-
Previous Article
${\sf {FAST}}$: Disk encryption and beyond
- AMC Home
- This Issue
-
Next Article
A note on the Signal-to-noise ratio of $ (n, m) $-functions
Information set decoding in the Lee metric with applications to cryptography
1. | Faculty of Mathematics and Statistics, University of St. Gallen, Bodanstr. 6, St. Gallen, Switzerland |
2. | Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland |
We convert Stern's information set decoding (ISD) algorithm to the ring $ \mathbb{Z}/4 \mathbb{Z} $ equipped with the Lee metric. Moreover, we set up the general framework for a McEliece and a Niederreiter cryptosystem over this ring. The complexity of the ISD algorithm determines the minimum key size in these cryptosystems for a given security level. We show that using Lee metric codes can substantially decrease the key size, compared to Hamming metric codes. In the end we explain how our results can be generalized to other Galois rings $ \mathbb{Z}/p^s\mathbb{Z} $.
References:
[1] |
E. F. Assmus and H. F. Mattson,
Error-correcting codes: An axiomatic approach, Information and Control, 6 (1963), 315-330.
doi: 10.1016/S0019-9958(63)80010-8. |
[2] |
A. Becker, A. Joux, A. May and A. Meurer, Decoding random binary linear codes in $2^{n/20}$: How 1+ 1 = 0 improves information set decoding, Advances in Cryptology–EUROCRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7237 (2012), 520–536.
doi: 10.1007/978-3-642-29011-4_31. |
[3] |
E. Berlekamp, Algebraic Coding Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
doi: 10.1142/9407. |
[4] |
D. J. Bernstein, Grover vs. McEliece, In International Workshop on Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, 6061 (2010), 73–80.
doi: 10.1007/978-3-642-12929-2_6. |
[5] |
D. J. Bernstein, T. Lange and C. Peters, Attacking and defending the McEliece cryptosystem, In International Workshop on Post-Quantum Cryptography, Springer, (2008), 31–46.
doi: 10.1007/978-3-540-88403-3_3. |
[6] |
D. J. Bernstein, T. Lange and C. Peters, Smaller decoding exponents: Ball-collision decoding, In Annual Cryptology Conference, Springer, (2011), 743–760.
doi: 10.1007/978-3-642-22792-9_42. |
[7] |
T. Blackford,
Cyclic codes over $\mathbb{Z}_4$ of oddly even length, Discrete Applied Mathematics, 128 (2003), 27-46.
doi: 10.1016/S0166-218X(02)00434-1. |
[8] |
I. F. Blake,
Codes over certain rings, Information and Control, 20 (1972), 396-404.
doi: 10.1016/S0019-9958(72)90223-9. |
[9] |
I. F. Blake,
Codes over integer residue rings, Information and Control, 29 (1975), 295-300.
doi: 10.1016/S0019-9958(75)80001-5. |
[10] |
E. Byrne,
Decoding a class of Lee metric codes over a Galois ring, IEEE Transactions on Information Theory, 48 (2002), 966-975.
doi: 10.1109/18.992804. |
[11] |
A. Canteaut and Hervé Chabanne, A Further Improvement of the Work Factor in an Attempt at Breaking McEliece's Cryptosystem, PhD thesis, INRIA, 1994. Google Scholar |
[12] |
A. Canteaut and F. Chabaud,
A new algorithm for finding minimum-weight words in a linear code: Application to McEliece's cryptosystem and to narrow-sense BCH codes of length 511, IEEE Transactions on Information Theory, 44 (1998), 367-378.
doi: 10.1109/18.651067. |
[13] |
A. Canteaut and N. Sendrier, Cryptanalysis of the original McEliece cryptosystem, In Advances in Cryptology ASIACRYPT'98 (Beijing), Lecture Notes in Comput. Sci., Springer, Berlin, 1514 (1998), 187–199.
doi: 10.1007/3-540-49649-1_16. |
[14] |
F. Chabaud, Asymptotic analysis of probabilistic algorithms for finding short codewords, Eurocode '92 (Udine, 1992), CISM Courses and Lect., Springer, Vienna, 339 (1993), 175–183. |
[15] |
J. T. Coffey and R. M. Goodman,
The complexity of information set decoding, IEEE Transactions on Information Theory, 36 (1990), 1031-1037.
doi: 10.1109/18.57202. |
[16] |
I. I. Dumer,
Two decoding algorithms for linear codes, Problemy Peredachi Informatsii, 25 (1989), 24-32.
|
[17] |
T. Etzion, A. Vardy and E. Yaakobi, Dense error-correcting codes in the Lee metric, In IEEE Information Theory Workshop, (2010), 1–5. Google Scholar |
[18] |
M. Finiasz and N. Sendrier, Security bounds for the design of code-based cryptosystems, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, (2009), 88–105.
doi: 10.1007/978-3-642-10366-7_6. |
[19] |
E. Gabidulin, A brief survey of metrics in coding theory, Mathematics of Distances and Applications, 66 (2012). Google Scholar |
[20] |
M. Greferath, An introduction to ring-linear coding theory, In Gröbner Bases, Coding, and Cryptography, Springer, (2009), 219–238.
doi: 10.1007/978-3-540-93806-4_13. |
[21] |
R. A. Hammons, V. P. Kumar, R. A. Calderbank, N. Sloane and P. Solé,
The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Transactions on Information Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[22] |
T. Helleseth and V. Zinoviev,
On $\mathbb{Z}_4$-linear Goethals codes and Kloosterman sums, Designs, Codes and Cryptography, 17 (1999), 269-288.
doi: 10.1023/A:1026491513009. |
[23] |
S. Hirose, May-Ozerov algorithm for nearest-neighbor problem over $\mathbb{F}_q$ and its application to information set decoding, In International Conference for Information Technology and Communications, Springer, (2016), 115–126. Google Scholar |
[24] |
C. Interlando, K. Khathuria, N. Rohrer, J. Rosenthal and V. Weger, Generalization of the Ball-Collision algorithm, preprint, arXiv: : 1812.10955, 2018. Google Scholar |
[25] |
D. S. Krotov,
$\mathbb{Z}_4$-linear Hadamard and extended perfect codes, Electron. Notes Discrete Math., Elsevier Sci. B. V., Amsterdam, 6 (2001), 107-112.
|
[26] |
E. A. Kruk, Decoding complexity bound for linear block codes, Problemy Peredachi Informatsii, 25 (1989), 103-107. Google Scholar |
[27] |
P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem, In Workshop on the Theory and Application of of Cryptographic Techniques, Springer, 330 (1988), 275–280.
doi: 10.1007/3-540-45961-8_25. |
[28] |
J. S. Leon,
A probabilistic algorithm for computing minimum weights of large error-correcting codes, IEEE Transactions on Information Theory, 34 (1988), 1354-1359.
doi: 10.1109/18.21270. |
[29] |
A. May, A. Meurer and E. Thomae, Decoding random linear codes in $\mathcal{O} (2^{0.054 n})$, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, 2011 (2011), 107–124.
doi: 10.1007/978-3-642-25385-0_6. |
[30] |
A. May and I. Ozerov, On computing nearest neighbors with applications to decoding of binary linear codes, In Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer, (2015), 203–228.
doi: 10.1007/978-3-662-46800-5_9. |
[31] |
R. J. McEliece, A Public-Key Cryptosystem Based on Algebraic Coding Theory, Technical report, DSN Progress report, Jet Propulsion Laboratory, Pasadena, 1978. Google Scholar |
[32] |
A. Meurer, A Coding-Theoretic Approach to Cryptanalysis, PhD thesis, Ruhr University Bochum, 2012. Google Scholar |
[33] |
A. A. Nechaev,
Kerdock code in a cyclic form, Discrete Mathematics and Applications, 1 (1991), 365-384.
doi: 10.1515/dma.1991.1.4.365. |
[34] |
R. Niebuhr, E. Persichetti, Pi erre-Louis Cayrel, S. Bulygin and J. Buchmann,
On lower bounds for information set decoding over $\mathbb{F}_q$ and on the effect of partial knowledge, International journal of information and Coding Theory, 4 (2017), 47-78.
doi: 10.1504/IJICOT.2017.081458. |
[35] |
C. Peters, Information-set decoding for linear codes over $\mathbb{F}_q$, In International Workshop on Post-Quantum Cryptography, Springer, 6061 (2010), 81–94.
doi: 10.1007/978-3-642-12929-2_7. |
[36] |
V. S. Pless and Z. Qian,
Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Transactions on Information Theory, 42 (1996), 1594-1600.
doi: 10.1109/18.532906. |
[37] |
E. Prange,
The use of information sets in decoding cyclic codes, IRE Transactions on Information Theory, 8 (1962), 5-9.
doi: 10.1109/tit.1962.1057777. |
[38] |
R. M. Roth and P. H. Siegel,
Lee-metric BCH codes and their application to constrained and partial-response channels, IEEE Transactions on Information Theory, 40 (1994), 1083-1096.
doi: 10.1109/18.335966. |
[39] |
C. Satyanarayana,
Lee metric codes over integer residue rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 250-254.
doi: 10.1109/TIT.1979.1056017. |
[40] |
P. Shankar,
On BCH codes over arbitrary integer rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 480-483.
doi: 10.1109/TIT.1979.1056063. |
[41] |
E. Spiegel,
Codes over $\mathbb{Z}_m$, Information and control, 35 (1977), 48-51.
doi: 10.1016/S0019-9958(77)90526-5. |
[42] |
J. Stern, A method for finding codewords of small weight, In International Colloquium on Coding Theory and Applications, Springer, 388 (1989), 106–113.
doi: 10.1007/BFb0019850. |
[43] |
I. Tal and R. M. Roth, On list decoding of alternant codes in the Hamming and Lee metrics, In IEEE International Symposium on Information Theory, 2003,364–364. Google Scholar |
[44] |
H. Tapia-Recillas,
A secret sharing scheme from a chain ring linear code, Congressus Numerantium, 186 (2007), 33-39.
|
[45] |
J. van Tilburg, On the McEliece public-key cryptosystem, In Conference on the Theory and Application of Cryptography, Springer, 403 (1990), 119–131.
doi: 10.1007/0-387-34799-2_10. |
[46] |
V. Weger, M. Battaglioni, P. Santini, F. Chiaraluce, M. Baldi and E. Persichetti, Information set decoding of Lee-metric codes over finite rings, arXiv preprint, arXiv: 2001.08425, 2020. Google Scholar |
[47] |
Y. Wu and C. N. Hadjicostis,
Decoding algorithm and architecture for BCH codes under the Lee metric, IEEE Transactions on Communications, 56 (2008), 2050-2059.
doi: 10.1109/TCOMM.2008.041227. |
show all references
References:
[1] |
E. F. Assmus and H. F. Mattson,
Error-correcting codes: An axiomatic approach, Information and Control, 6 (1963), 315-330.
doi: 10.1016/S0019-9958(63)80010-8. |
[2] |
A. Becker, A. Joux, A. May and A. Meurer, Decoding random binary linear codes in $2^{n/20}$: How 1+ 1 = 0 improves information set decoding, Advances in Cryptology–EUROCRYPT 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7237 (2012), 520–536.
doi: 10.1007/978-3-642-29011-4_31. |
[3] |
E. Berlekamp, Algebraic Coding Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
doi: 10.1142/9407. |
[4] |
D. J. Bernstein, Grover vs. McEliece, In International Workshop on Post-Quantum Cryptography, Lecture Notes in Comput. Sci., Springer, 6061 (2010), 73–80.
doi: 10.1007/978-3-642-12929-2_6. |
[5] |
D. J. Bernstein, T. Lange and C. Peters, Attacking and defending the McEliece cryptosystem, In International Workshop on Post-Quantum Cryptography, Springer, (2008), 31–46.
doi: 10.1007/978-3-540-88403-3_3. |
[6] |
D. J. Bernstein, T. Lange and C. Peters, Smaller decoding exponents: Ball-collision decoding, In Annual Cryptology Conference, Springer, (2011), 743–760.
doi: 10.1007/978-3-642-22792-9_42. |
[7] |
T. Blackford,
Cyclic codes over $\mathbb{Z}_4$ of oddly even length, Discrete Applied Mathematics, 128 (2003), 27-46.
doi: 10.1016/S0166-218X(02)00434-1. |
[8] |
I. F. Blake,
Codes over certain rings, Information and Control, 20 (1972), 396-404.
doi: 10.1016/S0019-9958(72)90223-9. |
[9] |
I. F. Blake,
Codes over integer residue rings, Information and Control, 29 (1975), 295-300.
doi: 10.1016/S0019-9958(75)80001-5. |
[10] |
E. Byrne,
Decoding a class of Lee metric codes over a Galois ring, IEEE Transactions on Information Theory, 48 (2002), 966-975.
doi: 10.1109/18.992804. |
[11] |
A. Canteaut and Hervé Chabanne, A Further Improvement of the Work Factor in an Attempt at Breaking McEliece's Cryptosystem, PhD thesis, INRIA, 1994. Google Scholar |
[12] |
A. Canteaut and F. Chabaud,
A new algorithm for finding minimum-weight words in a linear code: Application to McEliece's cryptosystem and to narrow-sense BCH codes of length 511, IEEE Transactions on Information Theory, 44 (1998), 367-378.
doi: 10.1109/18.651067. |
[13] |
A. Canteaut and N. Sendrier, Cryptanalysis of the original McEliece cryptosystem, In Advances in Cryptology ASIACRYPT'98 (Beijing), Lecture Notes in Comput. Sci., Springer, Berlin, 1514 (1998), 187–199.
doi: 10.1007/3-540-49649-1_16. |
[14] |
F. Chabaud, Asymptotic analysis of probabilistic algorithms for finding short codewords, Eurocode '92 (Udine, 1992), CISM Courses and Lect., Springer, Vienna, 339 (1993), 175–183. |
[15] |
J. T. Coffey and R. M. Goodman,
The complexity of information set decoding, IEEE Transactions on Information Theory, 36 (1990), 1031-1037.
doi: 10.1109/18.57202. |
[16] |
I. I. Dumer,
Two decoding algorithms for linear codes, Problemy Peredachi Informatsii, 25 (1989), 24-32.
|
[17] |
T. Etzion, A. Vardy and E. Yaakobi, Dense error-correcting codes in the Lee metric, In IEEE Information Theory Workshop, (2010), 1–5. Google Scholar |
[18] |
M. Finiasz and N. Sendrier, Security bounds for the design of code-based cryptosystems, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, (2009), 88–105.
doi: 10.1007/978-3-642-10366-7_6. |
[19] |
E. Gabidulin, A brief survey of metrics in coding theory, Mathematics of Distances and Applications, 66 (2012). Google Scholar |
[20] |
M. Greferath, An introduction to ring-linear coding theory, In Gröbner Bases, Coding, and Cryptography, Springer, (2009), 219–238.
doi: 10.1007/978-3-540-93806-4_13. |
[21] |
R. A. Hammons, V. P. Kumar, R. A. Calderbank, N. Sloane and P. Solé,
The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Transactions on Information Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[22] |
T. Helleseth and V. Zinoviev,
On $\mathbb{Z}_4$-linear Goethals codes and Kloosterman sums, Designs, Codes and Cryptography, 17 (1999), 269-288.
doi: 10.1023/A:1026491513009. |
[23] |
S. Hirose, May-Ozerov algorithm for nearest-neighbor problem over $\mathbb{F}_q$ and its application to information set decoding, In International Conference for Information Technology and Communications, Springer, (2016), 115–126. Google Scholar |
[24] |
C. Interlando, K. Khathuria, N. Rohrer, J. Rosenthal and V. Weger, Generalization of the Ball-Collision algorithm, preprint, arXiv: : 1812.10955, 2018. Google Scholar |
[25] |
D. S. Krotov,
$\mathbb{Z}_4$-linear Hadamard and extended perfect codes, Electron. Notes Discrete Math., Elsevier Sci. B. V., Amsterdam, 6 (2001), 107-112.
|
[26] |
E. A. Kruk, Decoding complexity bound for linear block codes, Problemy Peredachi Informatsii, 25 (1989), 103-107. Google Scholar |
[27] |
P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem, In Workshop on the Theory and Application of of Cryptographic Techniques, Springer, 330 (1988), 275–280.
doi: 10.1007/3-540-45961-8_25. |
[28] |
J. S. Leon,
A probabilistic algorithm for computing minimum weights of large error-correcting codes, IEEE Transactions on Information Theory, 34 (1988), 1354-1359.
doi: 10.1109/18.21270. |
[29] |
A. May, A. Meurer and E. Thomae, Decoding random linear codes in $\mathcal{O} (2^{0.054 n})$, In International Conference on the Theory and Application of Cryptology and Information Security, Springer, 2011 (2011), 107–124.
doi: 10.1007/978-3-642-25385-0_6. |
[30] |
A. May and I. Ozerov, On computing nearest neighbors with applications to decoding of binary linear codes, In Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer, (2015), 203–228.
doi: 10.1007/978-3-662-46800-5_9. |
[31] |
R. J. McEliece, A Public-Key Cryptosystem Based on Algebraic Coding Theory, Technical report, DSN Progress report, Jet Propulsion Laboratory, Pasadena, 1978. Google Scholar |
[32] |
A. Meurer, A Coding-Theoretic Approach to Cryptanalysis, PhD thesis, Ruhr University Bochum, 2012. Google Scholar |
[33] |
A. A. Nechaev,
Kerdock code in a cyclic form, Discrete Mathematics and Applications, 1 (1991), 365-384.
doi: 10.1515/dma.1991.1.4.365. |
[34] |
R. Niebuhr, E. Persichetti, Pi erre-Louis Cayrel, S. Bulygin and J. Buchmann,
On lower bounds for information set decoding over $\mathbb{F}_q$ and on the effect of partial knowledge, International journal of information and Coding Theory, 4 (2017), 47-78.
doi: 10.1504/IJICOT.2017.081458. |
[35] |
C. Peters, Information-set decoding for linear codes over $\mathbb{F}_q$, In International Workshop on Post-Quantum Cryptography, Springer, 6061 (2010), 81–94.
doi: 10.1007/978-3-642-12929-2_7. |
[36] |
V. S. Pless and Z. Qian,
Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Transactions on Information Theory, 42 (1996), 1594-1600.
doi: 10.1109/18.532906. |
[37] |
E. Prange,
The use of information sets in decoding cyclic codes, IRE Transactions on Information Theory, 8 (1962), 5-9.
doi: 10.1109/tit.1962.1057777. |
[38] |
R. M. Roth and P. H. Siegel,
Lee-metric BCH codes and their application to constrained and partial-response channels, IEEE Transactions on Information Theory, 40 (1994), 1083-1096.
doi: 10.1109/18.335966. |
[39] |
C. Satyanarayana,
Lee metric codes over integer residue rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 250-254.
doi: 10.1109/TIT.1979.1056017. |
[40] |
P. Shankar,
On BCH codes over arbitrary integer rings (corresp), IEEE Transactions on Information Theory, 25 (1979), 480-483.
doi: 10.1109/TIT.1979.1056063. |
[41] |
E. Spiegel,
Codes over $\mathbb{Z}_m$, Information and control, 35 (1977), 48-51.
doi: 10.1016/S0019-9958(77)90526-5. |
[42] |
J. Stern, A method for finding codewords of small weight, In International Colloquium on Coding Theory and Applications, Springer, 388 (1989), 106–113.
doi: 10.1007/BFb0019850. |
[43] |
I. Tal and R. M. Roth, On list decoding of alternant codes in the Hamming and Lee metrics, In IEEE International Symposium on Information Theory, 2003,364–364. Google Scholar |
[44] |
H. Tapia-Recillas,
A secret sharing scheme from a chain ring linear code, Congressus Numerantium, 186 (2007), 33-39.
|
[45] |
J. van Tilburg, On the McEliece public-key cryptosystem, In Conference on the Theory and Application of Cryptography, Springer, 403 (1990), 119–131.
doi: 10.1007/0-387-34799-2_10. |
[46] |
V. Weger, M. Battaglioni, P. Santini, F. Chiaraluce, M. Baldi and E. Persichetti, Information set decoding of Lee-metric codes over finite rings, arXiv preprint, arXiv: 2001.08425, 2020. Google Scholar |
[47] |
Y. Wu and C. N. Hadjicostis,
Decoding algorithm and architecture for BCH codes under the Lee metric, IEEE Transactions on Communications, 56 (2008), 2050-2059.
doi: 10.1109/TCOMM.2008.041227. |
1 | 2 | 3 | |
18 | 19 | 24 | 25 | 26 | ||
best |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | ||
best |
4 | 4 | 4 | 3 | 3 | 2 | 2 | 2 | ||
key size | 5198 | 5296 | 5390 | 6110 | 6160 | 6440 | 6446 | 6448 | ||
security level | 31 | 31 | 31 | 27 | 27 | 28 | 28 | 28 |
1 | 2 | 3 | |
18 | 19 | 24 | 25 | 26 | ||
best |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | ||
best |
4 | 4 | 4 | 3 | 3 | 2 | 2 | 2 | ||
key size | 5198 | 5296 | 5390 | 6110 | 6160 | 6440 | 6446 | 6448 | ||
security level | 31 | 31 | 31 | 27 | 27 | 28 | 28 | 28 |
[1] |
Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020129 |
[2] |
Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044 |
[3] |
Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045 |
[4] |
Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055 |
[5] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[6] |
Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049 |
[7] |
Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $ p $-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039 |
[8] |
Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020405 |
[9] |
Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325 |
[10] |
Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327 |
[11] |
Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021014 |
[12] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020451 |
[13] |
Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 |
[14] |
Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 |
[15] |
Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020124 |
[16] |
Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020458 |
[17] |
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 |
[18] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[19] |
Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051 |
[20] |
Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020118 |
2019 Impact Factor: 0.734
Tools
Metrics
Other articles
by authors
[Back to Top]