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On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$
1. | Interdisciplinary Graduate School of Science and Engineering, Shimane University, 1060 Nishikawatsu-cho, Matsue-shi, Shimane 690-8504, Japan |
2. | Security Research Department, Center of Technology Innovation-Systems Engineering, Hitachi, Ltd., 292, Yoshida-cho, Totsuka-ku, Yokohama-shi, Kanagawa 244-0817, Japan |
3. | Institute of Mathematics for Industry, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka-shi, Fukuoka 819-0395, Japan |
References:
[1] |
J.-M. Chen and T. T. Moh, On the Goubin-Courtois attack on TTM,, available at , (). Google Scholar |
[2] |
N. Courtois, The security of hidden field equations (HFE),, in Progr. Crypt., (2001), 266.
doi: 10.1007/3-540-45353-9_20. |
[3] |
J. Ding, J. E. Gower and D. S. Schmidt, Multivariate Public Key Cryptosystems,, Springer, (2006).
|
[4] |
J. Ding and T. Hodges, Cryptanalysis of an implementation scheme of TTM,, J. Algebra Appl., 3 (2004), 273.
doi: 10.1142/S0219498804000861. |
[5] |
J. Ding and D. Schmidt, The new TTM implementation is not secure,, in Workshop Coding Crypt. Combin., (2004), 113.
|
[6] |
V. Dubois, P.-A. Fouque, A. Shamir and J. Stern, Practical cryptanalysis of SFLASH,, in Adv. Crypt. - CRYPTO 2007, (2007), 1.
doi: 10.1007/978-3-540-74143-5_1. |
[7] |
V. Dubois, P.-A. Fouque and J. Stern, Cryptanalysis of SFLASH with slightly modified parameters,, in Adv. Crypt. - EUROCRYPT 2007, (2007), 264.
doi: 10.1007/978-3-540-72540-4_15. |
[8] |
A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture,, Birkhauser Verlag, (2000).
doi: 10.1007/978-3-0348-8440-2. |
[9] |
M. R. Garey and D. S. Johnson, Computer and Intractability: A Guide to the Theory of NP-completeness,, Freeman, (1979).
|
[10] |
L. Goubin and N. Courtois, Cryptanalysis of the TTM cryptosystem,, in Adv. Crypt. - ASIACRYPT 2000, (2000), 44.
doi: 10.1007/3-540-44448-3_4. |
[11] |
E.-M. G. M. Hubbers, Nilpotent Jacobians,, Ph.D thesis, (1998). Google Scholar |
[12] |
H. W. E. Jung, Über ganze birationale transformationen der ebene,, J. Reine Angew. Math., 184 (1942), 161.
|
[13] |
A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem,, in Adv. Crypt. - CRYPTO '99, (1999), 19.
doi: 10.1007/3-540-48405-1_2. |
[14] |
T. Kishimoto, A new proof of the non-tameness of the Nagata automorphism from the point of view of the Sarkisov program,, Compositio Math., 144 (2008), 963.
doi: 10.1112/S0010437X07003399. |
[15] |
W. van der Kulk, On polynomial rings in two variables,, Nieuw Archief voor Wiskunde, 3 (1953), 33.
|
[16] |
D. Lin, J.-C. Faugere, L. Perret and T. Wang, On enumeration of polynomial equivalence classes and their application to MPKC,, available at , ().
doi: 10.1016/j.ffa.2011.09.001. |
[17] |
T. Matsumoto and H. Imai, Public quadratic polynominal-tuples for efficient signature-verification and message-encryption,, in Adv. Crypt. - EUROCRYPT '88, (1988), 419.
doi: 10.1007/3-540-45961-8_39. |
[18] |
S. Maubach, Polynomial automorphisms over finite fields,, Serdica Math. J., 27 (2001), 343.
|
[19] |
T. T. Moh, A fast public key system with signature and master key functions,, Comm. Algebra, 27 (1999), 2207.
doi: 10.1080/00927879908826559. |
[20] |
T. T. Moh, An application of algebraic geometry to encryption: tame transformation method,, Rev. Mat. Iberoamericana, 19 (2003), 667.
doi: 10.4171/RMI/364. |
[21] |
T. T. Moh, J.-M. Chen, and B.-Y. Yang, Building instances of TTM immune to the Goubin-Courtois attack and the Ding-Schmidt attack,, available at , (). Google Scholar |
[22] |
M. Nagata, On Automorphism Group of $k[x, y]$,, Kinokuniya, (1972).
|
[23] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88,, in Adv. Crypt. - CRYPTO '95, (1995), 248.
doi: 10.1007/3-540-44750-4_20. |
[24] |
J. Patarin, Hidden field equations (HFE) and isomorphism of polynomials (IP): two new families of asymmetric algorithms,, in Adv. Crypt. - EUROCRYPT '96, (1996), 33. Google Scholar |
[25] |
J. Patarin, The oil and vinegar signature scheme,, in Dagstuhl Workshop on Cryptography, (1997). Google Scholar |
[26] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88,, Des. Codes Crypt., 20 (2000), 175.
doi: 10.1023/A:1008341625464. |
[27] |
J. Patarin, N. Courtois and L. Goubin, $C_{-+}^{*}$ and HM: variations around two schemes of T. Matsumoto and H. Imai,, in Adv. Crypt. - ASIACRYPT '98, (1998), 35. Google Scholar |
[28] |
J. Patarin, N. Courtois and L. Goubin, FLASH, a fast multivariate signature algorithm,, in Progr. Crypt., (2001), 297.
doi: 10.1007/3-540-45353-9_22. |
[29] |
J. Patarin and L. Goubin, Asymmetric cryptography with S-boxes,, in 1st Int. Conf. Inf. Sec. Crypt. - ICISC '97, (1997), 369. Google Scholar |
[30] |
K. Rusek, Polynomial Automorphisms,, Preprint 456, (1989). Google Scholar |
[31] |
I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials,, J. Amer. Math. Soc., 17 (2004), 181.
doi: 10.1090/S0894-0347-03-00438-7. |
[32] |
I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables,, J. Amer. Math. Soc., 17 (2004), 197.
doi: 10.1090/S0894-0347-03-00440-5. |
[33] |
P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring,, in 35th Ann. Symp. Found. Comp. Sci., (1994), 124.
doi: 10.1109/SFCS.1994.365700. |
[34] |
M. K. Smith, Stably tame automorphisms,, J. Pure Appl. Algebra, 58 (1989), 209.
doi: 10.1016/0022-4049(89)90158-8. |
[35] |
S. Spodzieja, On the Nagata automorphism,, Univ. Iagell. Acta Math., 1298 (2007), 131.
|
show all references
References:
[1] |
J.-M. Chen and T. T. Moh, On the Goubin-Courtois attack on TTM,, available at , (). Google Scholar |
[2] |
N. Courtois, The security of hidden field equations (HFE),, in Progr. Crypt., (2001), 266.
doi: 10.1007/3-540-45353-9_20. |
[3] |
J. Ding, J. E. Gower and D. S. Schmidt, Multivariate Public Key Cryptosystems,, Springer, (2006).
|
[4] |
J. Ding and T. Hodges, Cryptanalysis of an implementation scheme of TTM,, J. Algebra Appl., 3 (2004), 273.
doi: 10.1142/S0219498804000861. |
[5] |
J. Ding and D. Schmidt, The new TTM implementation is not secure,, in Workshop Coding Crypt. Combin., (2004), 113.
|
[6] |
V. Dubois, P.-A. Fouque, A. Shamir and J. Stern, Practical cryptanalysis of SFLASH,, in Adv. Crypt. - CRYPTO 2007, (2007), 1.
doi: 10.1007/978-3-540-74143-5_1. |
[7] |
V. Dubois, P.-A. Fouque and J. Stern, Cryptanalysis of SFLASH with slightly modified parameters,, in Adv. Crypt. - EUROCRYPT 2007, (2007), 264.
doi: 10.1007/978-3-540-72540-4_15. |
[8] |
A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture,, Birkhauser Verlag, (2000).
doi: 10.1007/978-3-0348-8440-2. |
[9] |
M. R. Garey and D. S. Johnson, Computer and Intractability: A Guide to the Theory of NP-completeness,, Freeman, (1979).
|
[10] |
L. Goubin and N. Courtois, Cryptanalysis of the TTM cryptosystem,, in Adv. Crypt. - ASIACRYPT 2000, (2000), 44.
doi: 10.1007/3-540-44448-3_4. |
[11] |
E.-M. G. M. Hubbers, Nilpotent Jacobians,, Ph.D thesis, (1998). Google Scholar |
[12] |
H. W. E. Jung, Über ganze birationale transformationen der ebene,, J. Reine Angew. Math., 184 (1942), 161.
|
[13] |
A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem,, in Adv. Crypt. - CRYPTO '99, (1999), 19.
doi: 10.1007/3-540-48405-1_2. |
[14] |
T. Kishimoto, A new proof of the non-tameness of the Nagata automorphism from the point of view of the Sarkisov program,, Compositio Math., 144 (2008), 963.
doi: 10.1112/S0010437X07003399. |
[15] |
W. van der Kulk, On polynomial rings in two variables,, Nieuw Archief voor Wiskunde, 3 (1953), 33.
|
[16] |
D. Lin, J.-C. Faugere, L. Perret and T. Wang, On enumeration of polynomial equivalence classes and their application to MPKC,, available at , ().
doi: 10.1016/j.ffa.2011.09.001. |
[17] |
T. Matsumoto and H. Imai, Public quadratic polynominal-tuples for efficient signature-verification and message-encryption,, in Adv. Crypt. - EUROCRYPT '88, (1988), 419.
doi: 10.1007/3-540-45961-8_39. |
[18] |
S. Maubach, Polynomial automorphisms over finite fields,, Serdica Math. J., 27 (2001), 343.
|
[19] |
T. T. Moh, A fast public key system with signature and master key functions,, Comm. Algebra, 27 (1999), 2207.
doi: 10.1080/00927879908826559. |
[20] |
T. T. Moh, An application of algebraic geometry to encryption: tame transformation method,, Rev. Mat. Iberoamericana, 19 (2003), 667.
doi: 10.4171/RMI/364. |
[21] |
T. T. Moh, J.-M. Chen, and B.-Y. Yang, Building instances of TTM immune to the Goubin-Courtois attack and the Ding-Schmidt attack,, available at , (). Google Scholar |
[22] |
M. Nagata, On Automorphism Group of $k[x, y]$,, Kinokuniya, (1972).
|
[23] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88,, in Adv. Crypt. - CRYPTO '95, (1995), 248.
doi: 10.1007/3-540-44750-4_20. |
[24] |
J. Patarin, Hidden field equations (HFE) and isomorphism of polynomials (IP): two new families of asymmetric algorithms,, in Adv. Crypt. - EUROCRYPT '96, (1996), 33. Google Scholar |
[25] |
J. Patarin, The oil and vinegar signature scheme,, in Dagstuhl Workshop on Cryptography, (1997). Google Scholar |
[26] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88,, Des. Codes Crypt., 20 (2000), 175.
doi: 10.1023/A:1008341625464. |
[27] |
J. Patarin, N. Courtois and L. Goubin, $C_{-+}^{*}$ and HM: variations around two schemes of T. Matsumoto and H. Imai,, in Adv. Crypt. - ASIACRYPT '98, (1998), 35. Google Scholar |
[28] |
J. Patarin, N. Courtois and L. Goubin, FLASH, a fast multivariate signature algorithm,, in Progr. Crypt., (2001), 297.
doi: 10.1007/3-540-45353-9_22. |
[29] |
J. Patarin and L. Goubin, Asymmetric cryptography with S-boxes,, in 1st Int. Conf. Inf. Sec. Crypt. - ICISC '97, (1997), 369. Google Scholar |
[30] |
K. Rusek, Polynomial Automorphisms,, Preprint 456, (1989). Google Scholar |
[31] |
I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials,, J. Amer. Math. Soc., 17 (2004), 181.
doi: 10.1090/S0894-0347-03-00438-7. |
[32] |
I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables,, J. Amer. Math. Soc., 17 (2004), 197.
doi: 10.1090/S0894-0347-03-00440-5. |
[33] |
P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring,, in 35th Ann. Symp. Found. Comp. Sci., (1994), 124.
doi: 10.1109/SFCS.1994.365700. |
[34] |
M. K. Smith, Stably tame automorphisms,, J. Pure Appl. Algebra, 58 (1989), 209.
doi: 10.1016/0022-4049(89)90158-8. |
[35] |
S. Spodzieja, On the Nagata automorphism,, Univ. Iagell. Acta Math., 1298 (2007), 131.
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