American Institute of Mathematical Sciences

Advanced
May  2021, 15(2): 347-363. doi: 10.3934/amc.2020070

Capacity-achieving private information retrieval scheme with a smaller sub-packetization

Wenqin Zhang 1,2, , Zhengchun Zhou 1, , Udaya Parampalli 3, and  Vladimir Sidorenko 4,

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China
2. 

State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China
3. 

Department of Computer Science and Software Engineering, University of Melbourne, Melbourne, Australia
4. 

Institute for Communications Engineering, Technical University of Munich, Germany

Received  April 2019 Revised  February 2020 Published  May 2021 Early access  April 2020

Fund Project: The work of W. Zhang and Z. Zhou was supported by the National Natural Science Foundation of China under Grant 61672028, and partially supported by National Cryptography Development Fund under Grant MMJJ20180103. Vladimir Sidorenko is on leave from Institute for Information Transmission Problems, Russian Academy of Sciences. His work is supported by the Russian Government, Contract No 14.W03.31.0019

Private information retrieval (PIR) allows a user to retrieve one out of $ M $ messages from $ N $ servers without revealing the identity of the desired message. Every message consists of $ L $ symbols (packets) from an additive group and the length $ L $ is called the sub-packetization. A PIR scheme with download cost (DC) $ D/L $ is implemented by querying $ D $ sums of the symbols to servers. We assume that each uncoded server can store up to $ tLM/N $ symbols, $ t\in\{1,2,\cdots,N\} $. The minimum DC of storage constrained PIR was determined by Attia et al. in 2018 to be $ DC_{min} = 1+1/t+1/t^{2}+\cdots+1/t^{M-1} $. The capacity of storage constrained PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired symbols that can be privately retrieved per bit of downloaded symbols. Tandon et al. designed a capacity-achieving PIR scheme with sub-packetization $ L' = {N\choose t}t^{M} $ of each message. In this paper, we design a PIR scheme with $ t $ times smaller sub-packetization $ L'/t $ and with the minimum DC for any parameters $ N, M, t $. We also prove that $ t^{M-1} $ is a factor of sub-packetization $ L $ for any capacity-achieving PIR scheme from storage constrained servers.

Keywords: Private information retrieval, minimum download cost, sub-packetization, storage constrained servers.
Mathematics Subject Classification: Primary: 06E75, 94A60, 11T23.
Citation: Wenqin Zhang, Zhengchun Zhou, Udaya Parampalli, Vladimir Sidorenko. Capacity-achieving private information retrieval scheme with a smaller sub-packetization. Advances in Mathematics of Communications, 2021, 15 (2) : 347-363. doi: 10.3934/amc.2020070
References:
[1]

M. A. Attia, D. Kumar and R. Tandon, The capacity of uncoded storage constrained PIR, 2018 IEEE International Symposium on Information Theory, (2018), 1959-1963.  doi: 10.1109/ISIT.2018.8437729.

[2]

K. Banawan and S. Ulukus, Multi-message private information retrieval: Capacity results and near-optimal schemes, IEEE Trans. Inform. Theory, 64 (2018), 6842-6862.  doi: 10.1109/TIT.2018.2828310.

[3]

K. Banawan and S. Ulukus, The capacity of private information retrieval from byzantine and colluding databases, IEEE Transactions on Information Theory, 65 (2018), 1206–1219, arXiv: 1706.01442. doi: 10.1109/TIT.2018.2869154.

[4]

K. Banawan and S. Ulukus, The capacity of private information retrieval from coded databases, IEEE Transactions Information Theory, 64 (2018), 1945-1956.  doi: 10.1109/TIT.2018.2791994.

[5]

Z. Chen, Z. Y. Wang and S. Jafar, The capacity of private information retrieval with private side information, (2017), arXiv: 1709.03022.

[6]

B. Chor, O. Goldreich, E. Kushilevitz and M. Sudan, Private information retrieval, 36th Annual Symposium on Foundations of Computer Science, (1995), 41-50.  doi: 10.1109/SFCS.1995.492461.

[7]

R. Freij-Hollanti, O. W. Gnilke, C. Hollanti and D. A. Karpuk, Private information retrieval from coded databases with colluding servers, SIAM Journal on Applied Algebra and Geometry, 1 (2017), 647-664.  doi: 10.1137/16M1102562.

[8]

S. Kadhe, B. Garcia, A. Heidarzadeh, S. El Rouayheb and A. Sprintson, Private information retrieval with side information, (2017), arXiv: 1709.00112.

[9]

H. Sun and S. A. Jafar, The capacity of private information retrieval, IEEE Transactions Information Theory, 63 (2017), 4075-4088.  doi: 10.1109/TIT.2017.2689028.

[10]

H. Sun and S. A. Jafar, The capacity of symmetric private information retrieval, 2016 IEEE Globecom Workshops, Washington, DC, USA, (2016), 1-5. 

[11]

H. Sun and S. A. Jafar, The capacity of private information retrieval with colluding databases, IEEE Trans. Inform. Theory, 64 (2018), 2361-2370.  doi: 10.1109/TIT.2017.2777490.

[12]

H. Sun and S. A. Jafar, Private information retrieval from MDS coded data with colluding servers: Settling a conjecture by Freij-Hollanti et al., IEEE Transactions Information Theory, 64 (2018), 1000-1022.  doi: 10.1109/TIT.2017.2779454.

[13]

H. Sun and S. A. Jafar, Optimal download cost of private information retrieval for arbitrary message length, IEEE Transactions Information Forensics and Security, 12 (2017), 2920-2932. 

[14]

R. Tajeddine, O. W. Gnilke and S. El Rouayheb, Private information retrieval from MDS coded data in distributed storage systems, IEEE Trans. Inform. Theory, 64 (2018), 7081-7093.  doi: 10.1109/TIT.2018.2815607.

[15]

R. Tandon, M. Abdul-Wahid, F. Almoualem and D. Kumar, PIR from storage constrained databases - coded caching meets PIR, 2018 IEEE International Conference on Communications, (2018), 1-7. 

[16]

Q. W. Wang and M. Skoglund, Symmetric private information retrieval for MDS coded distributed storage, IEEE International Conference on Communications, (2017), 1-6.  doi: 10.1109/ICC.2017.7997029.

[17]

N. Woolsey, R.-R. Chen and M. Y. Ji, A new design of private information retrieval for storage constrained databases, (2019), arXiv: 1901.07490. doi: 10.1109/ISIT.2019.8849767.

[18]

J. K. Xu and Z. F. Zhang, On sub-packetization of capacity-achieving PIR schemes for MDS coded databases, (2017), arXiv: 1712.02466.

[19]

Y. W. Zhang and G. N. Ge, A general private information retrieval scheme for MDS coded databases with colluding servers, (2017), arXiv: 1704.06785. doi: 10.1007/s10623-019-00640-x.

[20]

Y. W. Zhang and G. N. Ge, Multi-file private information retrieval from MDS coded databases with colluding servers, (2017), arXiv: 1705.03186.

[21]

Z. F. Zhang and J. K. Xu, Capacity-achieving PIR schemes with optimal sub-packetization, (2017), arXiv: 1710.11370.

show all references

References:
[1]

M. A. Attia, D. Kumar and R. Tandon, The capacity of uncoded storage constrained PIR, 2018 IEEE International Symposium on Information Theory, (2018), 1959-1963.  doi: 10.1109/ISIT.2018.8437729.

[2]

K. Banawan and S. Ulukus, Multi-message private information retrieval: Capacity results and near-optimal schemes, IEEE Trans. Inform. Theory, 64 (2018), 6842-6862.  doi: 10.1109/TIT.2018.2828310.

[3]

K. Banawan and S. Ulukus, The capacity of private information retrieval from byzantine and colluding databases, IEEE Transactions on Information Theory, 65 (2018), 1206–1219, arXiv: 1706.01442. doi: 10.1109/TIT.2018.2869154.

[4]

K. Banawan and S. Ulukus, The capacity of private information retrieval from coded databases, IEEE Transactions Information Theory, 64 (2018), 1945-1956.  doi: 10.1109/TIT.2018.2791994.

[5]

Z. Chen, Z. Y. Wang and S. Jafar, The capacity of private information retrieval with private side information, (2017), arXiv: 1709.03022.

[6]

B. Chor, O. Goldreich, E. Kushilevitz and M. Sudan, Private information retrieval, 36th Annual Symposium on Foundations of Computer Science, (1995), 41-50.  doi: 10.1109/SFCS.1995.492461.

[7]

R. Freij-Hollanti, O. W. Gnilke, C. Hollanti and D. A. Karpuk, Private information retrieval from coded databases with colluding servers, SIAM Journal on Applied Algebra and Geometry, 1 (2017), 647-664.  doi: 10.1137/16M1102562.

[8]

S. Kadhe, B. Garcia, A. Heidarzadeh, S. El Rouayheb and A. Sprintson, Private information retrieval with side information, (2017), arXiv: 1709.00112.

[9]

H. Sun and S. A. Jafar, The capacity of private information retrieval, IEEE Transactions Information Theory, 63 (2017), 4075-4088.  doi: 10.1109/TIT.2017.2689028.

[10]

H. Sun and S. A. Jafar, The capacity of symmetric private information retrieval, 2016 IEEE Globecom Workshops, Washington, DC, USA, (2016), 1-5. 

[11]

H. Sun and S. A. Jafar, The capacity of private information retrieval with colluding databases, IEEE Trans. Inform. Theory, 64 (2018), 2361-2370.  doi: 10.1109/TIT.2017.2777490.

[12]

H. Sun and S. A. Jafar, Private information retrieval from MDS coded data with colluding servers: Settling a conjecture by Freij-Hollanti et al., IEEE Transactions Information Theory, 64 (2018), 1000-1022.  doi: 10.1109/TIT.2017.2779454.

[13]

H. Sun and S. A. Jafar, Optimal download cost of private information retrieval for arbitrary message length, IEEE Transactions Information Forensics and Security, 12 (2017), 2920-2932. 

[14]

R. Tajeddine, O. W. Gnilke and S. El Rouayheb, Private information retrieval from MDS coded data in distributed storage systems, IEEE Trans. Inform. Theory, 64 (2018), 7081-7093.  doi: 10.1109/TIT.2018.2815607.

[15]

R. Tandon, M. Abdul-Wahid, F. Almoualem and D. Kumar, PIR from storage constrained databases - coded caching meets PIR, 2018 IEEE International Conference on Communications, (2018), 1-7. 

[16]

Q. W. Wang and M. Skoglund, Symmetric private information retrieval for MDS coded distributed storage, IEEE International Conference on Communications, (2017), 1-6.  doi: 10.1109/ICC.2017.7997029.

[17]

N. Woolsey, R.-R. Chen and M. Y. Ji, A new design of private information retrieval for storage constrained databases, (2019), arXiv: 1901.07490. doi: 10.1109/ISIT.2019.8849767.

[18]

J. K. Xu and Z. F. Zhang, On sub-packetization of capacity-achieving PIR schemes for MDS coded databases, (2017), arXiv: 1712.02466.

[19]

Y. W. Zhang and G. N. Ge, A general private information retrieval scheme for MDS coded databases with colluding servers, (2017), arXiv: 1704.06785. doi: 10.1007/s10623-019-00640-x.

[20]

Y. W. Zhang and G. N. Ge, Multi-file private information retrieval from MDS coded databases with colluding servers, (2017), arXiv: 1705.03186.

[21]

Z. F. Zhang and J. K. Xu, Capacity-achieving PIR schemes with optimal sub-packetization, (2017), arXiv: 1710.11370.

Table 1.  Sub-messages stored in the servers
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ A_{12}, A_{13} $ $ A_{12}, A_{23} $ $ A_{13}, A_{23} $
$ B_{12}, B_{13} $ $ B_{12}, B_{23} $ $ B_{13}, B_{23} $
$ C_{12}, C_{13} $ $ C_{12}, C_{23} $ $ C_{13}, C_{23} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ A_{12}, A_{13} $ $ A_{12}, A_{23} $ $ A_{13}, A_{23} $
$ B_{12}, B_{13} $ $ B_{12}, B_{23} $ $ B_{13}, B_{23} $
$ C_{12}, C_{13} $ $ C_{12}, C_{23} $ $ C_{13}, C_{23} $
Table 2.  Symbols stored in the servers
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ \underline{a^1_{12}},\; \underline{a^2_{12}} $ $ \underline{a^1_{12}},\; \underline{a^2_{12}} $ $ \underline{a^1_{13}},\; \underline{a^2_{13}} $
$ \underline{a^1_{13}},\; \underline{a^2_{13}} $ $ \underline{a^1_{23}},\; \underline{a^2_{23}} $ $ \underline{a^1_{23}},\; \underline{a^2_{23}} $
$ \underline{b^1_{12}},\; \underline{b^2_{12}} $ $ \underline{b^1_{12}},\; \underline{b^2_{12}} $ $ \underline{b^1_{13}},\; \underline{b^2_{13}} $
$ \underline{b^1_{13}},\; \underline{b^2_{13}} $ $ \underline{b^1_{23}},\; \underline{b^2_{23}} $ $ \underline{b^1_{23}},\; \underline{b^2_{23}} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ \underline{a^1_{12}},\; \underline{a^2_{12}} $ $ \underline{a^1_{12}},\; \underline{a^2_{12}} $ $ \underline{a^1_{13}},\; \underline{a^2_{13}} $
$ \underline{a^1_{13}},\; \underline{a^2_{13}} $ $ \underline{a^1_{23}},\; \underline{a^2_{23}} $ $ \underline{a^1_{23}},\; \underline{a^2_{23}} $
$ \underline{b^1_{12}},\; \underline{b^2_{12}} $ $ \underline{b^1_{12}},\; \underline{b^2_{12}} $ $ \underline{b^1_{13}},\; \underline{b^2_{13}} $
$ \underline{b^1_{13}},\; \underline{b^2_{13}} $ $ \underline{b^1_{23}},\; \underline{b^2_{23}} $ $ \underline{b^1_{23}},\; \underline{b^2_{23}} $
Table 3.  Queries to download message $ A $
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ a^1_{12} $ $ a^1_{23} $ $ a^1_{13} $
$ b^1_{12} $ $ b^1_{23} $ $ b^1_{13} $
$ a^2_{13}+ b^1_{13} $ $ a^2_{12}+ b^1_{12} $ $ a^2_{23}+ b^1_{23} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ a^1_{12} $ $ a^1_{23} $ $ a^1_{13} $
$ b^1_{12} $ $ b^1_{23} $ $ b^1_{13} $
$ a^2_{13}+ b^1_{13} $ $ a^2_{12}+ b^1_{12} $ $ a^2_{23}+ b^1_{23} $
Table 4.  Queries to download message $ B $
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ b^1_{12} $ $ b^1_{23} $ $ b^1_{13} $
$ a^1_{12} $ $ a^1_{23} $ $ a^1_{13} $
$ b^2_{13}+a^1_{13} $ $ b^2_{12}+a^1_{12} $ $ b^2_{23}+a^1_{23} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $
$ b^1_{12} $ $ b^1_{23} $ $ b^1_{13} $
$ a^1_{12} $ $ a^1_{23} $ $ a^1_{13} $
$ b^2_{13}+a^1_{13} $ $ b^2_{12}+a^1_{12} $ $ b^2_{23}+a^1_{23} $
Table 5.  Sub-messages stored in the servers
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $ $ Serv^4 $ $ Serv^5 $
$ A_{12}, A_{13} $ $ A_{12}, A_{23} $ $ A_{13}, A_{23} $ $ A_{14}, A_{24} $ $ A_{15}, A_{25} $
$ A_{14}, A_{15} $ $ A_{24}, A_{25} $ $ A_{34}, A_{35} $ $ A_{34}, A_{45} $ $ A_{35}, A_{45} $
$ B_{12}, B_{13} $ $ B_{12}, B_{23} $ $ B_{13}, B_{23} $ $ B_{14}, B_{24} $ $ B_{15}, B_{25} $
$ B_{14}, B_{15} $ $ B_{24}, B_{25} $ $ B_{34}, B_{35} $ $ B_{34}, B_{45} $ $ B_{35}, B_{45} $
$ C_{12}, C_{13} $ $ C_{12}, C_{23} $ $ C_{13}, C_{23} $ $ C_{14}, C_{24} $ $ C_{15}, C_{25} $
$ C_{14}, C_{15} $ $ C_{24}, C_{25} $ $ C_{34}, C_{35} $ $ C_{34}, C_{45} $ $ C_{35}, C_{45} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $ $ Serv^4 $ $ Serv^5 $
$ A_{12}, A_{13} $ $ A_{12}, A_{23} $ $ A_{13}, A_{23} $ $ A_{14}, A_{24} $ $ A_{15}, A_{25} $
$ A_{14}, A_{15} $ $ A_{24}, A_{25} $ $ A_{34}, A_{35} $ $ A_{34}, A_{45} $ $ A_{35}, A_{45} $
$ B_{12}, B_{13} $ $ B_{12}, B_{23} $ $ B_{13}, B_{23} $ $ B_{14}, B_{24} $ $ B_{15}, B_{25} $
$ B_{14}, B_{15} $ $ B_{24}, B_{25} $ $ B_{34}, B_{35} $ $ B_{34}, B_{45} $ $ B_{35}, B_{45} $
$ C_{12}, C_{13} $ $ C_{12}, C_{23} $ $ C_{13}, C_{23} $ $ C_{14}, C_{24} $ $ C_{15}, C_{25} $
$ C_{14}, C_{15} $ $ C_{24}, C_{25} $ $ C_{34}, C_{35} $ $ C_{34}, C_{45} $ $ C_{35}, C_{45} $
Table 6.  Queries to download message $ A $
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $ $ Serv^4 $ $ Serv^5 $
Step $ 1 $ $ a^1_{12},\; a^1_{13} $ $ a^1_{23},\; a^1_{24} $ $ a^1_{34},\; a^1_{35} $ $ a^1_{14},\; a^1_{45} $ $ a^1_{15},\; a^1_{25} $
$ b^1_{12},\; b^1_{13} $ $ b^1_{23},\; b^1_{24} $ $ b^1_{34},\; b^1_{35} $ $ b^1_{14},\; b^1_{45} $ $ b^1_{15},\; b^1_{25} $
$ c^1_{12},\; c^1_{13} $ $ c^1_{23},\; c^1_{24} $ $ c^1_{34},\; c^1_{35} $ $ c^1_{14},\; c^1_{45} $ $ c^1_{15},\; c^1_{25} $
Step $ 2 $ $ a^2_{14}+b^1_{14} $ $ a^2_{12}+b^1_{12} $ $ a^2_{13}+b^1_{13} $ $ a^2_{24}+b^1_{24} $ $ a^2_{35}+b^1_{35} $
$ a^3_{14}+c^1_{14} $ $ a^3_{12}+c^1_{12} $ $ a^3_{13}+c^1_{13} $ $ a^3_{24}+c^1_{24} $ $ a^3_{35}+c^1_{35} $
$ a^2_{15}+b^1_{15} $ $ a^2_{25}+b^1_{25} $ $ a^2_{23}+b^1_{23} $ $ a^2_{34}+b^1_{34} $ $ a^2_{45}+b^1_{45} $
$ a^3_{15}+c^1_{15} $ $ a^3_{25}+c^1_{25} $ $ a^3_{23}+c^1_{23} $ $ a^3_{34}+c^1_{34} $ $ a^3_{45}+c^1_{45} $
$ b^2_{14}+c^2_{14} $ $ b^2_{12}+c^2_{12} $ $ b^2_{13}+c^2_{13} $ $ b^2_{24}+c^2_{24} $ $ b^2_{35}+c^2_{35} $
$ b^2_{15}+c^2_{15} $ $ b^2_{25}+c^2_{25} $ $ b^2_{23}+c^2_{23} $ $ b^2_{34}+c^2_{34} $ $ b^2_{45}+c^2_{45} $
Step $ 3 $ $ a^4_{12}+b^2_{12}+c^2_{12} $ $ a^4_{23}+b^2_{23}+c^2_{23} $ $ a^4_{34}+b^2_{34}+c^2_{34} $ $ a^4_{14}+b^2_{14}+c^2_{14} $ $ a^4_{15}+b^2_{15}+c^2_{15} $
$ a^4_{13}+b^2_{13}+c^2_{13} $ $ a^4_{24}+b^2_{24}+c^2_{24} $ $ a^4_{35}+b^2_{35}+c^2_{35} $ $ a^4_{45}+b^2_{45}+c^2_{45} $ $ a^4_{25}+b^2_{25}+c^2_{25} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $ $ Serv^4 $ $ Serv^5 $
Step $ 1 $ $ a^1_{12},\; a^1_{13} $ $ a^1_{23},\; a^1_{24} $ $ a^1_{34},\; a^1_{35} $ $ a^1_{14},\; a^1_{45} $ $ a^1_{15},\; a^1_{25} $
$ b^1_{12},\; b^1_{13} $ $ b^1_{23},\; b^1_{24} $ $ b^1_{34},\; b^1_{35} $ $ b^1_{14},\; b^1_{45} $ $ b^1_{15},\; b^1_{25} $
$ c^1_{12},\; c^1_{13} $ $ c^1_{23},\; c^1_{24} $ $ c^1_{34},\; c^1_{35} $ $ c^1_{14},\; c^1_{45} $ $ c^1_{15},\; c^1_{25} $
Step $ 2 $ $ a^2_{14}+b^1_{14} $ $ a^2_{12}+b^1_{12} $ $ a^2_{13}+b^1_{13} $ $ a^2_{24}+b^1_{24} $ $ a^2_{35}+b^1_{35} $
$ a^3_{14}+c^1_{14} $ $ a^3_{12}+c^1_{12} $ $ a^3_{13}+c^1_{13} $ $ a^3_{24}+c^1_{24} $ $ a^3_{35}+c^1_{35} $
$ a^2_{15}+b^1_{15} $ $ a^2_{25}+b^1_{25} $ $ a^2_{23}+b^1_{23} $ $ a^2_{34}+b^1_{34} $ $ a^2_{45}+b^1_{45} $
$ a^3_{15}+c^1_{15} $ $ a^3_{25}+c^1_{25} $ $ a^3_{23}+c^1_{23} $ $ a^3_{34}+c^1_{34} $ $ a^3_{45}+c^1_{45} $
$ b^2_{14}+c^2_{14} $ $ b^2_{12}+c^2_{12} $ $ b^2_{13}+c^2_{13} $ $ b^2_{24}+c^2_{24} $ $ b^2_{35}+c^2_{35} $
$ b^2_{15}+c^2_{15} $ $ b^2_{25}+c^2_{25} $ $ b^2_{23}+c^2_{23} $ $ b^2_{34}+c^2_{34} $ $ b^2_{45}+c^2_{45} $
Step $ 3 $ $ a^4_{12}+b^2_{12}+c^2_{12} $ $ a^4_{23}+b^2_{23}+c^2_{23} $ $ a^4_{34}+b^2_{34}+c^2_{34} $ $ a^4_{14}+b^2_{14}+c^2_{14} $ $ a^4_{15}+b^2_{15}+c^2_{15} $
$ a^4_{13}+b^2_{13}+c^2_{13} $ $ a^4_{24}+b^2_{24}+c^2_{24} $ $ a^4_{35}+b^2_{35}+c^2_{35} $ $ a^4_{45}+b^2_{45}+c^2_{45} $ $ a^4_{25}+b^2_{25}+c^2_{25} $
Table 7.  The number of symbols downloaded in each step of the scheme
Steps Tuple Number of Total symbols Number of Useful symbols
Step 1 Single $ N{{M}\choose{1}}\lambda $ $ N\lambda $
Step 2 Pair $ N{{M}\choose{2}}\lambda(t-1) $ $ N{{M-1}\choose{2-1}}\lambda(t-1) $
Step 3 Triple $ N{{M}\choose{3}}\lambda(t-1)^2 $ $ N{{M-1}\choose{3-1}}\lambda(t-1)^2 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step k k-tuple $ N{{M}\choose{k}}\lambda(t-1)^{k-1} $ $ N{{M-1}\choose{k-1}}\lambda(t-1)^{k-1} $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step M M-tuple $ N{{M}\choose{M}}\lambda(t-1)^{M-1} $ $ N{{M-1}\choose{M-1}}\lambda(t-1)^{M-1} $
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Steps Tuple Number of Total symbols Number of Useful symbols
Step 1 Single $ N{{M}\choose{1}}\lambda $ $ N\lambda $
Step 2 Pair $ N{{M}\choose{2}}\lambda(t-1) $ $ N{{M-1}\choose{2-1}}\lambda(t-1) $
Step 3 Triple $ N{{M}\choose{3}}\lambda(t-1)^2 $ $ N{{M-1}\choose{3-1}}\lambda(t-1)^2 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step k k-tuple $ N{{M}\choose{k}}\lambda(t-1)^{k-1} $ $ N{{M-1}\choose{k-1}}\lambda(t-1)^{k-1} $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step M M-tuple $ N{{M}\choose{M}}\lambda(t-1)^{M-1} $ $ N{{M-1}\choose{M-1}}\lambda(t-1)^{M-1} $
Table 8.  Queries to download message $ A $
$ Serv^1 $ $ Serv^2 $ $ Serv^3 $ $ Serv^4 $
$ a^1_{12},a^1_{14} $ $ a^1_{23},a^1_{24} $ $ a^1_{13} $ $ a^1_{34} $
$ b^1_{12},b^1_{14} $ $ b^1_{23},b^1_{24} $ $ b^1_{13} $ $ b^1_{34} $
$ a^2_{13}+b^1_{13} $ $ a^2_{12}+b^1_{12} $ $ a^2_{23}+b^1_{23} $ $ a^2_{14}+b^1_{14} $
$ a^2_{34}+b^1_{34} $ $ a^2_{24}+b^1_{24} $
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$ Serv^1 $ $ Serv^2 $ $ Serv^3 $ $ Serv^4 $
$ a^1_{12},a^1_{14} $ $ a^1_{23},a^1_{24} $ $ a^1_{13} $ $ a^1_{34} $
$ b^1_{12},b^1_{14} $ $ b^1_{23},b^1_{24} $ $ b^1_{13} $ $ b^1_{34} $
$ a^2_{13}+b^1_{13} $ $ a^2_{12}+b^1_{12} $ $ a^2_{23}+b^1_{23} $ $ a^2_{14}+b^1_{14} $
$ a^2_{34}+b^1_{34} $ $ a^2_{24}+b^1_{24} $
Table 9.  The number of symbols downloaded in each step of the scheme
Steps Tuple Number of Total symbols (all servers) Number of Useful symbols (all servers)
Step 1 Single $ {{{M}\choose{1}}[da+t(N-d)]} $ $ da+t(N-d) $
Step 2 Pair $ {{M}\choose{2}}\frac{d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)}{{{N}\choose{t}}^{0}} $ $ {{M-1}\choose{2-1}}\frac{d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)}{{{N}\choose{t}}^0} $
Step 3 Triple $ {{M}\choose{3}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^2}{{{N}\choose{t}}^{1}} $ $ {{M-1}\choose{3-1}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^2}{{{N}\choose{t}}^1} $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step k k-tuple $ {{M}\choose{k}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{k-1}}{{{N}\choose{t}}^{k-2}} $ $ {{M-1}\choose{k-1}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{k-1}}{{{N}\choose{t}}^{k-2}} $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step M M-tuple $ {{M}\choose{M}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{M-1}}{{{N}\choose{t}}^{M-2}} $ $ {{M-1}\choose{M-1}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{M-1}}{{{N}\choose{t}}^{M-2}} $
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Steps Tuple Number of Total symbols (all servers) Number of Useful symbols (all servers)
Step 1 Single $ {{{M}\choose{1}}[da+t(N-d)]} $ $ da+t(N-d) $
Step 2 Pair $ {{M}\choose{2}}\frac{d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)}{{{N}\choose{t}}^{0}} $ $ {{M-1}\choose{2-1}}\frac{d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)}{{{N}\choose{t}}^0} $
Step 3 Triple $ {{M}\choose{3}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^2}{{{N}\choose{t}}^{1}} $ $ {{M-1}\choose{3-1}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^2}{{{N}\choose{t}}^1} $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step k k-tuple $ {{M}\choose{k}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{k-1}}{{{N}\choose{t}}^{k-2}} $ $ {{M-1}\choose{k-1}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{k-1}}{{{N}\choose{t}}^{k-2}} $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
Step M M-tuple $ {{M}\choose{M}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{M-1}}{{{N}\choose{t}}^{M-2}} $ $ {{M-1}\choose{M-1}}\frac{{[d({{N-1}\choose{t-1}}-a)+(N-d)({{N-1}\choose{t-1}}-t)]}^{M-1}}{{{N}\choose{t}}^{M-2}} $
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