springer (1998)

    Google Scholar 

  20. Blomer, J., May, A.: New partial key exposure attacks on RSA. In: Boneh, D. (ed.) CRYPTO Lecture Notes in Computer Science, vol. 2729, pp. 27–43. Springer, New York (2003)

    Google Scholar 

  21. Ernst, M., Jochemsz, E., May, A., deWeger, B.: Partial key exposure attacks on RSA up to full size exponents. In: Cramer, R. (ed.) EUROCRYPT, Lecture Notes in Computer Science, vol. 3494, pp. 371–386. Springer (2005)

    Google Scholar 

  22. Sarkar, S., Maitra, S.: Partial key exposure attacks on RSA and its variant by guessing a few bits of one of the prime factors. Bull. Korean Math. Soc. 46(4), 721–741 (2009)

    Article  MathSciNet  Google Scholar 

  23. Aono, Y.: A new lattice construction for partial key exposure attack for RSA. In: Jarecki, S., Tsudik, G. (eds.) Public Key Cryptography, Lecture Notes in Computer Science, vol. 5443, pp. 34–53. Springer (2009)

    Google Scholar 

  24. Takayasu, A., Kunihiro, N.: Partial key exposure attacks on RSA: achieving the Boneh-Durfee bound. In: Joux, A., Youssef, A.M. (eds.) Selected Areas in Cryptography—SAC 2014—21st International Conference Lecture Notes in Computer Science, vol. 8781, pp. 345–362. Springer (2014)

    Google Scholar 

  25. Joye, M., Le Point, L.: Partial Key Exposure on RSA with private exponents larger than N. In: Proceedings of the 8th International Conference on Information Security Practice and Experience, vol. 7232, pp. 369–380 (2012)

  26. Takayasu, A., Kunihiro, N.: A tool kit for partial key exposure attacks on RSA. In: Handschuh, H. (ed.) Topics in Cryptology—CT-RSA 2017—The Cryptographers’ Track at the RSA Conference 2017, Lecture Notes in Computer Science, vol. 10159, pp. 58–73. Springer (2017)

    Google Scholar 

  27. Alex Halderman, J., Schoen, D., Heninger, N., Clarkson, W., Paul, W., Calandrino, J., Feldman, J., Appelbaum, J., Felten, W.: Lest we remember: cold boot attacks on encryption keys. In: 17th USENIX Security Symposium, San Jose, CA (2008)

  28. Sarkar, S.: Partial key exposure: generalized framework to attack RSA. In: Berstein, J., Chattarjee, S. (eds.) INDOCRYPT, Lecture Notes in Computer Science, vol. 7107, pp. 76–92. Springer (2011)

    Google Scholar 

  29. Boneh, D.: Twenty years of attacks on the RSA cryptosystem. Not. Am. Math. Soc. 46(2), 203–213 (1999)

    MathSciNet  Google Scholar 

  30. Mumtaz, M., Ping, L.: Forty years of attacks on the RSA cryptosystem: a brief survey. J. Discrete Math. Sci. Cryptogr. 22(1), 9–29 (2019)

    Article  Google Scholar 

  31. Mumtaz, M., Ping, L.: An improved cryptanalysis for large RSA decryption exponent with constrained secret key. Int. J. Inf. Comput. Secur. 14(2), 102–117 (2019)

    Google Scholar 

  32. Bernstein, D.J., et al.: Factoring RSA keys from certified smart cards: coppersmith in the wild. In: Sako, K., Sarkar, P. (eds.) Advances in Cryptology—ASIACRYPT 2013. ASIACRYPT 2013. Lecture Notes in Computer Science, vol. 8270. Springer (2013)

    Google Scholar 

  33. Proos, J.A.: Imperfect Decryption and Partial Information Attacks in Cryptography. Ph.D. thesis, University of Waterloo (2003)

  34. Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Darnell, M. (ed.) IMA International Conference, Volume 1355 of Lecture Notes in Computer Science, pp. 131–142. Springer (1997)

    Google Scholar 

  35. Coron, J.-S.: Finding small roots of bivariate integer polynomial equations revisited. In: Cachin, C., Camenisch, J. (eds.) EUROCRYPT, Volume 3027 of Lecture Notes in Computer Science, pp. 492–505. Springer (2004)

    Google Scholar 

  36. Jochemsz, E.: Cryptanalysis of RSA Variants Using Small roots of Polynomials. Ph.D. thesis, Technische Universiteit Eindhoven (2007)

  37. Jochmesz, E., deWeger, B.: A Strategy for finding roots of multivariate polynomials with new applications in attacking RSA variants. In: Lai, X., Chen, K. (eds.) ASIACRYPT, Lecture Notes in Computer Science, vol. 4284, pp. 267–282. Springer (2007)

    Google Scholar 

  38. Hermann, M., May, A.: On factoring arbitrary integers with known bits. Cryptology ePrint Archive, report 374 (2007)

  39. Suzuki, K., Takayasu, A., Kunihiro, N.: Extended partial key exposure attacks on RSA: improvement up to full size decryption exponents. Theoret. Comput. Sci. 841, 62–83 (2020)

    Article  MathSciNet  Google Scholar 

  40. Developers, T.S., Stein, W., Joyner, D., Kohel, D., Cremona, J., Eröcal, B.: SageMath. http://www.sagemath.org

Download references

Author information

Authors and Affiliations

  1. Vasavi College of Engineering, Hyderabad, India

    Santosh Kumar Ravva

  2. CVR College of Engineering, Hyderabad, India

    K. L. N. C. Prakash

  3. Chaitanya Bharathi Institute of Technology, Hyderabad, India

    S. R. M. Krishna

Authors
  1. Santosh Kumar Ravva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. L. N. C. Prakash
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. R. M. Krishna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Santosh Kumar Ravva.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ravva, S., Prakash, K.L.N.C. & Krishna, S.R.M. Partial key exposure attack on RSA using some private key blocks. J Comput Virol Hack Tech 20, 185–193 (2024). https://doi.org/10.1007/s11416-023-00507-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11416-023-00507-9

Keywords

Search

Navigation