Abstract
The RSA cryptosystem, invented in 1977 is the most popular public cryptosystem for electronic commerce. Its three inventors Rivest, Shamir and Adleman received the Year 2002 Turing Award, the equivalent Nobel Prize in Computer Science. RSA offers both encryption and digital signatures and is deployed in many commercial systems. The security of RSA is based on the assumption that factoring large integers is difficult. However, most successful attacks on RSA are not based on factoring. Rather, they exploit additional information that may be encoded in the parameters of RSA and in the particular way in which RSA is used. In this chapter, we give a survey of the mathematics of the RSA cryptosystem focussing on the cryptanalysis of RSA using a variety of diophantine methods and lattice-reduction based techniques.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ajtai, M.: The shortest vector problem in L2 is NP-hard for randomized reductions. In: STOC 1998, pp. 10–19 (1998)
Blömer, J., May, A.: A Generalized Wiener Attack on RSA. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 1–13. Springer, Heidelberg (2004)
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)
Boneh, D.: Twenty years of attacks on the RSA cyptosystem. Notices of the AMS 46(2), 203–213 (1999)
Boneh, D., Durfee, G.: Cryptanalysis of RSA with Private Key d Less than N 0.292. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 1–11. Springer, Heidelberg (1999)
Cachin, C., Micali, S., Stadler, M.A.: Computationally Private Information Retrieval with Polylogarithmic Communication. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 402–414. Springer, Heidelberg (1999)
Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer (1993)
Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. Journal of Cryptology 10(4), 233–260
Diffie, W., Hellman, E.: New directions in cryptography. IEEE Transactions on Information Theory 22(5), 644–654 (1976)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, London
Herrmann, M.: Improved Cryptanalysis of the Multi-Prime φ - Hiding Assumption. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 92–99. Springer, Heidelberg (2011)
Herrmann, M., May, A.: Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 406–424. Springer, Heidelberg (2008)
Hinek, M.: Cryptanalysis of RSA and Its Variants. Cryptography and Network Security Series. Chapman, Hall/CRC, Boca Raton (2009)
Howgrave-Graham, N.: Finding Small Roots of Univariate Modular Equations Revisited. In: Darnell, M.J. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)
Howgrave-Graham, N.: Approximate Integer Common Divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)
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)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen, vol. 261, pp. 513–534.
May, A.: New RSA Vulnerabilities Using Lattice Reduction Methods. Ph.D. thesis, Paderborn (2003), http://www.cits.rub.de/imperia/md/content/may/paper/bp.ps
Nassr, D.I., Bahig, H.M., Bhery, A., Daoud, S.S.: A new RSA vulnerability using continued fractions. In: Proceedings of AICCSA, pp. 694–701 (2008)
Nitaj, A.: Cryptanalysis of RSA Using the Ratio of the Primes. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 98–115. Springer, Heidelberg (2009)
Rivest, R., Shamir, A., Adleman, L.: A Method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21(2), 120–126
de Weger, B.: Cryptanalysis of RSA with small prime difference. Applicable Algebra in Engineering, Communication and Computing 13(1), 17–28
Wiener, M.: Cryptanalysis of short RSA secret exponents. IEEE Transactions on Information Theory 36, 553–558
Yan, S.Y.: Cryptanalytic Attacks on RSA. Springer (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag GmbH Berlin Heidelberg
About this chapter
Cite this chapter
Nitaj, A. (2013). Diophantine and Lattice Cryptanalysis of the RSA Cryptosystem. In: Yang, XS. (eds) Artificial Intelligence, Evolutionary Computing and Metaheuristics. Studies in Computational Intelligence, vol 427. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29694-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-29694-9_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29693-2
Online ISBN: 978-3-642-29694-9
eBook Packages: EngineeringEngineering (R0)