Public quadratic polynomial-tuples for efficient signature-verification and message-encryption
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization
CRYPTO '99 Proceedings of the 19th 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
Asymmetric Cryptography with a Hidden Monomial
CRYPTO '96 Proceedings of the 16th Annual International Cryptology Conference on Advances in Cryptology
Attacking the Affine Parts of SFLASH
Proceedings of the 8th IMA International Conference on Cryptography and Coding
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Unbalanced oil and vinegar signature schemes
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
l-invertible cycles for multivariate quadratic (MQ) public key cryptography
PKC'07 Proceedings of the 10th international conference on Practice and theory in public-key cryptography
Key recovery on hidden monomial multivariate schemes
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
Total break of the l-IC signature scheme
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
Cryptanalysis of multivariate and odd-characteristic HFE variants
PKC'11 Proceedings of the 14th international conference on Practice and theory in public key cryptography conference on Public key cryptography
Implementing minimized multivariate PKC on low-resource embedded systems
SPC'06 Proceedings of the Third international conference on Security in Pervasive Computing
Practical key-recovery for all possible parameters of SFLASH
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
Solving underdetermined systems of multivariate quadratic equations revisited
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Cryptanalysis of enhanced TTS, STS and all its variants, or: why cross-terms are important
AFRICACRYPT'12 Proceedings of the 5th international conference on Cryptology in Africa
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In this article, we investigate the question of equivalent keys for two ${\mathcal M}$ultivariate ${\mathcal Q}$uadratic public key schemes HFE and C*−− and improve over a previously known result, which appeared at PKC 2005. Moreover, we show a new non-trivial extension of these results to the classes HFE-, HFEv, HFEv-, and C*−−, which are cryptographically stronger variants of the original HFE and C* schemes. In particular, we are able to reduce the size of the private — and hence the public — key space by at least one order of magnitude and several orders of magnitude on average. While the results are of independent interest themselves as they broaden our understanding of ${\mathcal M}$ultivariate ${\mathcal Q}$uadratic schemes, we also see applications both in cryptanalysis and in memory efficient implementations.