American Institute of Mathematical Sciences

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doi: 10.3934/amc.2020126

A practicable timing attack against HQC and its countermeasure

Guillaume Wafo-Tapa 1, , Slim Bettaieb 1, , Loïc Bidoux 1, , Philippe Gaborit 2, and  Etienne Marcatel 3,

1. 

Worldline, ZI Rue de la pointe, 59113 Seclin, France
2. 

University of Limoges, XLIM-DMI, 123, Av. Albert Thomas, 87060 Limoges, France
3. 

Atos Trustway, Avenue Jean Jaurès, 78340 Les Clayes-sous-Bois, University of Grenoble Alpes, CNRS, IF, 38000 Grenoble, France

Received  August 2019 Revised  January 2020 Published  December 2020

In this paper, we present a practicable chosen ciphertext timing attack retrieving the secret key of HQC. The attack exploits a correlation between the weight of the error to be decoded and the running time of the decoding algorithm of BCH codes. For the 128-bit security parameters of HQC, the attack runs in less than a minute on a desktop computer using roughly 6000 decoding requests and has a success probability of approximately 93 percent. To prevent this attack, we provide an implementation of a constant time algorithm for the decoding of BCH codes. Our implementation of the countermeasure achieves a constant time execution of the decoding process without a significant performance penalty.

Keywords: HQC, BCH decoding, timing attack, constant time implementation.
Mathematics Subject Classification: Primary: 11T71, 94B35; Secondary: 94B05.
Citation: Guillaume Wafo-Tapa, Slim Bettaieb, Loïc Bidoux, Philippe Gaborit, Etienne Marcatel. A practicable timing attack against HQC and its countermeasure. Advances in Mathematics of Communications, doi: 10.3934/amc.2020126
References:
[1]

C. Aguilar-Melchor, N. Aragon, S. Bettaieb, L. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit, E. Persichetti and G. Zémor, Hamming Quasi-Cyclic (HQC), 2017. Google Scholar

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[5]

E. R. Berlekamp, Non-binary BCH Decoding, Technical report, North Carolina State University. Dept. of Statistics, 1966. Google Scholar

[6]

D. J. Bernstein, T. Chou and P. Schwabe, Mcbits: Fast constant-time code-based cryptography, in International Workshop on Cryptographic Hardware and Embedded Systems, Springer, (2013), 250–272. doi: 10.1007/978-3-642-40349-1_15.  Google Scholar

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T. Chou, McBits revisited, in International Conference on Cryptographic Hardware and Embedded Systems, Springer, (2017), 213–231. doi: 10.1007/978-3-319-66787-4_11.  Google Scholar

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R. Chandra. Bose and D. K. Ray-Chaudhuri, On a class of error correcting binary group codes, Information and Control, 3 (1960), 68-79.  doi: 10.1016/S0019-9958(60)90287-4.  Google Scholar

[10]

J.-P. D'Anvers, F. Vercauteren and Ingrid Verbauwhede, On the impact of decryption failures on the security of LWE/LWR based schemes, IACR Cryptology ePrint Archive, (2018), 1089. Google Scholar

[11]

È. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.   Google Scholar

[12]

S. Gao and T. Mateer, Additive fast fourier transforms over finite fields, IEEE Transactions on Information Theory, 56 (2010), 6265-6272.  doi: 10.1109/TIT.2010.2079016.  Google Scholar

[13]

A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.   Google Scholar

[14]

D. Hofheinz, K. Hövelmanns and E. Kiltz, A modular analysis of the Fujisaki-Okamoto transformation, in Theory of Cryptography Conference, Springer, (2017), 341–371.  Google Scholar

[15]

L. L. Joiner and J. J. Komo, Decoding binary BCH codes, in Southeastcon '95, 1995. doi: 10.1109/SECON.1995.513059.  Google Scholar

[16]

S. Lin and D. J. Costello, in Error control coding, Prentice Hall Englewood Cliffs, (2004)., Google Scholar

[17]

X. Lu, Y. Liu, Z. Zhang, D. Jia, H. Xue, J. He, B. Li, K. Wang, Z. Liu and H. Yang, LAC: Practical Ring-LWE based Public-Key Encryption with Byte-Level Modulus, IACR Cryptology ePrint Archive, (2018), 1009. Google Scholar

[18]

M. Walters and S. Sinha Roy, Constant-time BCH Error-Correcting Code, in 2020 IEEE International Symposium on Circuits and Systems (ISCAS), Sevilla, (2020), 1–5. doi: 10.1109/ISCAS45731.2020.9180846.  Google Scholar

[19]

Y. Xu, Implementation of Berlekamp-Massey algorithm without inversion, IEE Proceedings I-Communications, Speech and Vision, 138 (1991), 138-140.   Google Scholar

show all references

References:
[1]

C. Aguilar-Melchor, N. Aragon, S. Bettaieb, L. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit, E. Persichetti and G. Zémor, Hamming Quasi-Cyclic (HQC), 2017. Google Scholar

[2]

C. Aguilar-Melchor, N. Aragon, S. Bettaieb, L. Bidoux, O. Blazy, J.-C. Deneuville, P. Gaborit and G. Zémor, Rank Quasi-Cyclic (RQC), 2017. Google Scholar

[3]

C. Aguilar-Melchor, O. Blazy, J.-C. Deneuville, P. Gaborit and G. Zémor, Efficient encryption from random quasi-cyclic codes, IEEE Transactions on Information Theory, 64 (2018), 3927-3943.  doi: 10.1109/TIT.2018.2804444.  Google Scholar

[4]

S. Bettaieb, L. Bidoux, P. Gaborit and E. Marcatel, Preventing timing attacks against RQC using constant time decoding of Gabidulin codes, in International Conference on Post-Quantum Cryptography, Springer, (2019), 371–386. doi: 10.1007/978-3-030-25510-7_20.  Google Scholar

[5]

E. R. Berlekamp, Non-binary BCH Decoding, Technical report, North Carolina State University. Dept. of Statistics, 1966. Google Scholar

[6]

D. J. Bernstein, T. Chou and P. Schwabe, Mcbits: Fast constant-time code-based cryptography, in International Workshop on Cryptographic Hardware and Embedded Systems, Springer, (2013), 250–272. doi: 10.1007/978-3-642-40349-1_15.  Google Scholar

[7]

D. J. Bernstein and B-Y. Yang, Fast constant-time gcd computation and modular inversion, in IACR Transactions on Cryptographic Hardware and Embedded Systems, (2019), 340–398. doi: 10.46586/tches.v2019.i3.340-398.  Google Scholar

[8]

T. Chou, McBits revisited, in International Conference on Cryptographic Hardware and Embedded Systems, Springer, (2017), 213–231. doi: 10.1007/978-3-319-66787-4_11.  Google Scholar

[9]

R. Chandra. Bose and D. K. Ray-Chaudhuri, On a class of error correcting binary group codes, Information and Control, 3 (1960), 68-79.  doi: 10.1016/S0019-9958(60)90287-4.  Google Scholar

[10]

J.-P. D'Anvers, F. Vercauteren and Ingrid Verbauwhede, On the impact of decryption failures on the security of LWE/LWR based schemes, IACR Cryptology ePrint Archive, (2018), 1089. Google Scholar

[11]

È. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.   Google Scholar

[12]

S. Gao and T. Mateer, Additive fast fourier transforms over finite fields, IEEE Transactions on Information Theory, 56 (2010), 6265-6272.  doi: 10.1109/TIT.2010.2079016.  Google Scholar

[13]

A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.   Google Scholar

[14]

D. Hofheinz, K. Hövelmanns and E. Kiltz, A modular analysis of the Fujisaki-Okamoto transformation, in Theory of Cryptography Conference, Springer, (2017), 341–371.  Google Scholar

[15]

L. L. Joiner and J. J. Komo, Decoding binary BCH codes, in Southeastcon '95, 1995. doi: 10.1109/SECON.1995.513059.  Google Scholar

[16]

S. Lin and D. J. Costello, in Error control coding, Prentice Hall Englewood Cliffs, (2004)., Google Scholar

[17]

X. Lu, Y. Liu, Z. Zhang, D. Jia, H. Xue, J. He, B. Li, K. Wang, Z. Liu and H. Yang, LAC: Practical Ring-LWE based Public-Key Encryption with Byte-Level Modulus, IACR Cryptology ePrint Archive, (2018), 1009. Google Scholar

[18]

M. Walters and S. Sinha Roy, Constant-time BCH Error-Correcting Code, in 2020 IEEE International Symposium on Circuits and Systems (ISCAS), Sevilla, (2020), 1–5. doi: 10.1109/ISCAS45731.2020.9180846.  Google Scholar

[19]

Y. Xu, Implementation of Berlekamp-Massey algorithm without inversion, IEE Proceedings I-Communications, Speech and Vision, 138 (1991), 138-140.   Google Scholar

Figure 1.  Description of $\textsf{HQC.PKE}$ [1]
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Figure 2.  Decoding execution times (in CPU cycles) of various BCH codes for different error weights with field mutliplication implemented by lookup tables (variant 1)
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Figure 3.  Decoding execution times (in CPU cycles) of various BCH codes for different error weights with field multiplication implemented via $\textsf{pclmulqdq}$ instruction (variant 2)
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Table 1.  Parameter sets for HQC [1]
Instance $ n_1 $ $ n_2 $ $ n $ $ k $ $ t $ $ w $ $ w_\mathbf{r} = w_\mathbf{e} $ security
HQC-128-1 796 31 24, 677 256 60 67 77 128
HQC-192-1 766 57 43, 669 256 57 101 117 192
HQC-192-2 766 61 46, 747 256 57 101 117 192
HQC-256-1 766 83 63, 587 256 57 133 153 256
HQC-256-2 796 85 67, 699 256 60 133 153 256
HQC-256-3 796 89 70, 853 256 60 133 153 256
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Instance $ n_1 $ $ n_2 $ $ n $ $ k $ $ t $ $ w $ $ w_\mathbf{r} = w_\mathbf{e} $ security
HQC-128-1 796 31 24, 677 256 60 67 77 128
HQC-192-1 766 57 43, 669 256 57 101 117 192
HQC-192-2 766 61 46, 747 256 57 101 117 192
HQC-256-1 766 83 63, 587 256 57 133 153 256
HQC-256-2 796 85 67, 699 256 60 133 153 256
HQC-256-3 796 89 70, 853 256 60 133 153 256
Table 2.  The percentage of failed attacks against HQC-128-1 in function of the number of repeated requests to $ \mathcal{O}^{HQC}_{\text{Time}} $ for $ p = 100 $ for $ 1000 $ attacks
Number of repetition of each request
$ 1 $ $ 3 $ $ 5 $ $ 7 $ $ 9 $
Percentage of failed attacks $ 100\% $ $ 99, 8\% $ $ 49, 7\% $ $ 8, 3\% $ $ 7, 6\% $
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Number of repetition of each request
$ 1 $ $ 3 $ $ 5 $ $ 7 $ $ 9 $
Percentage of failed attacks $ 100\% $ $ 99, 8\% $ $ 49, 7\% $ $ 8, 3\% $ $ 7, 6\% $
Table 3.  Attack complexity and bandwidth cost against HQC
Complexity upper bound Requests
$ 128-1 $ $ 192-2 $ $ 256-3 $ $ 128-1 $ $ 192-2 $ $ 256-3 $
Oracle $\textsf{Init}$ ($ p=1 $) $ 2^{25} $ $ 2^{26} $ $ 2^{27} $ $ 2 $ $ 2 $ $ 2 $
First iteration $ 2^{35} $ $ 2^{36} $ $ 2^{37} $ $ 1793 $ $ 1936 $ $ 2257 $
Second iteration - phase 1 $ 2^{31} $ $ 2^{34} $ $ 2^{35} $ $ 198 $ $ 350 $ $ 528 $
Second iteration - phase 2 $ 2^{35} $ $ 2^{37} $ $ 2^{38} $ $ 3448 $ $ 3564 $ $ 3844 $
Total $ 2^{36} $ $ 2^{38} $ $ 2^{39} $ $ 5441 $ $ 5852 $ $ 6631 $
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Complexity upper bound Requests
$ 128-1 $ $ 192-2 $ $ 256-3 $ $ 128-1 $ $ 192-2 $ $ 256-3 $
Oracle $\textsf{Init}$ ($ p=1 $) $ 2^{25} $ $ 2^{26} $ $ 2^{27} $ $ 2 $ $ 2 $ $ 2 $
First iteration $ 2^{35} $ $ 2^{36} $ $ 2^{37} $ $ 1793 $ $ 1936 $ $ 2257 $
Second iteration - phase 1 $ 2^{31} $ $ 2^{34} $ $ 2^{35} $ $ 198 $ $ 350 $ $ 528 $
Second iteration - phase 2 $ 2^{35} $ $ 2^{37} $ $ 2^{38} $ $ 3448 $ $ 3564 $ $ 3844 $
Total $ 2^{36} $ $ 2^{38} $ $ 2^{39} $ $ 5441 $ $ 5852 $ $ 6631 $
Table 4.  Running time (CPU cycles) and overhead when original or constant time BCH decoding is used in the decapsulation step of HQC
HQC.$\textsf{Decaps}$ Overhead
Original BCH Constant time BCH
HQC-128-1 $ 507285 $ $ 563414 $ $ 11.06\% $
HQC-192-1 $ 947552 $ $ 995272 $ $ 5.05\% $
HQC-192-2 $ 992057 $ $ 1047054 $ $ 5.54\% $
HQC-256-1 $ 1490993 $ $ 1538824 $ $ 3.21\% $
HQC-256-2 $ 1562207 $ $ 1616673 $ $ 3.49\% $
HQC-256-3 $ 1617269 $ $ 1675195 $ $ 3.58\% $
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HQC.$\textsf{Decaps}$ Overhead
Original BCH Constant time BCH
HQC-128-1 $ 507285 $ $ 563414 $ $ 11.06\% $
HQC-192-1 $ 947552 $ $ 995272 $ $ 5.05\% $
HQC-192-2 $ 992057 $ $ 1047054 $ $ 5.54\% $
HQC-256-1 $ 1490993 $ $ 1538824 $ $ 3.21\% $
HQC-256-2 $ 1562207 $ $ 1616673 $ $ 3.49\% $
HQC-256-3 $ 1617269 $ $ 1675195 $ $ 3.58\% $
Table 5.  The mean of the decoding execution time (in CPU cycles) of various BCH codes for errors of weight 0 and 1 with field multiplication implemented by lookup tables (variant 1)
BCH code $ [n, k, d] $ (variant 1) Number of errors to be decoded
$ 0 $ error $ 1 $ error
[$ 766, 256, 115 $] $ 103466 $ $ 103436 $
[$ 796, 256, 121 $] $ 105349 $ $ 105332 $
[$ 4095, 418, 1003 $] $ 2811954 $ $ 2812663 $
[$ 8191, 7580, 95 $] $ 827943 $ $ 828040 $
[$ 16383, 14598, 261 $] $ 2028480 $ $ 2028889 $
[$ 32767, 16412, 2631 $] $ 24529993 $ $ 26599523 $
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BCH code $ [n, k, d] $ (variant 1) Number of errors to be decoded
$ 0 $ error $ 1 $ error
[$ 766, 256, 115 $] $ 103466 $ $ 103436 $
[$ 796, 256, 121 $] $ 105349 $ $ 105332 $
[$ 4095, 418, 1003 $] $ 2811954 $ $ 2812663 $
[$ 8191, 7580, 95 $] $ 827943 $ $ 828040 $
[$ 16383, 14598, 261 $] $ 2028480 $ $ 2028889 $
[$ 32767, 16412, 2631 $] $ 24529993 $ $ 26599523 $
Table 6.  The mean of the decoding execution time (in CPU cycles) of various BCH codes for errors of weight $ 0 $ and $ 1 $ with field multiplication implemented via $\textsf{pclmulqdq}$ instruction (variant 2)
BCH code $ [n, k, d] $ (variant 2) Number of errors to be decoded
$ 0 $ error $ 1 $ error
[$ 766, 256, 115 $] $ 128391 $
[$ 796, 256, 121 $] $ 133984 $ $ 134056 $
[$ 4095, 418, 1003 $] $ 5110719 $ $ 5110691 $
[$ 8191, 7580, 95 $] $ 1252564 $ $ 1252542 $
[$ 16383, 14598, 261 $] $ 3397224 $ $ 3396703 $
[$ 32767, 16412, 2631 $] $ 19205561 $ $ 19205407 $
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BCH code $ [n, k, d] $ (variant 2) Number of errors to be decoded
$ 0 $ error $ 1 $ error
[$ 766, 256, 115 $] $ 128391 $
[$ 796, 256, 121 $] $ 133984 $ $ 134056 $
[$ 4095, 418, 1003 $] $ 5110719 $ $ 5110691 $
[$ 8191, 7580, 95 $] $ 1252564 $ $ 1252542 $
[$ 16383, 14598, 261 $] $ 3397224 $ $ 3396703 $
[$ 32767, 16412, 2631 $] $ 19205561 $ $ 19205407 $
Table 7.  Decoding of some BCH codes with multiplication by tables
BCH code $ [n, k, d] $ Running time (in CPU cycles)
LuT Syndromes ELP Roots Total
$ [766, 256, 115] $ 0 34240 30089 26778 91873
$ [796, 256, 121] $ 0 34646 33359 27086 95861
$ [4095, 418, 1003] $ 82491 291827 2145899 187004 2711521
$ [8191, 7580, 95] $ 124587 278191 23216 186407 616569
$ [16383, 14598, 261] $ 245850 789651 166062 552630 1760773
$ [32767, 16412, 2631] $ 503337 2531258 17361393 1786677 22217535
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BCH code $ [n, k, d] $ Running time (in CPU cycles)
LuT Syndromes ELP Roots Total
$ [766, 256, 115] $ 0 34240 30089 26778 91873
$ [796, 256, 121] $ 0 34646 33359 27086 95861
$ [4095, 418, 1003] $ 82491 291827 2145899 187004 2711521
$ [8191, 7580, 95] $ 124587 278191 23216 186407 616569
$ [16383, 14598, 261] $ 245850 789651 166062 552630 1760773
$ [32767, 16412, 2631] $ 503337 2531258 17361393 1786677 22217535
Table 8.  Decoding of some BCH codes with multiplication by $\textsf{pclmulqdq}$
BCH code $ [n, k, d] $ Running time (in CPU cycles)
LuT Syndromes ELP Roots Total
$ [766, 256, 115] $ 0 42799 50735 34017 128226
$ [796, 256, 121] $ 0 43560 55562 34404 134157
$ [4095, 418, 1003] $ 96997 474817 4585893 321102 5482880
$ [8191, 7580, 95] $ 134176 443016 61288 311542 953739
$ [16383, 14598, 261] $ 260450 1501411 474177 1106680 3352090
$ [32767, 16412, 2631] $ 484200 2143567 14832791 1514189 18996691
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BCH code $ [n, k, d] $ Running time (in CPU cycles)
LuT Syndromes ELP Roots Total
$ [766, 256, 115] $ 0 42799 50735 34017 128226
$ [796, 256, 121] $ 0 43560 55562 34404 134157
$ [4095, 418, 1003] $ 96997 474817 4585893 321102 5482880
$ [8191, 7580, 95] $ 134176 443016 61288 311542 953739
$ [16383, 14598, 261] $ 260450 1501411 474177 1106680 3352090
$ [32767, 16412, 2631] $ 484200 2143567 14832791 1514189 18996691
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