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
Weakness in Quaternion Signatures
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
New Public-Key Cryptosystem Using Braid Groups
CRYPTO '00 Proceedings of the 20th Annual International Cryptology Conference on Advances in Cryptology
Cryptoanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt'88
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
Cryptanalysis of the Oil & Vinegar Signature Scheme
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Cryptanalysis of the TTM Cryptosystem
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
An efficient signature scheme based on quadratic equations
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Infinite grobner bases and noncommutative polly cracker cryptosystems
Infinite grobner bases and noncommutative polly cracker cryptosystems
Multivariate Public Key Cryptosystems (Advances in Information Security)
Multivariate Public Key Cryptosystems (Advances in Information Security)
Unbalanced oil and vinegar signature schemes
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
New differential-algebraic attacks and reparametrization of rainbow
ACNS'08 Proceedings of the 6th international conference on Applied cryptography and network security
Building secure tame-like multivariate public-key cryptosystems: the new TTS
ACISP'05 Proceedings of the 10th Australasian conference on Information Security and Privacy
Rainbow, a new multivariable polynomial signature scheme
ACNS'05 Proceedings of the Third international conference on Applied Cryptography and Network Security
SCN'06 Proceedings of the 5th international conference on Security and Cryptography for Networks
Selecting parameters for the rainbow signature scheme
PQCrypto'10 Proceedings of the Third international conference on Post-Quantum Cryptography
Reducing the key size of rainbow using non-commutative rings
CT-RSA'12 Proceedings of the 12th conference on Topics in Cryptology
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In 1984, Ong, Schnorr and Shamir proposed an efficient signature scheme (OSS signature scheme) using a bivariate quadratic equation. Its security was believed to be based on the difficulty of integer factorization. However, an efficient attack without integer factorization was subsequently found. In 2008, Hashimoto and Sakurai proposed an extended scheme (HS scheme), based on OSS signature scheme that used multivariate and non-commutative ring. HS scheme uses a composite number as a modulus in the same manner as OSS signature scheme. In this paper, we redefine HS scheme in such a way that it deals with not only integers modulo a composite number, but also elements of a finite field. In the case of a finite field, it becomes a scheme in the multivariate public key cryptosystem. In fact, its public key is constructed by a version of Rainbow in which all the components in the parameter are equal. (We call such a Rainbow a uniformly-layered Rainbow.) In particular, our scheme is a candidate for post-quantum cryptography. If a non-commutative ring used in the proposed scheme is chosen by the group ring associated to dihedral group, the speed of the signature generation can be accelerated by about 50% in comparison with the corresponding Rainbow. We analyze the security of the extended HS scheme against some attacks and conclude that if its base field is GF(256), then the dimension of a non-commutative ring must be more than 10 in order to be secure.