Efficient public-key operation in multivariate schemes

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May 2019, 13(2): 343-371. doi: 10.3934/amc.2019023

Efficient public-key operation in multivariate schemes

Felipe Cabarcas ^1,, , Daniel Cabarcas ^2, and John Baena ^2,

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

^* Corresponding author

Received July 2018 Revised January 2019 Published February 2019

Fund Project: This work was partially supported by "Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas", Colciencias (Colombia), Project No. 111865842333 and Contract No. 049-2015

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.

Keywords: Multivariate public-key cryptography, encryption schemes, signature schemes, ZHFE, UOV, Rainbow, algebraic attack, x86 architecture, SIMD, PCLMULQDQ.

Mathematics Subject Classification: Primary: 11T71, 94A60; Secondary: 14G50.

Citation: Felipe Cabarcas, Daniel Cabarcas, John Baena. Efficient public-key operation in multivariate schemes. Advances in Mathematics of Communications, 2019, 13 (2) : 343-371. doi: 10.3934/amc.2019023

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.Google Scholar
[2]	D. J. Bernstein, J. Buchmann and E. Dahmen, Post-Quantum Cryptography, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-88702-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[17]	Intel®, Intel®64 and IA-32 Architectures Optimization Reference Manual, Technical report, Intel®, 2015.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar

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.Google Scholar
[2]	D. J. Bernstein, J. Buchmann and E. Dahmen, Post-Quantum Cryptography, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-88702-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[17]	Intel®, Intel®64 and IA-32 Architectures Optimization Reference Manual, Technical report, Intel®, 2015.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. Public-key-operation throughput for different number of variables for $ m = n $. The register type used in the program to hold the message is fixed for each finite field and shown in brackets. For GF(2) it was a vector each of 4 (64-bit-unsigned int); for the GF(3) it was 16-bit-unsigned integers; for 13 and 73, it was 32-bit-unsigned integers; for 919 and 117659 it was 64-bit-unsigned integers. Finally, for the $ 2^{64} $ and $ 2^{128} $ it was used 128-bit registers (coded with assembler intrinsics.)

$q$	2	2(wfe)	3	3(wfe)	13	$\beta_0$	0.23	0.12	0.05	0.02
$\beta_0$	0.76	0.55	0.61	0.50	0.27	$q$	13(wfe)	73	919	117659

$q$	$c$	$b$
$2$	$1.43*10^{-4}$	$0.61$
$2$ wfe	$1.58*10^{-5}$	$0.58$
$3$	$8.62*10^{-6}$	$0.95$
$3$ wfe	$2.55*10^{-6}$	$0.98$
$13$	$2.32*10^{-6}$	$1.53$
$13$ wfe	$3.16*10^{-5}$	$1.35$
$73$	$1.30*10^{-6}$	$1.76$
$919$	$2.73*10^{-5}$	$1.63$
$117659$	$2.50*10^{-3}$	$1.66$
$2^{64}$	$3.97*10^{-7}$	$2.27$
$2^{128}$	$1.20*10^{-6}$	$2.18$

	$q$
bit sec. level	$2$	$3$	$13$	$73$	$919$	$117659$	$2^{64}$	$2^{128}$
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

	$q$
bit sec.level	$2$	$3$	$13$	$73$	$919$	$117659$	$2^{64}$	$2^{128}$
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

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