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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 , ().
|
[2] |
N. Courtois, The security of hidden field equations (HFE), in Progr. Crypt., CT-RSA '01, Springer-Verlag, 2001, 266-281.
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), 273-282.
doi: 10.1142/S0219498804000861. |
[5] |
J. Ding and D. Schmidt, The new TTM implementation is not secure, in Workshop Coding Crypt. Combin., CCC2003, Birkhauser Verlag, 2004, 113-128. |
[6] |
V. Dubois, P.-A. Fouque, A. Shamir and J. Stern, Practical cryptanalysis of SFLASH, in Adv. Crypt. - CRYPTO 2007, Springer-Verlag, 2007, 1-12.
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, Springer-Verlag, 2007, 264-275.
doi: 10.1007/978-3-540-72540-4_15. |
[8] |
A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhauser Verlag, Basel, 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, New York, 1979. |
[10] |
L. Goubin and N. Courtois, Cryptanalysis of the TTM cryptosystem, in Adv. Crypt. - ASIACRYPT 2000, Springer-Verlag, 2000, 44-57.
doi: 10.1007/3-540-44448-3_4. |
[11] |
E.-M. G. M. Hubbers, Nilpotent Jacobians, Ph.D thesis, Univ. Nijmegen, 1998. |
[12] |
H. W. E. Jung, Über ganze birationale transformationen der ebene, J. Reine Angew. Math., 184 (1942), 161-174. |
[13] |
A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem, in Adv. Crypt. - CRYPTO '99, Springer-Verlag, 1999, 19-30.
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), 963-977.
doi: 10.1112/S0010437X07003399. |
[15] |
W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, 3 (1953), 33-41. |
[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, Springer-Verlag, 1988, 419-453.
doi: 10.1007/3-540-45961-8_39. |
[18] |
S. Maubach, Polynomial automorphisms over finite fields, Serdica Math. J., 27 (2001), 343-350. |
[19] |
T. T. Moh, A fast public key system with signature and master key functions, Comm. Algebra, 27 (1999), 2207-2222.
doi: 10.1080/00927879908826559. |
[20] |
T. T. Moh, An application of algebraic geometry to encryption: tame transformation method, Rev. Mat. Iberoamericana, 19 (2003), 667-685.
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 , ().
|
[22] |
M. Nagata, On Automorphism Group of $k[x, y]$, Kinokuniya, Tokyo, 1972. |
[23] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88, in Adv. Crypt. - CRYPTO '95, Springer-Verlag, 1995, 248-261.
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, Springer-Verlag, 1996, 33-48. |
[25] |
J. Patarin, The oil and vinegar signature scheme, in Dagstuhl Workshop on Cryptography, 1997. |
[26] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88, Des. Codes Crypt., 20 (2000), 175-209.
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, Springer-Verlag, 1998, 35-50. |
[28] |
J. Patarin, N. Courtois and L. Goubin, FLASH, a fast multivariate signature algorithm, in Progr. Crypt., CT-RSA '01, Springer-Verlag, 2001, 297-307.
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, Springer-Verlag, 1997, 369-380. |
[30] |
K. Rusek, Polynomial Automorphisms, Preprint 456, Inst. Math., Polish Acad. Sci., IMPAN, Warsaw, 1989. |
[31] |
I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc., 17 (2004), 181-196.
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), 197-227.
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., IEEE, 1994, 124-134.
doi: 10.1109/SFCS.1994.365700. |
[34] |
M. K. Smith, Stably tame automorphisms, J. Pure Appl. Algebra, 58 (1989), 209-212.
doi: 10.1016/0022-4049(89)90158-8. |
[35] |
S. Spodzieja, On the Nagata automorphism, Univ. Iagell. Acta Math., 1298 (2007), 131-136. |
show all references
References:
[1] |
J.-M. Chen and T. T. Moh, On the Goubin-Courtois attack on TTM,, available at , ().
|
[2] |
N. Courtois, The security of hidden field equations (HFE), in Progr. Crypt., CT-RSA '01, Springer-Verlag, 2001, 266-281.
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), 273-282.
doi: 10.1142/S0219498804000861. |
[5] |
J. Ding and D. Schmidt, The new TTM implementation is not secure, in Workshop Coding Crypt. Combin., CCC2003, Birkhauser Verlag, 2004, 113-128. |
[6] |
V. Dubois, P.-A. Fouque, A. Shamir and J. Stern, Practical cryptanalysis of SFLASH, in Adv. Crypt. - CRYPTO 2007, Springer-Verlag, 2007, 1-12.
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, Springer-Verlag, 2007, 264-275.
doi: 10.1007/978-3-540-72540-4_15. |
[8] |
A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhauser Verlag, Basel, 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, New York, 1979. |
[10] |
L. Goubin and N. Courtois, Cryptanalysis of the TTM cryptosystem, in Adv. Crypt. - ASIACRYPT 2000, Springer-Verlag, 2000, 44-57.
doi: 10.1007/3-540-44448-3_4. |
[11] |
E.-M. G. M. Hubbers, Nilpotent Jacobians, Ph.D thesis, Univ. Nijmegen, 1998. |
[12] |
H. W. E. Jung, Über ganze birationale transformationen der ebene, J. Reine Angew. Math., 184 (1942), 161-174. |
[13] |
A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem, in Adv. Crypt. - CRYPTO '99, Springer-Verlag, 1999, 19-30.
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), 963-977.
doi: 10.1112/S0010437X07003399. |
[15] |
W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, 3 (1953), 33-41. |
[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, Springer-Verlag, 1988, 419-453.
doi: 10.1007/3-540-45961-8_39. |
[18] |
S. Maubach, Polynomial automorphisms over finite fields, Serdica Math. J., 27 (2001), 343-350. |
[19] |
T. T. Moh, A fast public key system with signature and master key functions, Comm. Algebra, 27 (1999), 2207-2222.
doi: 10.1080/00927879908826559. |
[20] |
T. T. Moh, An application of algebraic geometry to encryption: tame transformation method, Rev. Mat. Iberoamericana, 19 (2003), 667-685.
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 , ().
|
[22] |
M. Nagata, On Automorphism Group of $k[x, y]$, Kinokuniya, Tokyo, 1972. |
[23] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88, in Adv. Crypt. - CRYPTO '95, Springer-Verlag, 1995, 248-261.
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, Springer-Verlag, 1996, 33-48. |
[25] |
J. Patarin, The oil and vinegar signature scheme, in Dagstuhl Workshop on Cryptography, 1997. |
[26] |
J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt '88, Des. Codes Crypt., 20 (2000), 175-209.
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, Springer-Verlag, 1998, 35-50. |
[28] |
J. Patarin, N. Courtois and L. Goubin, FLASH, a fast multivariate signature algorithm, in Progr. Crypt., CT-RSA '01, Springer-Verlag, 2001, 297-307.
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, Springer-Verlag, 1997, 369-380. |
[30] |
K. Rusek, Polynomial Automorphisms, Preprint 456, Inst. Math., Polish Acad. Sci., IMPAN, Warsaw, 1989. |
[31] |
I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc., 17 (2004), 181-196.
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), 197-227.
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., IEEE, 1994, 124-134.
doi: 10.1109/SFCS.1994.365700. |
[34] |
M. K. Smith, Stably tame automorphisms, J. Pure Appl. Algebra, 58 (1989), 209-212.
doi: 10.1016/0022-4049(89)90158-8. |
[35] |
S. Spodzieja, On the Nagata automorphism, Univ. Iagell. Acta Math., 1298 (2007), 131-136. |
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