Abstract
In 1996, Coppersmith introduced two lattice reduction based techniques to find small roots in polynomial equations. One technique works for modular univariate polynomials, the other for bivariate polynomials over the integers. Since then, these methods have been used in a huge variety of cryptanalytic applications. Some applications also use extensions of Coppersmith’s techniques on more variables. However, these extensions are heuristic methods. In the present paper, we present and analyze a new variation of Coppersmith’s algorithm on three variables over the integers. We also study the applicability of our method to short RSA exponents attacks. In addition to lattice reduction techniques, our method also uses Gröbner bases computations. Moreover, at least in principle, it can be generalized to four or more variables.
Chapter PDF
Similar content being viewed by others
References
Bardet, M.: Etude de sytèmes algébriques surdéterminés. Applications aux codes correcteurs et à la cryptographie. PhD thesis, University of Paris 6 (2004)
Blömer, J., May, A.: Low Secret Exponent RSA Revisited. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 4–19. Springer, Heidelberg (2001)
Blömer, J., May, A.: A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 251–267. Springer, Heidelberg (2005)
Boneh, D., Durfee, G.: Cryptanalysis of RSA with Private Key Less Than N 0.292. IEEE Transactions on Information Theory 46, 1339–1349 (2000)
Coppersmith, D.: Finding a Small Root of a Bivariate Integer Equation; Factoring with high bits known. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 178–189. Springer, Heidelberg (1996)
Coppersmith, D.: Finding a Small Root of a Univariate Modular Equation. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 155–165. Springer, Heidelberg (1996)
Coppersmith, D.: Finding Small Solutions to Small Degree Polynomials. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, p. 20. Springer, Heidelberg (2001)
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)
Ernst, M., Jochemsz, E., May, A., de Weger, B.: Partial Key Exposure Attacks on RSA up to Full Size Exponents. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 371–386. Springer, Heidelberg (2005)
Faugère, J.-C.: A New Efficient Algorithm for Computing Gröbner Bases (F4). Journal of Pure and Applied Algebra 139, 61–88 (1999)
Hinek, M.J.: New partial key exposure attacks on RSA revisited. Technical report, CACR, Centre for Applied Cryptographic Research, University of Waterloo (2004)
Hinek, M.J.: Small Private Exponent Partial Key-Exposure Attacks on Multiprime RSA. Technical report, CACR, Centre for Applied Cryptographic Research, University of Waterloo (2005)
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)
Lenstra, J.A.K., Lenstra, H.W., Lovasz, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 513–534 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bauer, A., Joux, A. (2007). Toward a Rigorous Variation of Coppersmith’s Algorithm on Three Variables. In: Naor, M. (eds) Advances in Cryptology - EUROCRYPT 2007. EUROCRYPT 2007. Lecture Notes in Computer Science, vol 4515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72540-4_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-72540-4_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72539-8
Online ISBN: 978-3-540-72540-4
eBook Packages: Computer ScienceComputer Science (R0)