An efficient solution of the congruence x2+ky2=m (modn)
IEEE Transactions on Information Theory
Efficient signature schemes based on birational permutations
CRYPTO '93 Proceedings of the 13th annual international cryptology conference on Advances in cryptology
An efficient signature scheme based on quadratic equations
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
A secure method for signature delegation to mobile agents
Proceedings of the 2004 ACM symposium on Applied computing
On the security of stepwise triangular systems
Designs, Codes and Cryptography
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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.