Skip to main content
Springer Nature Link
Log in

An Improved Algorithm to Solve the Systems of Univariate Modular Equations

  • Conference paper
  • First Online:
Information Security Applications (WISA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10763))

Included in the following conference series:

  • 1116 Accesses

Abstract

In this paper, we propose an improved algorithm to solve the univariate modular equations with mutually co-prime moduli problem. This problem was first proposed in Håstad’s original RSA broadcast attack. At PKC 2008, May and Ritzenhofen improved Håstad’s result by using a slightly different transformation from polynomial systems to a single polynomial. In this work, we propose a new construction method to combine all the k equations into a single equation \(f(x)\equiv 0 \mod \prod _{i=1}^{k}N_i\). Our improved algorithm possesses two advantages compared with the two previous ones. Compared with Håstad’s approach, our algorithm only needs fewer number of equations which suffice for a recovery of all common roots. Compared with May and Ritzenhofen’s, our method obtains the single equation f(x) with a smaller degree. The benefit is that this new algorithm will find the small solution \(x_0\) more efficiently when we invoke Coppersmith’s algorithm.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    It is worth mentioning that the time complexity is not correct in the Theorem 1 of Sect. 2 in [14], neither is the time complexity estimation of solving the SMUPE-problem in [14].

References

  1. Bi, J., Coron, J.-S., Faugère, J.-C., Nguyen, P.Q., Renault, G., Zeitoun, R.: Rounding and chaining LLL: finding faster small roots of univariate polynomial congruences. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 185–202. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54631-0_11

    Chapter  Google Scholar 

  2. Bach, E., Shallit, J.: Algorithmic Number Theory: Efficient Algorithms - Volume 1. The MIT Press, Cambridge (1996)

    MATH  Google Scholar 

  3. Boneh, D.: Twenty years of attacks on the RSA cryptosystem. Not. AMS 46, 203–213 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Boneh, D., Venkatesan, R.: Breaking RSA may not be equivalent to factoring. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 59–71. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054117

    Chapter  Google Scholar 

  5. Brown, D.: Breaking RSA may be as difficult as factoring. Cryptology ePrint Archive Report 2005/380 (2005)

    Google Scholar 

  6. Coppersmith, D.: Finding small solutions to small degree polynomials. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 20–31. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44670-2_3

    Chapter  Google Scholar 

  7. Coppersmith, D.: Small solutions to polynomial equations and low exponent vulnerabilities. J. Cryptol. 10(4), 223–260 (1997)

    Article  MathSciNet  Google Scholar 

  8. Coppersmith, D., Franklin, M., Patarin, J., Reiter, M.: Low-exponent rsa with related messages. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 1–9. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_1

    Chapter  Google Scholar 

  9. Hastad, J.: N using RSA with low exponent in a public key network. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 403–408. Springer, Heidelberg (1986). https://doi.org/10.1007/3-540-39799-X_29

    Chapter  Google Scholar 

  10. Håstad, J.: Solving simultaneous modular equations of low degree. SIAM J. Comput. 17(2), 336–341 (1988)

    Article  MathSciNet  Google Scholar 

  11. Leander, G., Rupp, A.: On the equivalence of RSA and factoring regarding generic ring algorithms. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 241–251. Springer, Heidelberg (2006). https://doi.org/10.1007/11935230_16

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. May, A.: Using LLL-reduction for solving RSA and factorization problems: a survey. In: LLL+25 Conference in Honour of the 25th Birthday of the LLL Algorithm (2007)

    Google Scholar 

  14. May, A., Ritzenhofen, M.: Solving systems of modular equations in one variable: how many RSA-encrypted messages does eve need to know? In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 37–46. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78440-1_3

    Chapter  Google Scholar 

  15. Nguên, P.Q., Stehlé, D.: Floating-point LLL revisited. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 215–233. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_13

    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. Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  18. Simmons, G.: A weak privacy protocol using the RSA crypto algorithm. Cryptologia 7(2), 180–182 (1983)

    Article  Google Scholar 

Download references

Acknowledgments

This paper is partially supported by: NSF of China under grants No. 61502269 & 61672019, 973 Program grant 2013CB834205 and the national key research and development program of China grant No. 2017YFA0303903.

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Tsinghua University, Beijing, 100084, China

    Jingguo Bi

  2. State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China

    Jingguo Bi

  3. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Mingqiang Wang

  4. China Information Technology Security Evaluation Center, Beijing, 10085, China

    Wei Wei

Authors
  1. Jingguo Bi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingqiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingqiang Wang .

Editor information

Editors and Affiliations

  1. KAIST, Daejeon, Korea (Republic of)

    Brent ByungHoon Kang

  2. Georgia Institute of Technology, Atlanta, Georgia, USA

    Taesoo Kim

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bi, J., Wang, M., Wei, W. (2018). An Improved Algorithm to Solve the Systems of Univariate Modular Equations. In: Kang, B., Kim, T. (eds) Information Security Applications. WISA 2017. Lecture Notes in Computer Science(), vol 10763. Springer, Cham. https://doi.org/10.1007/978-3-319-93563-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-93563-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93562-1

  • Online ISBN: 978-3-319-93563-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics