American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020126
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 Early access 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

Figure 1.  Description of $\textsf{HQC.PKE}$ [1]
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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\% $
Table Options

Download as excel

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 $
Table Options

Download as excel

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\% $
Table Options

Download as excel

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 $
Table Options

Download as excel

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 $
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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
[1]

Angelot Behajaina. On BCH split metacyclic codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021045
[2]

Chandan Dey, Sumit Kumar Pandey, Tapabrata Roy, Santanu Sarkar. Differential faultt attack on DEFAULT. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022035
[3]

José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound. Advances in Mathematics of Communications, 2016, 10 (2) : 459-474. doi: 10.3934/amc.2016018
[4]

Xinmei Huang, Qin Yue, Yansheng Wu, Xiaoping Shi. Ternary Primitive LCD BCH codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021014
[5]

Edoardo Beretta, Dimitri Breda. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences & Engineering, 2011, 8 (4) : 931-952. doi: 10.3934/mbe.2011.8.931
[6]

Yacine Chitour, Benedetto Piccoli. Traffic circles and timing of traffic lights for cars flow. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 599-630. doi: 10.3934/dcdsb.2005.5.599
[7]

Haode Yan, Hao Liu, Chengju Li, Shudi Yang. Parameters of LCD BCH codes with two lengths. Advances in Mathematics of Communications, 2018, 12 (3) : 579-594. doi: 10.3934/amc.2018034
[8]

Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007
[9]

Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173
[10]

Mohamed Baouch, Juan Antonio López-Ramos, Blas Torrecillas, Reto Schnyder. An active attack on a distributed Group Key Exchange system. Advances in Mathematics of Communications, 2017, 11 (4) : 715-717. doi: 10.3934/amc.2017052
[11]

Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443
[12]

Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
[13]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[14]

Violetta Weger, Karan Khathuria, Anna-Lena Horlemann, Massimo Battaglioni, Paolo Santini, Edoardo Persichetti. On the hardness of the Lee syndrome decoding problem. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022029
[15]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[16]

Per Danzl, Ali Nabi, Jeff Moehlis. Charge-balanced spike timing control for phase models of spiking neurons. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1413-1435. doi: 10.3934/dcds.2010.28.1413
[17]

Diego F. Aranha, Ricardo Dahab, Julio López, Leonardo B. Oliveira. Efficient implementation of elliptic curve cryptography in wireless sensors. Advances in Mathematics of Communications, 2010, 4 (2) : 169-187. doi: 10.3934/amc.2010.4.169
[18]

Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger, Rachid Mchich. Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2163-2181. doi: 10.3934/dcdss.2021055
[19]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[20]

Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (504)
  • HTML views (672)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy