Cryptanalysis of RSA Variants with Modified Euler Quotient

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10831)

Abstract

The standard RSA scheme provides the key equation \(ed\equiv 1\pmod {\varphi (N)}\) for \(N=pq\), where \(\varphi (N)=(p-1)(q-1)\) is Euler quotient (or Euler’s totient function), e and d are the public and private keys, respectively. It has been extended to the following variants with modified Euler quotient \(\omega (N)=(p^2-1)(q^2-1)\), which in turn indicates the modified key equation is \(ed\equiv 1\pmod {\omega (N)}\).

  • An RSA-type scheme based on singular cubic curves \(y^2\equiv x^3+bx^2\pmod {N}\) for \(N=pq\).

  • An extended RSA scheme based on the field of Gaussian integers for \(N=PQ\), where P, Q are Gaussian primes with \(p=|P|\), \(q=|Q|\).

  • A scheme working in quadratic field quotients using Lucas sequences with an RSA modulus \(N=pq\).

In this paper, we investigate some key-related attacks on such RSA variants using lattice-based techniques. To be specific, small private key attack, multiple private keys attack, and partial key exposure attack are proposed. Furthermore, we provide the first results for multiple private keys attack and partial key exposure attack when analyzing the RSA variants with modified Euler quotient.

Keywords

RSA variants Modified Euler quotient Lattice Multiple private keys attack Partial key exposure attack 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This work was partially supported by National Natural Science Foundation of China (Grant Nos. 61522210, 61632013).

References

  1. 1.
    Aono, Y.: Minkowski sum based lattice construction for multivariate simultaneous Coppersmith’s technique and applications to RSA. In: Boyd, C., Simpson, L. (eds.) ACISP 2013. LNCS, vol. 7959, pp. 88–103. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39059-3_7CrossRefGoogle Scholar
  2. 2.
    Blömer, J., May, A.: New partial key exposure attacks on RSA. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 27–43. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45146-4_2CrossRefGoogle Scholar
  3. 3.
    Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N0.292. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 1–11. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48910-X_1CrossRefGoogle Scholar
  4. 4.
    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).  https://doi.org/10.1007/3-540-49649-1_3CrossRefGoogle Scholar
  5. 5.
    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).  https://doi.org/10.1007/978-3-319-40367-0_16CrossRefGoogle Scholar
  6. 6.
    Bunder, M., Tonien, J.: New attack on the RSA cryptosystem based on continued fractions. Malays. J. Math. Sci. 11(S3), 45–57 (2017)MathSciNetGoogle Scholar
  7. 7.
    Castagnos, G.: An efficient probabilistic public-key cryptosystem over quadratic fields quotients. Finite Fields Appl. 13(3), 563–576 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Coppersmith, D.: Finding a small root of a bivariate integer equation; factoring with high bits known. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 178–189. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-68339-9_16CrossRefGoogle Scholar
  9. 9.
    Coppersmith, D.: Finding a small root of a univariate modular equation. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 155–165. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-68339-9_14CrossRefGoogle Scholar
  10. 10.
    Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Cryptol. 10(4), 233–260 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Coron, J.-S.: Finding small roots of bivariate integer polynomial equations revisited. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 492–505. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_29CrossRefGoogle Scholar
  12. 12.
    Coron, J.-S.: Finding small roots of bivariate integer polynomial equations: a direct approach. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 379–394. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74143-5_21CrossRefGoogle Scholar
  13. 13.
    Elkamchouchi, H., Elshenawy, K., Shaban, H.: Extended RSA cryptosystem and digital signature schemes in the domain of Gaussian integers. In: ICCS 2002, vol. 1, pp. 91–95. IEEE (2002)Google Scholar
  14. 14.
    Ernst, M., Jochemsz, E., May, A., de Weger, B.: Partial key exposure attacks on RSA up to full size exponents. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 371–386. Springer, Heidelberg (2005).  https://doi.org/10.1007/11426639_22CrossRefGoogle Scholar
  15. 15.
    Fiat, A.: Batch RSA. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 175–185. Springer, New York (1990).  https://doi.org/10.1007/0-387-34805-0_17CrossRefGoogle Scholar
  16. 16.
    Halderman, J.A., Schoen, S.D., Heninger, N., Clarkson, W., Paul, W., Calandrino, J.A., Feldman, A.J., Appelbaum, J., Felten, E.W.: Lest we remember: cold-boot attacks on encryption keys. Commun. ACM 52(5), 91–98 (2009)CrossRefGoogle Scholar
  17. 17.
    Herrmann, M., May, A.: Maximizing small root bounds by linearization and applications to small secret exponent RSA. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 53–69. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13013-7_4CrossRefGoogle Scholar
  18. 18.
    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/BFb0024458CrossRefGoogle Scholar
  19. 19.
    Howgrave-Graham, N., Seifert, J.-P.: Extending Wiener’s attack in the presence of many decrypting exponents. CQRE 1999. LNCS, vol. 1740, pp. 153–166. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-46701-7_14CrossRefGoogle Scholar
  20. 20.
    Jochemsz, E., May, A.: A strategy for finding roots of multivariate polynomials with new applications in attacking RSA variants. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 267–282. Springer, Heidelberg (2006).  https://doi.org/10.1007/11935230_18CrossRefGoogle Scholar
  21. 21.
    Jochemsz, E., May, A.: A polynomial time attack on RSA with private CRT-exponents smaller than N0.073. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 395–411. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74143-5_22CrossRefGoogle Scholar
  22. 22.
    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).  https://doi.org/10.1007/3-540-68697-5_9Google Scholar
  23. 23.
    Kuwakado, H., Koyama, K., Tsuruoka, Y.: New RSA-type scheme based on singular cubic curves \({y^2\equiv x^3+bx^2}\) (mod \({n}\)). IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E78–A(1), 27–33 (1995)Google Scholar
  24. 24.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    May, A.: Using LLL-reduction for solving RSA and factorization problems. In: Nguyen, P.Q., Vallée, B. (eds.) The LLL Algorithm - Survey and Applications. ISC, pp. 315–348. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-02295-1_10Google Scholar
  26. 26.
    Peng, L., Hu, L., Lu, Y., Sarkar, S., Xu, J., Huang, Z.: Cryptanalysis of variants of RSA with multiple small secret exponents. In: Biryukov, A., Goyal, V. (eds.) INDOCRYPT 2015. LNCS, vol. 9462, pp. 105–123. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-26617-6_6CrossRefGoogle Scholar
  27. 27.
    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).  https://doi.org/10.1007/978-3-319-54705-3_9CrossRefGoogle Scholar
  28. 28.
    Quisquater, J.J., Couvreur, C.: Fast decipherment algorithm for RSA public-key cryptosystem. Electron. Lett. 18(21), 905–907 (1982)CrossRefGoogle Scholar
  29. 29.
    Ristenpart, T., Tromer, E., Shacham, H., Savage, S.: Hey, you, get off of my cloud: exploring information leakage in third-party compute clouds. In: Al-Shaer, E., Jha, S., Keromytis, A.D. (eds.) ACM CCS 2009, pp. 199–212. ACM Press, Chicago (2009)Google Scholar
  30. 30.
    Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sarkar, S.: Small secret exponent attack on RSA variant with modulus \(N=p^r q\). Des. Codes Cryptogr. 73(2), 383–392 (2014)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Sarkar, S.: Revisiting prime power RSA. Discrete Appl. Math. 203, 127–133 (2016)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Sarkar, S., Maitra, S.: Cryptanalytic results on ‘Dual CRT’ and ‘Common Prime’ RSA. Des. Codes Cryptogr. 66(1–3), 157–174 (2013)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Sarkar, S., Venkateswarlu, A.: Partial key exposure attack on CRT-RSA. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 255–264. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13039-2_15Google Scholar
  35. 35.
    Takagi, T.: Fast RSA-type cryptosystem modulo pkq. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 318–326. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0055738CrossRefGoogle Scholar
  36. 36.
    Takayasu, A., Kunihiro, N.: Cryptanalysis of RSA with multiple small secret exponents. In: Susilo, W., Mu, Y. (eds.) ACISP 2014. LNCS, vol. 8544, pp. 176–191. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-08344-5_12Google Scholar
  37. 37.
    Takayasu, A., Kunihiro, N.: Partial key exposure attacks on RSA: achieving the Boneh-Durfee bound. In: Joux, A., Youssef, A. (eds.) SAC 2014. LNCS, vol. 8781, pp. 345–362. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13051-4_21CrossRefGoogle Scholar
  38. 38.
    Takayasu, A., Kunihiro, N.: Partial key exposure attacks on RSA with multiple exponent pairs. In: Liu, J.K., Steinfeld, R. (eds.) ACISP 2016. LNCS, vol. 9723, pp. 243–257. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40367-0_15CrossRefGoogle Scholar
  39. 39.
    Takayasu, A., Kunihiro, N.: A tool kit for partial key exposure attacks on RSA. In: Handschuh, H. (ed.) CT-RSA 2017. LNCS, vol. 10159, pp. 58–73. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-52153-4_4CrossRefGoogle Scholar
  40. 40.
    Wiener, M.J.: Cryptanalysis of short RSA secret exponents. IEEE Trans. Inf. Theory 36(3), 553–558 (1990)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zheng, M., Hu, H.: Cryptanalysis of prime power RSA with two private exponents. Sci. China Inf. Sci. 58(11), 1–8 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CAS Key Laboratory of Electromagnetic Space InformationUniversity of Science and Technology of ChinaHefeiChina
  2. 2.The University of TokyoTokyoJapan

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