Skip to main content
SpringerLink
Log in

Cryptanalysis of RSA Variants with Primes Sharing Most Significant Bits

  • Conference paper
  • First Online:
Information Security (ISC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13118))

Included in the following conference series:

Abstract

We consider four variants of the RSA cryptosystem with an RSA modulus \(N=pq\) where the public exponent e and the private exponent d satisfy an equation of the form \(ed-k\left( p^2-1\right) \left( q^2-1\right) =1\). We show that, if the prime numbers p and q share most significant bits, that is, if the prime difference \(|p-q|\) is sufficiently small, then one can solve the equation for larger values of d, and factor the RSA modulus, which makes the systems insecure.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key \(d\) less than \(N^{0.292}\). In: Advances in Cryptology-Eurocrypt 1999, Lecture Notes in Computer Science, vol. 1592, Springer-Verlag, pp. 1–11 (1999)

    Google Scholar 

  2. Boneh, D.: Twenty years of attacks on the RSA cryptosystem. Notices Amer. Math. Soc. 46(2), 203–213 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Bunder, M., Nitaj, A., Susilo, W., Tonien, J.: A new attack on three variants of the RSA cryptosystem. In: Liu, J.K., Steinfeld, R. (eds.) ACISP 2016. LNCS, vol. 9723, pp. 258–268. Springer, Cham (2016)

    Google Scholar 

  4. Bunder, M., Nitaj, A., Susilo, W., Tonien, J.: A generalized attack on RSA type cryptosystems. Theoretical Comput. Sci. 704, 74–81 (2017)

    Article  MathSciNet  Google Scholar 

  5. Castagnos, G.: An efficient probabilistic public-key cryptosystem over quadratic field quotients, 2007, Finite Fields and Their Applications, 13(3–13), p. 563–576 (2007). http://www.math.u-bordeaux1.fr/~gcastagn/publi/crypto_quad.pdf

  6. Collins, T., Hopkins, D., Langford, S., Sabin, M.: Public key cryptographic apparatus and Method. US Patent 5,848,159, Jan 1997

    Google Scholar 

  7. Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Crypt. 10(4), 233–260 (1997)

    Article  MathSciNet  Google Scholar 

  8. Elkamchouchi, H., Elshenawy, K., Shaban, H.: Extended RSA cryptosystem and digital signature schemes in the domain of Gaussian integers. In: Proceedings of the 8th International Conference on Communication Systems, pp. 91–95 (2002)

    Google Scholar 

  9. Hardy, G.H., Wright, E.M.: An Introduction to Theory of Numbers, 5th edn. The Clarendon Press, Oxford University Press, New York (1979)

    MATH  Google Scholar 

  10. Hinek, M.: Cryptanalysis of RSA and Its Variants. Chapman & Hall/CRC, Cryptography and Network Security Series, Boca Raton (2009)

    Book  Google Scholar 

  11. Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Darnell, M. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0024458

    Chapter  Google Scholar 

  12. Kuwakado, H., Koyama, K. Tsuruoka, Y.: A new RSA-type scheme based on singular cubic curves \(y^2=x^3+bx^2~({\rm mod}\; n)\). IEICE Transactions on Fundamentals, vol. E78-A, pp. 27–33 (1995)

    Google Scholar 

  13. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 513–534 (1982)

    Article  MathSciNet  Google Scholar 

  14. May, A.: New RSA Vulnerabilities Using Lattice Reduction Methods, PhD Thesis, University of Paderborn (2003)

    Google Scholar 

  15. Nitaj, A., Pan, Y., Tonien, J.: A generalized attack on some variants of the RSA cryptosystem. In: Cid, C., Jacobson, M., Jr. (eds.) SAC 2018. LNCS, vol. 11349, pp. 421–433. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-10970-7_19

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Smith, P.J., Lennon, G.J.J.: LUC: A New Public-Key Cryptosystem, pp. 103–117. Elsevier Science Publishers, Ninth IFIP Symposium on Computer Science Security (1993)

    Google Scholar 

  18. Peng, L., Hu, L., Lu, Y., Wei, H.: An improved analysis on three variants of the RSA cryptosystem. In: Chen, K., Lin, D., Yung, M. (eds.) Inscrypt 2016. LNCS, vol. 10143, pp. 140–149. Springer, Cham (2017)

    Google Scholar 

  19. Quisquater, J.J., Couvreur, C.: Fast decipherment algorithm for RSA public key cryptosystem. Electron. Lett. 18, 905–907 (1982)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Zheng, M., Kunihiro, N., Hu, H.: Cryptanalysis of RSA Variants with Modified Euler Quotient. In: Joux A., Nitaj A., Rachidi T. (eds.) Progress in Cryptology-AFRICACRYPT 2018. AFRICACRYPT 2018. Lecture Notes in Computer Science, vol. 10831. Springer, Cham (2018)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ICES Team, ENSIAS, Mohammed V University in Rabat, Rabat, Morocco

    Meryem Cherkaoui-Semmouni

  2. Normandie University, UNICAEN, CNRS, LMNO, 14000, Caen, France

    Abderrahmane Nitaj

  3. Institute of Cybersecurity and Cryptology, School of Computing and Information Technology, University of Wollongong, Wollongong, Australia

    Willy Susilo & Joseph Tonien

Authors
  1. Meryem Cherkaoui-Semmouni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abderrahmane Nitaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Willy Susilo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joseph Tonien
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abderrahmane Nitaj .

Editor information

Editors and Affiliations

  1. Monash University, Clayton, VIC, Australia

    Joseph K. Liu

  2. Norwegian University of Science and Technology, Gjøvik, Norway

    Sokratis Katsikas

  3. Technical University of Denmark, Kongens Lyngby, Denmark

    Weizhi Meng

  4. University of Wollongong, Wollongong, NSW, Australia

    Willy Susilo

  5. Petra Christian University, Surabaya, Indonesia

    Rolly Intan

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cherkaoui-Semmouni, M., Nitaj, A., Susilo, W., Tonien, J. (2021). Cryptanalysis of RSA Variants with Primes Sharing Most Significant Bits. In: Liu, J.K., Katsikas, S., Meng, W., Susilo, W., Intan, R. (eds) Information Security. ISC 2021. Lecture Notes in Computer Science(), vol 13118. Springer, Cham. https://doi.org/10.1007/978-3-030-91356-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-91356-4_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-91355-7

  • Online ISBN: 978-3-030-91356-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation