Abstract
Shamir presents in [3] a family of cryptographic signature schemes based on birational permutations of the integers modulo a large integer N of unknown factorization. These schemes are attractive because of the low computational requirements, both for signature generation and signature verification. However, the two schemes presented in Shamir’s paper are weak. We show here how to break the first scheme, by first reducing it algebraically to the earlier Ong-Schnorr-Shamir signature scheme, and then applying the Pollard solution to that scheme. We then show some attacks on the second scheme. These attacks give ideas which can be applied to schemes in this general family.
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References
H. Ong, C. P. Schnorr, and A. Shamir: A fast signature scheme based on quadratic equations. Proc. 16th ACM Symp. Theory of Computing, pp.208–216; 1984.
J. M. Pollard and C. P. Schnorr: An efficient solution of the congruence x 2 + ky 2 = m(mod n). IEEE Trans. Inform. Theory vol IT-33 no 5, pp.702–709; Sept., 1987.
A. Shamir: Efficient signature schemes based on birational permutations. Manuscript March 1993. To appear, Crypto 93.
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© 1994 Springer-Verlag Berlin Heidelberg
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Coppersmith, D., Stern, J., Vaudenay, S. (1994). Attacks on the Birational Permutation Signature Schemes. In: Stinson, D.R. (eds) Advances in Cryptology — CRYPTO’ 93. CRYPTO 1993. Lecture Notes in Computer Science, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48329-2_37
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DOI: https://doi.org/10.1007/3-540-48329-2_37
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