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Cryptanalysis and enhancement of multi factor remote user authentication scheme based on signcryption
Combining subspace codes
1. | Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Contrada Macchia Romana, 85100, Potenza, Italy |
2. | Mathematisches Institut, Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany |
3. | Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Complesso Universitario di Monte Sant'Angelo, Cupa Nuova Cintia 21, 80126, Napoli, Italy |
4. | Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy |
In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size $ A_q(n, d; k) $ of codes whose codewords are $ k $-subspaces of $ {\mathbb F}_q^n $ with minimum subspace distance $ d $. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant-dimension subspace codes for many parameters, including $ A_q(10, 4; 5) $, $ A_q(12, 4; 4) $, $ A_q(12, 6, 6) $ and $ A_q(16, 4; 4) $.
References:
[1] |
J. André,
Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z., 60 (1954), 156-186.
doi: 10.1007/BF01187370. |
[2] |
A. Beutelspacher,
On parallelisms in finite projective spaces, Geometriae Dedicata, 3 (1974), 35-40.
doi: 10.1007/BF00181359. |
[3] |
M. Braun, P. R. J. Östergård and A. Wassermann,
New lower bounds for binary constant-dimension subspace codes, Exp. Math., 27 (2018), 179-183.
doi: 10.1080/10586458.2016.1239145. |
[4] |
H. Chen, X. He, J. Weng and L. Xu,
New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inform. Theory, 66 (2020), 5317-5321.
doi: 10.1109/TIT.2020.2975776. |
[5] |
A. Cossidente, G. Marino and F. Pavese, Subspace code constructions, Ric. di Mat., (2020). to appear.
doi: 10.1007/s11587-020-00521-9. |
[6] |
A. Cossidente and F. Pavese,
On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.
doi: 10.1007/s10623-014-0018-6. |
[7] |
A. Cossidente and F. Pavese,
Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.
doi: 10.1007/s10623-015-0166-3. |
[8] |
A. Cossidente and F. Pavese,
Subspace codes in ${\rm{PG(2N - 1, Q)}}$, Combinatorica, 37 (2017), 1073-1095.
doi: 10.1007/s00493-016-3354-5. |
[9] |
A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 107-129. |
[10] |
P. Delsarte,
Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[11] |
R. H. F. Denniston,
Some packings of projective spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 52 (1972), 36-40.
|
[12] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[13] |
T. Etzion and N. Silberstein,
Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.
doi: 10.1109/TIT.2012.2220119. |
[14] |
P. Frankl and V. Rödl,
Near perfect coverings in graphs and hypergraphs, European J. Combin., 6 (1985), 317-326.
doi: 10.1016/S0195-6698(85)80045-7. |
[15] |
M. Gadouleau and Z. Yan,
Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.
doi: 10.1109/TIT.2010.2048447. |
[16] |
H. Gluesing-Luerssen and C. Troha,
Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.
doi: 10.3934/amc.2016023. |
[17] |
X. He, Construction of constant dimension code from two parallel versions of linkage construction, IEEE Communi. Lett., 24 (2020), 2392-2395. Google Scholar |
[18] |
X. He and Y. Chen, Construction of constant dimension codes from several parallel lifted MRD code, preprint, arXiv: 1911.00154. Google Scholar |
[19] |
D. Heinlein,
New LMRD code bounds for constant dimension codes and improved constructions, IEEE Trans. Inform. Theory, 65 (2019), 4822-4830.
doi: 10.1109/TIT.2019.2905002. |
[20] |
D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inform. Theory, 67 (2020), 705-715. Google Scholar |
[21] |
D. Heinlein, T. Honold, M. Kiermaier, S. Kurz and A. Wassermann,
Classifying optimal binary subspace codes of length $8$, constant dimension $4$ and minimum distance $6$, Des. Codes Cryptogr., 87 (2019), 375-391.
doi: 10.1007/s10623-018-0544-8. |
[22] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864. Google Scholar |
[23] |
D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in Coding Theory and Applications : 5th International Castle Meeting, ICMCTA 2017, Vihula, Estonia, August 28-31, 2017, Proceedings, vol. 10495 of Lecture Notes in Computer Science, Springer International Publishing, Cham, (2017), 163-191.
doi: 10.1007/978-3-319-66278-7_15. |
[24] |
D. Heinlein and S. Kurz,
Coset construction for subspace codes, IEEE Trans. Inform. Theory, 63 (2017), 7651-7660.
doi: 10.1109/TIT.2017.2753822. |
[25] |
T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday, World Scientific, (2016), 141-175. |
[26] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, Topics in Finite Fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 632 (2015), 157-176.
doi: 10.1090/conm/632/12627. |
[27] |
T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, Springer, Cham, (2018), 131-170. |
[28] |
A.-L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 25-42. |
[29] |
M. Kiermaier and S. Kurz,
On the lengths of divisible codes, IEEE Trans. Inform. Theory, 66 (2020), 4051-4060.
doi: 10.1109/TIT.2020.2968832. |
[30] |
R. Koetter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[31] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical methods in computer science, Springer, (2008), 31-42.
doi: 10.1007/978-3-540-89994-5_4. |
[32] |
S. Kurz,
Packing vector spaces into vector spaces, Australas. J. Combin., 68 (2017), 122-130.
|
[33] |
S. Kurz, A note on the linkage construction for constant dimension codes, preprint, arXiv: 1906.09780. Google Scholar |
[34] |
S. Kurz,
Subspaces intersecting in at most a point, Des. Codes Cryptogr., 88 (2020), 595-599.
doi: 10.1007/s10623-019-00699-6. |
[35] |
S. Liu, Y. Chang and T. Feng,
Parallel multilevel construction for constant dimension codes, IEEE Trans. Inform. Theory, 66 (2020), 6884-6897.
doi: 10.1109/TIT.2020.3004315. |
[36] |
E. L. Năstase and P. A. Sissokho,
The maximum size of a partial spread in a finite projective space, J. Combin. Theory Ser. A, 152 (2017), 353-362.
doi: 10.1016/j.jcta.2017.06.012. |
[37] |
B. Segre,
Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.
doi: 10.1007/BF02410047. |
[38] |
J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2019.
doi: 10.1515/9783110642094-013. |
[39] |
N. Silberstein and A.-L. Trautmann,
Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.
doi: 10.1109/TIT.2015.2435743. |
[40] |
D. Silva, F. R. Kschischang and R. Köetter,
A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[41] |
H. Wang, C. Xing and R. Safavi-Naini,
Linear authentication codes: Bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.
doi: 10.1109/TIT.2003.809567. |
[42] |
S.-T. Xia and F.-W. Fu,
Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.
doi: 10.1007/s10623-008-9221-7. |
[43] |
L. Xu and H. Chen,
New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inform. Theory, 64 (2018), 6315-6319.
doi: 10.1109/TIT.2018.2839596. |
show all references
References:
[1] |
J. André,
Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z., 60 (1954), 156-186.
doi: 10.1007/BF01187370. |
[2] |
A. Beutelspacher,
On parallelisms in finite projective spaces, Geometriae Dedicata, 3 (1974), 35-40.
doi: 10.1007/BF00181359. |
[3] |
M. Braun, P. R. J. Östergård and A. Wassermann,
New lower bounds for binary constant-dimension subspace codes, Exp. Math., 27 (2018), 179-183.
doi: 10.1080/10586458.2016.1239145. |
[4] |
H. Chen, X. He, J. Weng and L. Xu,
New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inform. Theory, 66 (2020), 5317-5321.
doi: 10.1109/TIT.2020.2975776. |
[5] |
A. Cossidente, G. Marino and F. Pavese, Subspace code constructions, Ric. di Mat., (2020). to appear.
doi: 10.1007/s11587-020-00521-9. |
[6] |
A. Cossidente and F. Pavese,
On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.
doi: 10.1007/s10623-014-0018-6. |
[7] |
A. Cossidente and F. Pavese,
Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.
doi: 10.1007/s10623-015-0166-3. |
[8] |
A. Cossidente and F. Pavese,
Subspace codes in ${\rm{PG(2N - 1, Q)}}$, Combinatorica, 37 (2017), 1073-1095.
doi: 10.1007/s00493-016-3354-5. |
[9] |
A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 107-129. |
[10] |
P. Delsarte,
Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[11] |
R. H. F. Denniston,
Some packings of projective spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 52 (1972), 36-40.
|
[12] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[13] |
T. Etzion and N. Silberstein,
Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.
doi: 10.1109/TIT.2012.2220119. |
[14] |
P. Frankl and V. Rödl,
Near perfect coverings in graphs and hypergraphs, European J. Combin., 6 (1985), 317-326.
doi: 10.1016/S0195-6698(85)80045-7. |
[15] |
M. Gadouleau and Z. Yan,
Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.
doi: 10.1109/TIT.2010.2048447. |
[16] |
H. Gluesing-Luerssen and C. Troha,
Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.
doi: 10.3934/amc.2016023. |
[17] |
X. He, Construction of constant dimension code from two parallel versions of linkage construction, IEEE Communi. Lett., 24 (2020), 2392-2395. Google Scholar |
[18] |
X. He and Y. Chen, Construction of constant dimension codes from several parallel lifted MRD code, preprint, arXiv: 1911.00154. Google Scholar |
[19] |
D. Heinlein,
New LMRD code bounds for constant dimension codes and improved constructions, IEEE Trans. Inform. Theory, 65 (2019), 4822-4830.
doi: 10.1109/TIT.2019.2905002. |
[20] |
D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inform. Theory, 67 (2020), 705-715. Google Scholar |
[21] |
D. Heinlein, T. Honold, M. Kiermaier, S. Kurz and A. Wassermann,
Classifying optimal binary subspace codes of length $8$, constant dimension $4$ and minimum distance $6$, Des. Codes Cryptogr., 87 (2019), 375-391.
doi: 10.1007/s10623-018-0544-8. |
[22] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864. Google Scholar |
[23] |
D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in Coding Theory and Applications : 5th International Castle Meeting, ICMCTA 2017, Vihula, Estonia, August 28-31, 2017, Proceedings, vol. 10495 of Lecture Notes in Computer Science, Springer International Publishing, Cham, (2017), 163-191.
doi: 10.1007/978-3-319-66278-7_15. |
[24] |
D. Heinlein and S. Kurz,
Coset construction for subspace codes, IEEE Trans. Inform. Theory, 63 (2017), 7651-7660.
doi: 10.1109/TIT.2017.2753822. |
[25] |
T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday, World Scientific, (2016), 141-175. |
[26] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, Topics in Finite Fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 632 (2015), 157-176.
doi: 10.1090/conm/632/12627. |
[27] |
T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, Springer, Cham, (2018), 131-170. |
[28] |
A.-L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 25-42. |
[29] |
M. Kiermaier and S. Kurz,
On the lengths of divisible codes, IEEE Trans. Inform. Theory, 66 (2020), 4051-4060.
doi: 10.1109/TIT.2020.2968832. |
[30] |
R. Koetter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[31] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical methods in computer science, Springer, (2008), 31-42.
doi: 10.1007/978-3-540-89994-5_4. |
[32] |
S. Kurz,
Packing vector spaces into vector spaces, Australas. J. Combin., 68 (2017), 122-130.
|
[33] |
S. Kurz, A note on the linkage construction for constant dimension codes, preprint, arXiv: 1906.09780. Google Scholar |
[34] |
S. Kurz,
Subspaces intersecting in at most a point, Des. Codes Cryptogr., 88 (2020), 595-599.
doi: 10.1007/s10623-019-00699-6. |
[35] |
S. Liu, Y. Chang and T. Feng,
Parallel multilevel construction for constant dimension codes, IEEE Trans. Inform. Theory, 66 (2020), 6884-6897.
doi: 10.1109/TIT.2020.3004315. |
[36] |
E. L. Năstase and P. A. Sissokho,
The maximum size of a partial spread in a finite projective space, J. Combin. Theory Ser. A, 152 (2017), 353-362.
doi: 10.1016/j.jcta.2017.06.012. |
[37] |
B. Segre,
Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.
doi: 10.1007/BF02410047. |
[38] |
J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2019.
doi: 10.1515/9783110642094-013. |
[39] |
N. Silberstein and A.-L. Trautmann,
Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.
doi: 10.1109/TIT.2015.2435743. |
[40] |
D. Silva, F. R. Kschischang and R. Köetter,
A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[41] |
H. Wang, C. Xing and R. Safavi-Naini,
Linear authentication codes: Bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.
doi: 10.1109/TIT.2003.809567. |
[42] |
S.-T. Xia and F.-W. Fu,
Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.
doi: 10.1007/s10623-008-9221-7. |
[43] |
L. Xu and H. Chen,
New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inform. Theory, 64 (2018), 6315-6319.
doi: 10.1109/TIT.2018.2839596. |
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