Public quadratic polynomial-tuples for efficient signature-verification and message-encryption
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
Efficient signature schemes based on birational permutations
CRYPTO '93 Proceedings of the 13th annual international cryptology conference on Advances in cryptology
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Algebraic Methods for Constructing Asymmetric Cryptosystems
AAECC-3 Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes
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
Attacking the Affine Parts of SFLASH
Proceedings of the 8th IMA International Conference on Cryptography and Coding
A Fast and Secure Implementation of Sflash
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
Square-Vinegar Signature Scheme
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Square, a New Multivariate Encryption Scheme
CT-RSA '09 Proceedings of the The Cryptographers' Track at the RSA Conference 2009 on Topics in Cryptology
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
Practical cryptanalysis of SFLASH
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
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
Differential cryptanalysis for multivariate schemes
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Inverting HFE is quasipolynomial
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Polynomial equivalence problems: algorithmic and theoretical aspects
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
Ring signature scheme based on multivariate public key cryptosystems
Computers & Mathematics with Applications
Secure variants of the square encryption scheme
PQCrypto'10 Proceedings of the Third international conference on Post-Quantum Cryptography
Roots of square: cryptanalysis of double-layer square and square+
PQCrypto'11 Proceedings of the 4th international conference on Post-Quantum Cryptography
On the differential security of multivariate public key cryptosystems
PQCrypto'11 Proceedings of the 4th international conference on Post-Quantum Cryptography
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Following the cryptanalyses of the encryption scheme HFE and of the signature scheme SFLASH, no serious alternative multivariate cryptosystems remained, except maybe the signature schemes UOV and HFE*** ***. Recently, two proposals have been made to build highly efficient multivariate cryptosystems around a quadratic internal transformation: the first one is a signature scheme called square-vinegar and the second one is an encryption scheme called square introduced at CT-RSA 2009. In this paper, we present a total break of both the square-vinegar signature scheme and the square encryption scheme. For the practical parameters proposed by the authors of these cryptosystems, the complexity of our attacks is about 235 operations. All the steps of the attack have been implemented in the Magma computer algebra system and allowed to experimentally assess the results presented in this paper.