Abstract
Recently Ong, Schnorr, and Shamir [OSS1, OSS2] have presented new public key signature schemes based on quadratic equations. We will refer to these as the OSS schemes. The security of the schemes rest in part on the difficulty of finding solutions to
where n is the product of two large rational primes. In the original OSS scheme [OSS1], K, M, X, and Y were to be rational integers. However, when this version succumbed to an attack by Pollard [PS,S1], a new version was introduced [OSS2], where M, X, and Y were to be quadratic integers, i. e. elements of the ring \( Z[\sqrt d ] \). In this paper we will show that the OSS system in \( Z[\sqrt d ] \) is also breakable The method by which we do this is to reduce the problem of solving the congruence over the ring \( Z[\sqrt d ] \) to the problem of solving the congruence over the integers, for which we can use Pollard’s algorithm.
Research sponsored by NSF Grant #53-4510-2651
Download to read the full chapter text
Chapter PDF
References
H. Ong, C. P. Schnorr, and A. Shamir, “An Efficient Signature Scheme Based on Quadratic Equations,” Proc. 16th ACM Symp. Theor. Comput. (1984) 208–216.
H. Ong, C. P. Schnorr, and A. Shamir, “Efficient Signature Schemes based on Polynomial Equations,” to appear in Crypto 84, Lecture Notes in Computer Science, Springer-Verlag, N. Y., 1984.
J. M. Pollard and C.-P. Schnorr, “Solution of x2 + ky2 ≡ m (mod n), with applications to digital signatures”, preprint, 1985.
J. Shallit, “An Exposition of Pollard’s Algorithm for Quadratic Congruences,” Technical Report 84-006, Department of Computer Science, University of Chicago, Dec. 1984.
M. O. Rabin, “Digitalized signatures and public-key functions as intractable as factorization,” M.I.T. Laboratory for Computer Science, Technical report LCS/TR-212, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Estes, D., Adleman, L.M., Kompella, K., McCurley, K.S., Miller, G.L. (1986). Breaking the Ong-Schnorr-Shamir Signature Scheme for Quadratic Number Fields. In: Williams, H.C. (eds) Advances in Cryptology — CRYPTO ’85 Proceedings. CRYPTO 1985. Lecture Notes in Computer Science, vol 218. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-39799-X_1
Download citation
DOI: https://doi.org/10.1007/3-540-39799-X_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16463-0
Online ISBN: 978-3-540-39799-1
eBook Packages: Springer Book Archive