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Constructions of optimal balanced $ (m, n, \{4, 5\}, 1) $-OOSPCs
Efficient public-key operation in multivariate schemes
1. | Grupo SISTEMIC, Departamento de Ingeniería Electrónica, Universidad de Antioquia (UdeA), Calle 70 No. 52-21, Medellín, Colombia |
2. | Universidad Nacional de Colombia Sede Medellín, Calle 59 A N 63-20, Medellín, Colombia |
The public-key operation in multivariate encryption and signature schemes evaluates $ m $ quadratic polynomials in $ n $ variables. In this paper we analyze how fast this simple operation can be made. We optimize it for different finite fields on modern architectures. We provide an objective and inherent efficiency measure of our implementations, by comparing their performance with the peak performance of the CPU. In order to provide a fair comparison for different parameter sets, we also analyze the expected security based on the algebraic attack taking into consideration the hybrid approach. We compare the attack's efficiency for different finite fields and establish trends. We detail the role that the field equations play in the attack. We then provide a broad picture of efficiency of MQ-public-key operation against security.
References:
[1] |
C. Berbain, O. Billet and H. Gilbert, Efficient implementations of multivariate quadratic systems, in Selected Areas in Cryptography (eds. E. Biham and A. Youssef), vol. 4356 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2007,174–187. |
[2] |
D. J. Bernstein, J. Buchmann and E. Dahmen, Post-Quantum Cryptography, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-88702-7. |
[3] |
L. Bettale, J.-C. Faugère and L. Perret, Cryptanalysis of the TRMS Signature Scheme of PKC'05, Progress in cryptology – AFRICACRYPT 2008, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008,143–155.
doi: 10.1007/978-3-540-68164-9_10. |
[4] |
L. Bettale, J.-C. Faugere and L. Perret,
Hybrid approach for solving multivariate systems over finite fields, Journal of Mathematical Cryptology, 3 (2009), 177-197.
doi: 10.1515/JMC.2009.009. |
[5] |
L. Bettale, J.-C. Faugère and L. Perret, Solving polynomial systems over finite fields: Improved analysis of the hybrid approach, in ISSAC 2012 - 37th International Symposium on Symbolic and Algebraic Computation, ACM, Grenoble, France, 2012, 67–74.
doi: 10.1145/2442829.2442843. |
[6] |
L. Bettale, J.-C. Faugère and L. Perret,
Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic, Designs, Codes and Cryptography, 69 (2013), 1-52.
doi: 10.1007/s10623-012-9617-2. |
[7] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[8] |
C. Bouillaguet, H.-C. Chen, C.-M. Cheng, T. Chou, R. Niederhagen, A. Shamir and B.-Y. Yang, Fast exhaustive search for polynomial systems in $\mathbb{F}_2$, in Cryptographic Hardware and Embedded Systems, CHES 2010 (eds. S. Mangard and F.-X. Standaert), Springer Berlin Heidelberg, Berlin, Heidelberg, 2010, 203–218. |
[9] |
A.-T. Chen, M.-S. Chen, T.-R. Chen, C.-M. Cheng, J. Ding, E.-H. Kuo, F.-S. Lee and B.-Y. Yang, Sse implementation of multivariate pkcs on modern x86 cpus, in Cryptographic Hardware and Embedded Systems - CHES 2009 (eds. C. Clavier and K. Gaj), vol. 5747 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2009, 33–48.
doi: 10.1007/978-3-642-04138-9_3. |
[10] |
J. Ding, A. Petzoldt and L.-c. Wang, The cubic simple matrix encryption scheme, in Post-Quantum Cryptography: 6th International Workshop, PQCrypto 2014, Waterloo, ON, Canada, October 1-3, 2014. Proceedings (ed. M. Mosca), Springer International Publishing, Cham, 2014, 76–87. |
[11] |
J. Ding and D. Schmidt, Rainbow, a new multivariable polynomial signature scheme, in Applied Cryptography and Network Security (eds. J. Ioannidis, A. Keromytis and M. Yung), vol. 3531 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2005,164–175.
doi: 10.1007/11496137_12. |
[12] |
J. Ding, D. Schmidt and F. Werner, Algebraic attack on HFE revisited, in Information Security, 11th International Conference, ISC 2008, Taipei, Taiwan, September 15-18, 2008. Proceedings (eds. T.-C. Wu, C.-L. Lei, V. Rijmen and D.-T. Lee), vol. 5222 of Lecture Notes in Computer Science, Springer, 2008,215–227.
doi: 10.1007/978-3-540-85886-7_15. |
[13] |
V. Dubois, P.-A. Fouque, A. Shamir and J. Stern,
Practical cryptanalysis of Sflash, CRYPTO, 4622 (2007), 1-12.
doi: 10.1007/978-3-540-74143-5_1. |
[14] |
J.-C. Faugère and A. Joux, Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases, in Advances in cryptology–-CRYPTO 2003, vol. 2729 of Lecture Notes in Comput. Sci., Springer, Berlin, 2003, 44–60.
doi: 10.1007/978-3-540-45146-4_3. |
[15] |
S. Gueron and M. E. Kounavis, White Paper: Intel Carry-Less Multiplication Instruction and its Usage for Computing the GCM Mode, Technical report, Intel, 2010. |
[16] |
N. Howgrave-Graham, A hybrid lattice-reduction and meet-in-the-middle attack against ntru, in Advances in Cryptology - CRYPTO 2007 (ed. A. Menezes), vol. 4622 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2007,150–169.
doi: 10.1007/978-3-540-74143-5_9. |
[17] |
Intel®, Intel®64 and IA-32 Architectures Optimization Reference Manual, Technical report, Intel®, 2015. |
[18] |
A. Kipnis, J. Patarin and L. Goubin, Unbalanced oil and vinegar signature schemes, in Advances in cryptology–-EUROCRYPT '99 (Prague), vol. 1592 of Lecture Notes in Comput. Sci., Springer, Berlin, 1999,206–222.
doi: 10.1007/3-540-48910-X_15. |
[19] |
A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem by relinearization, in Advances in cryptology–-CRYPTO '99 (Santa Barbara, CA), vol. 1666 of Lecture Notes in Comput. Sci., Springer, Berlin, 1999, 19–30.
doi: 10.1007/3-540-48405-1_2. |
[20] |
T. Matsumoto and H. Imai, Public quadratic polynomial-tuples for efficient signature-verification and message-encryption, in Advances in cryptology–-EUROCRYPT '88 (Davos, 1988), vol. 330 of Lecture Notes in Comput. Sci., Springer, Berlin, 1988,419–453.
doi: 10.1007/3-540-45961-8_39. |
[21] |
NIST, Proposed submission requirements and evaluation criteria for the post-quantum cryptography standardization process, http://csrc.nist.gov/groups/ST/post-quantum-crypto/call-for-proposals-2016.html, 2016, Accessed: 08/26/2016. |
[22] |
J. Patarin, Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): Two
new families of asymmetric algorithms, in Advances in Cryptology—EUROCRYPT 96 (ed.
U. Maurer), vol. 1070 of Lecture Notes in Computer Science, Springer-Verlag, 1996, 33–48. |
[23] |
J. Patarin, N. Courtois and L. Goubin, FLASH, a fast multivariate signature algorithm, in Topics in Cryptology–-CT-RSA 2001 (San Francisco, CA), vol. 2020 of Lecture Notes in Comput. Sci., Springer, Berlin, 2001,298–307.
doi: 10.1007/3-540-45353-9_22. |
[24] |
J. Patarin, N. Courtois and L. Goubin, QUARTZ, 128-bit long digital signatures, in Topics in cryptology–-CT-RSA 2001 (San Francisco, CA), vol. 2020 of Lecture Notes in Comput. Sci., Springer, Berlin, 2001,282–297.
doi: 10.1007/3-540-45353-9_21. |
[25] |
J. Patarin, L. Goubin and N. Courtois, C*-+ and HM: Variations around two schemes of T. Matsumoto and H. Imai, in ASIACRYPT '98: Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security, Springer-Verlag, London, UK, 1998, 35–50.
doi: 10.1007/3-540-49649-1_4. |
[26] |
J. Porras, J. Baena and J. Ding, ZHFE, a new multivariate public key encryption scheme, in Post-Quantum Cryptography (ed. M. Mosca), vol. 8772 of Lecture Notes in Computer Science, Springer International Publishing, 2014,229–245.
doi: 10.1007/978-3-319-11659-4_14. |
[27] |
K. Sakumoto, T. Shirai and H. Hiwatari, Public-key identification schemes based on multivariate quadratic polynomials, in Advances in Cryptology - CRYPTO 2011 (ed. P. Rogaway), vol. 6841 of Lecture Notes in Computer Science, Springer Berlin / Heidelberg, 2011,706–723.
doi: 10.1007/978-3-642-22792-9_40. |
[28] |
C. Tao, A. Diene, S. Tang and J. Ding, Simple matrix scheme for encryption, in Post-Quantum Cryptography: 5th International Workshop, PQCrypto 2013, Limoges, France, June 4-7, 2013. Proceedings (ed. P. Gaborit), Springer Berlin Heidelberg, Berlin, Heidelberg, 6841 (2013), 231–242. |
[29] |
E. Thomae and C. Wolf, Solving underdetermined systems of multivariate quadratic equations revisited, in Public Key Cryptography – PKC 2012 (eds. M. Fischlin, J. Buchmann and M. Manulis), 7293 (2012), 156–171.
doi: 10.1007/978-3-642-30057-8_10. |
[30] |
R. C. Whaley and A. Petitet,
Minimizing development and maintenance costs in supporting persistently optimized BLAS, Software: Practice and Experience, 35 (2005), 101-121.
doi: 10.1002/spe.626. |
show all references
References:
[1] |
C. Berbain, O. Billet and H. Gilbert, Efficient implementations of multivariate quadratic systems, in Selected Areas in Cryptography (eds. E. Biham and A. Youssef), vol. 4356 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2007,174–187. |
[2] |
D. J. Bernstein, J. Buchmann and E. Dahmen, Post-Quantum Cryptography, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-88702-7. |
[3] |
L. Bettale, J.-C. Faugère and L. Perret, Cryptanalysis of the TRMS Signature Scheme of PKC'05, Progress in cryptology – AFRICACRYPT 2008, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008,143–155.
doi: 10.1007/978-3-540-68164-9_10. |
[4] |
L. Bettale, J.-C. Faugere and L. Perret,
Hybrid approach for solving multivariate systems over finite fields, Journal of Mathematical Cryptology, 3 (2009), 177-197.
doi: 10.1515/JMC.2009.009. |
[5] |
L. Bettale, J.-C. Faugère and L. Perret, Solving polynomial systems over finite fields: Improved analysis of the hybrid approach, in ISSAC 2012 - 37th International Symposium on Symbolic and Algebraic Computation, ACM, Grenoble, France, 2012, 67–74.
doi: 10.1145/2442829.2442843. |
[6] |
L. Bettale, J.-C. Faugère and L. Perret,
Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic, Designs, Codes and Cryptography, 69 (2013), 1-52.
doi: 10.1007/s10623-012-9617-2. |
[7] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[8] |
C. Bouillaguet, H.-C. Chen, C.-M. Cheng, T. Chou, R. Niederhagen, A. Shamir and B.-Y. Yang, Fast exhaustive search for polynomial systems in $\mathbb{F}_2$, in Cryptographic Hardware and Embedded Systems, CHES 2010 (eds. S. Mangard and F.-X. Standaert), Springer Berlin Heidelberg, Berlin, Heidelberg, 2010, 203–218. |
[9] |
A.-T. Chen, M.-S. Chen, T.-R. Chen, C.-M. Cheng, J. Ding, E.-H. Kuo, F.-S. Lee and B.-Y. Yang, Sse implementation of multivariate pkcs on modern x86 cpus, in Cryptographic Hardware and Embedded Systems - CHES 2009 (eds. C. Clavier and K. Gaj), vol. 5747 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2009, 33–48.
doi: 10.1007/978-3-642-04138-9_3. |
[10] |
J. Ding, A. Petzoldt and L.-c. Wang, The cubic simple matrix encryption scheme, in Post-Quantum Cryptography: 6th International Workshop, PQCrypto 2014, Waterloo, ON, Canada, October 1-3, 2014. Proceedings (ed. M. Mosca), Springer International Publishing, Cham, 2014, 76–87. |
[11] |
J. Ding and D. Schmidt, Rainbow, a new multivariable polynomial signature scheme, in Applied Cryptography and Network Security (eds. J. Ioannidis, A. Keromytis and M. Yung), vol. 3531 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2005,164–175.
doi: 10.1007/11496137_12. |
[12] |
J. Ding, D. Schmidt and F. Werner, Algebraic attack on HFE revisited, in Information Security, 11th International Conference, ISC 2008, Taipei, Taiwan, September 15-18, 2008. Proceedings (eds. T.-C. Wu, C.-L. Lei, V. Rijmen and D.-T. Lee), vol. 5222 of Lecture Notes in Computer Science, Springer, 2008,215–227.
doi: 10.1007/978-3-540-85886-7_15. |
[13] |
V. Dubois, P.-A. Fouque, A. Shamir and J. Stern,
Practical cryptanalysis of Sflash, CRYPTO, 4622 (2007), 1-12.
doi: 10.1007/978-3-540-74143-5_1. |
[14] |
J.-C. Faugère and A. Joux, Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases, in Advances in cryptology–-CRYPTO 2003, vol. 2729 of Lecture Notes in Comput. Sci., Springer, Berlin, 2003, 44–60.
doi: 10.1007/978-3-540-45146-4_3. |
[15] |
S. Gueron and M. E. Kounavis, White Paper: Intel Carry-Less Multiplication Instruction and its Usage for Computing the GCM Mode, Technical report, Intel, 2010. |
[16] |
N. Howgrave-Graham, A hybrid lattice-reduction and meet-in-the-middle attack against ntru, in Advances in Cryptology - CRYPTO 2007 (ed. A. Menezes), vol. 4622 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2007,150–169.
doi: 10.1007/978-3-540-74143-5_9. |
[17] |
Intel®, Intel®64 and IA-32 Architectures Optimization Reference Manual, Technical report, Intel®, 2015. |
[18] |
A. Kipnis, J. Patarin and L. Goubin, Unbalanced oil and vinegar signature schemes, in Advances in cryptology–-EUROCRYPT '99 (Prague), vol. 1592 of Lecture Notes in Comput. Sci., Springer, Berlin, 1999,206–222.
doi: 10.1007/3-540-48910-X_15. |
[19] |
A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem by relinearization, in Advances in cryptology–-CRYPTO '99 (Santa Barbara, CA), vol. 1666 of Lecture Notes in Comput. Sci., Springer, Berlin, 1999, 19–30.
doi: 10.1007/3-540-48405-1_2. |
[20] |
T. Matsumoto and H. Imai, Public quadratic polynomial-tuples for efficient signature-verification and message-encryption, in Advances in cryptology–-EUROCRYPT '88 (Davos, 1988), vol. 330 of Lecture Notes in Comput. Sci., Springer, Berlin, 1988,419–453.
doi: 10.1007/3-540-45961-8_39. |
[21] |
NIST, Proposed submission requirements and evaluation criteria for the post-quantum cryptography standardization process, http://csrc.nist.gov/groups/ST/post-quantum-crypto/call-for-proposals-2016.html, 2016, Accessed: 08/26/2016. |
[22] |
J. Patarin, Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): Two
new families of asymmetric algorithms, in Advances in Cryptology—EUROCRYPT 96 (ed.
U. Maurer), vol. 1070 of Lecture Notes in Computer Science, Springer-Verlag, 1996, 33–48. |
[23] |
J. Patarin, N. Courtois and L. Goubin, FLASH, a fast multivariate signature algorithm, in Topics in Cryptology–-CT-RSA 2001 (San Francisco, CA), vol. 2020 of Lecture Notes in Comput. Sci., Springer, Berlin, 2001,298–307.
doi: 10.1007/3-540-45353-9_22. |
[24] |
J. Patarin, N. Courtois and L. Goubin, QUARTZ, 128-bit long digital signatures, in Topics in cryptology–-CT-RSA 2001 (San Francisco, CA), vol. 2020 of Lecture Notes in Comput. Sci., Springer, Berlin, 2001,282–297.
doi: 10.1007/3-540-45353-9_21. |
[25] |
J. Patarin, L. Goubin and N. Courtois, C*-+ and HM: Variations around two schemes of T. Matsumoto and H. Imai, in ASIACRYPT '98: Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security, Springer-Verlag, London, UK, 1998, 35–50.
doi: 10.1007/3-540-49649-1_4. |
[26] |
J. Porras, J. Baena and J. Ding, ZHFE, a new multivariate public key encryption scheme, in Post-Quantum Cryptography (ed. M. Mosca), vol. 8772 of Lecture Notes in Computer Science, Springer International Publishing, 2014,229–245.
doi: 10.1007/978-3-319-11659-4_14. |
[27] |
K. Sakumoto, T. Shirai and H. Hiwatari, Public-key identification schemes based on multivariate quadratic polynomials, in Advances in Cryptology - CRYPTO 2011 (ed. P. Rogaway), vol. 6841 of Lecture Notes in Computer Science, Springer Berlin / Heidelberg, 2011,706–723.
doi: 10.1007/978-3-642-22792-9_40. |
[28] |
C. Tao, A. Diene, S. Tang and J. Ding, Simple matrix scheme for encryption, in Post-Quantum Cryptography: 5th International Workshop, PQCrypto 2013, Limoges, France, June 4-7, 2013. Proceedings (ed. P. Gaborit), Springer Berlin Heidelberg, Berlin, Heidelberg, 6841 (2013), 231–242. |
[29] |
E. Thomae and C. Wolf, Solving underdetermined systems of multivariate quadratic equations revisited, in Public Key Cryptography – PKC 2012 (eds. M. Fischlin, J. Buchmann and M. Manulis), 7293 (2012), 156–171.
doi: 10.1007/978-3-642-30057-8_10. |
[30] |
R. C. Whaley and A. Petitet,
Minimizing development and maintenance costs in supporting persistently optimized BLAS, Software: Practice and Experience, 35 (2005), 101-121.
doi: 10.1002/spe.626. |














| 2 | 2(wfe) | 3 | 3(wfe) | 13 | 0.23 | 0.12 | 0.05 | 0.02 | |
0.76 | 0.55 | 0.61 | 0.50 | 0.27 |
|
13(wfe) | 73 | 919 | 117659 |
| 2 | 2(wfe) | 3 | 3(wfe) | 13 | 0.23 | 0.12 | 0.05 | 0.02 | |
0.76 | 0.55 | 0.61 | 0.50 | 0.27 |
|
13(wfe) | 73 | 919 | 117659 |
|
||
|
||
| ||
| ||
bit sec. level | ||||||||
80 | 88 | 54 | 38 | 31 | 32 | 28 | 25 | 25 |
100 | 112 | 68 | 48 | 39 | 40 | 36 | 31 | 31 |
112 | 126 | 77 | 54 | 43 | 45 | 41 | 34 | 35 |
128 | 145 | 88 | 62 | 50 | 52 | 48 | 39 | 40 |
192 | 222 | 133 | 95 | 75 | 79 | 75 | 59 | 61 |
bit sec. level | ||||||||
80 | 88 | 54 | 38 | 31 | 32 | 28 | 25 | 25 |
100 | 112 | 68 | 48 | 39 | 40 | 36 | 31 | 31 |
112 | 126 | 77 | 54 | 43 | 45 | 41 | 34 | 35 |
128 | 145 | 88 | 62 | 50 | 52 | 48 | 39 | 40 |
192 | 222 | 133 | 95 | 75 | 79 | 75 | 59 | 61 |
bit sec. level | ||||||||
80 | 97 | 61 | 55 | 55 | 55 | 54 | 51 | 49 |
100 | 123 | 77 | 69 | 69 | 69 | 68 | 65 | 63 |
112 | 138 | 86 | 77 | 77 | 77 | 76 | 73 | 71 |
128 | 158 | 99 | 88 | 88 | 88 | 87 | 84 | 81 |
192 | 240 | 150 | 133 | 132 | 132 | 130 | 127 | 124 |
bit sec. level | ||||||||
80 | 97 | 61 | 55 | 55 | 55 | 54 | 51 | 49 |
100 | 123 | 77 | 69 | 69 | 69 | 68 | 65 | 63 |
112 | 138 | 86 | 77 | 77 | 77 | 76 | 73 | 71 |
128 | 158 | 99 | 88 | 88 | 88 | 87 | 84 | 81 |
192 | 240 | 150 | 133 | 132 | 132 | 130 | 127 | 124 |
bit sec.level | ||||||||
80 | 480 | 315 | 135 | 98 | 100 | 89 | 81 | 82 |
100 | 600 | 394 | 169 | 121 | 125 | 114 | 100 | 102 |
112 | 672 | 441 | 189 | 135 | 141 | 129 | 111 | 113 |
128 | 768 | 504 | 216 | 154 | 161 | 149 | 126 | 129 |
192 | 1152 | 756 | 324 | 230 | 243 | 229 | 187 | 192 |
bit sec.level | ||||||||
80 | 480 | 315 | 135 | 98 | 100 | 89 | 81 | 82 |
100 | 600 | 394 | 169 | 121 | 125 | 114 | 100 | 102 |
112 | 672 | 441 | 189 | 135 | 141 | 129 | 111 | 113 |
128 | 768 | 504 | 216 | 154 | 161 | 149 | 126 | 129 |
192 | 1152 | 756 | 324 | 230 | 243 | 229 | 187 | 192 |
bit sec.level | ||||||||
80 | 176 | 121 | 61 | 47 | 48 | 44 | 41 | 42 |
100 | 216 | 147 | 72 | 55 | 56 | 52 | 48 | 48 |
112 | 240 | 163 | 79 | 59 | 61 | 57 | 51 | 52 |
128 | 272 | 184 | 88 | 66 | 68 | 64 | 56 | 57 |
192 | 400 | 268 | 124 | 91 | 95 | 91 | 77 | 78 |
bit sec.level | ||||||||
80 | 176 | 121 | 61 | 47 | 48 | 44 | 41 | 42 |
100 | 216 | 147 | 72 | 55 | 56 | 52 | 48 | 48 |
112 | 240 | 163 | 79 | 59 | 61 | 57 | 51 | 52 |
128 | 272 | 184 | 88 | 66 | 68 | 64 | 56 | 57 |
192 | 400 | 268 | 124 | 91 | 95 | 91 | 77 | 78 |
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