Public quadratic polynomial-tuples for efficient signature-verification and message-encryption
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
QUARTZ, 128-Bit Long Digital Signatures
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
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
EUROCRYPT '02 Proceedings of the International Conference on the Theory and Applications of Cryptographic Techniques: Advances in Cryptology
C*-+ and HM: Variations Around Two Schemes of T. Matsumoto and H. Imai
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: 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
On the Security of HFE, HFEv- and Quartz
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Efficient algorithms for solving overdefined systems of multivariate polynomial equations
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Differential cryptanalysis for multivariate schemes
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Inoculating multivariate schemes against differential attacks
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Cryptanalysis of HFE with internal perturbation
PKC'07 Proceedings of the 10th international conference on Practice and theory in public-key cryptography
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
Inoculating multivariate schemes against differential attacks
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
General fault attacks on multivariate public key cryptosystems
PQCrypto'11 Proceedings of the 4th international conference on Post-Quantum Cryptography
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Though the Perturbed Matsumoto-Imai (PMI) cryptosystem is considered insecure due to the recent differential attack of Fouque, Granboulan, and Stern, even more recently Ding and Gower showed that PMI can be repaired with the Plus (+) method of externally adding as few as 10 randomly chosen quadratic polynomials. Since relatively few extra polynomials are added, the attack complexity of a Gröbner basis attack on PMI+ will be roughly equal to that of PMI. Using Magma's implementation of the F4 Gröbner basis algorithm, we attack PMI with parameters q = 2, 0 ≤ r ≤ 10, and 14 ≤ n ≤ 59. Here, q is the number of field elements, n the number of equations/variables, and r the perturbation dimension. Based on our experimental results, we give estimates for the running time for such an attack. We use these estimates to judge the security of some proposed schemes, and we suggest more efficient schemes. In particular, we estimate that an attack using F4 against the parameters q = 2, r = 5, n = 96 (suggested in [7]) has a time complexity of less than 250 3-DES computations, which would be considered insecure for practical applications.