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November  2019, 13(4): 629-643. doi: 10.3934/amc.2019039

Embedding cover-free families and cryptographical applications

Thais Bardini Idalino and  Lucia Moura

Department of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada

Received  October 2018 Published  June 2019

Fund Project: Thais Bardini Idalino acknowledges funding granted from CNPq-Brazil [233697/2014-4] and OGS. Lucia Moura was supported by an NSERC discovery grant.

Cover-free families are set systems used as solutions for a large variety of problems, and in particular, problems where we deal with $ n $ elements and want to identify $ d $ defective ones among them by performing only $ t $ tests ($ t \leq n $). We are especially interested in cryptographic problems, and we note that some of these problems need cover-free families with an increasing size $ n $. Solutions that propose the increase of $ n $, such as monotone families and nested families, have been recently considered in the literature. In this paper, we propose a generalization that we call embedding families, which allows us to increase both $ n $ and $ d $. We propose constructions of embedding families using polynomials over finite fields embedded via extension fields; we study how different parameter combinations can be used to prioritize increase of $ d $ or of the compression ratio as $ n $ grows. We also provide new constructions for monotone families with improved compression ratio. Finally, we show how to use embedded sequences of orthogonal arrays and packing arrays to build embedding families.

Keywords: Combinatorial cryptography, combinatorial group testing, cover-free family, embedding cover-free family, polynomial over finite field, separating hash family.
Mathematics Subject Classification: Primary: 05B99, 94C12; Secondary: 11T71.
Citation: Thais Bardini Idalino, Lucia Moura. Embedding cover-free families and cryptographical applications. Advances in Mathematics of Communications, 2019, 13 (4) : 629-643. doi: 10.3934/amc.2019039
References:
[1]

D. Boneh, C. Gentry, B. Lynn and H. Shacham, Aggregate and verifiably encrypted signatures from bilinear maps, In: Biham E. (eds) Advances in Cryptology – EUROCRYPT 2003. Lecture Notes in Comput. Sci., vol 2656, Springer, Berlin, Heidelberg, 2003, 416–432. doi: 10.1007/3-540-39200-9_26.  Google Scholar

[2]

K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Statistics, 23 (1952), 426-434.  doi: 10.1214/aoms/1177729387.  Google Scholar

[3]

C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, 2007.  Google Scholar

[4] D. Z. Du and F. K. Hwang, Combinatorial Group Testing and Its Applications, World Scientific, 2000.   Google Scholar
[5]

P. Erdös, P. Frankl and Z. Füredi, Families of finite sets in which no set is covered by the union of r others, Israel J. Math., 51 (1985), 79-89.  doi: 10.1007/BF02772959.  Google Scholar

[6]

Z. Füredi, On r-Cover-free Families, J. Combin. Theory Ser. A, 73 (1996), 172-173.  doi: 10.1006/jcta.1996.0012.  Google Scholar

[7]

E. Gafni, J. Staddon and Y. L. Yin, Efficient Methods for Integrating Traceability and Broadcast Encryption, In: Wiener M. (eds) Advances in Cryptology – CRYPTO 1999. Lecture Notes in Comput. Sci., vol 1666, Springer, Berlin, Heidelberg, 1999, 372–387. doi: 10.1007/3-540-48405-1_24.  Google Scholar

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G. Hartung, B. Kaidel, A. Koch, J. Koch and A. Rupp, Fault-tolerant aggregate signatures., In Public-Key Cryptography – PKC 2016. Lecture Notes in Comput. Sci., vol 9614, Springer, Cham, 2016, 331–356. doi: 10.1007/978-3-662-49384-7_13.  Google Scholar

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T. B. Idalino and L. Moura, Efficient unbounded fault-tolerant aggregate signatures using nested cover-free families, In: Iliopoulos C., Leong H., Sung WK. (eds) Combinatorial Algorithms – IWOCA 2018. Lecture Notes in Comput. Sci., vol 10979, Springer, Cham, 2018, 52–64. doi: 10.1007/978-3-319-94667-2_5.  Google Scholar

[10]

T. B. Idalino, L. Moura, R. F. Custódio and D. Panario, Locating modifications in signed data for partial data integrity, Inform. Process. Lett., 115 (2015), 731-737.  doi: 10.1016/j.ipl.2015.02.014.  Google Scholar

[11]

K. M. Kim, Perfect Hash Families: Constructions and Applications, Master's thesis, University of Waterloo, 2003. Google Scholar

[12]

P. C. Li, G. H. J. van Rees and R. Wei, Constructions of 2-cover-free families and related separating hash families, J. Combin. Des., 14 (2006), 423-440.  doi: 10.1002/jcd.20109.  Google Scholar

[13]

S. Ling, H. Wang and C. Xing, Cover-Free Families and Their Applications, In: Security in Distributed and Networking Systems, chapter 4, 2007. Google Scholar

[14]

J. Pastuszak, J. Pieprzyk and J. Seberry, Codes identifying bad signature in batches, In: Roy B., Okamoto E. (eds) Progress in Cryptology – INDOCRYPT 2000. Lecture Notes in Comput. Sci., vol 1977, Springer, Berlin, Heidelberg, 2000, 143–154. doi: 10.1007/3-540-44495-5_13.  Google Scholar

[15]

J. Pieprzyk, H. Wang and C. Xing, Multiple-time signature schemes against adaptive chosen message attacks, In: Matsui M., Zuccherato R.J. (eds) Selected Areas in Cryptography – SAC 2003. Lecture Notes in Comput. Sci., vol 3006, Springer, Berlin, Heidelberg, 2004, 88–100. doi: 10.1007/978-3-540-24654-1_7.  Google Scholar

[16]

I. S. Reed and G. Solomon, Polynomial codes over certain finite fields, J. Soc. Indust. Appl. Math., 8 (1960), 300-304.  doi: 10.1137/0108018.  Google Scholar

[17]

R. Safavi-Naini and H. Wang, New results on multi-receiver authentication codes, In: Nyberg K. (eds) Advances in Cryptology – EUROCRYPT 1998. Lecture Notes in Comput. Sci., vol 1403, Springer, Berlin, Heidelberg, 1998, 527–541. doi: 10.1007/BFb0054151.  Google Scholar

[18]

J. N. Staddon, D. R. Stinson and R. Wei, Combinatorial properties of frameproof and traceability codes, IEEE Trans. Inform. Theory, 47 (2001), 1042-1049.  doi: 10.1109/18.915661.  Google Scholar

[19]

B. Stevens and E. Mendelsohn, Packing arrays, Theoret. Comput. Sci., 321 (2004), 25-148.  doi: 10.1016/j.tcs.2003.06.004.  Google Scholar

[20]

D. R. Stinson and R. Wei, Combinatorial properties and constructions of traceability schemes and frameproof codes, SIAM J. Discrete Math., 11 (1998), 41-53.  doi: 10.1137/S0895480196304246.  Google Scholar

[21]

D. R. Stinson and R. Wei, Generalized cover-free families, Discrete Math., 279 (2004), 463-477.  doi: 10.1016/S0012-365X(03)00287-5.  Google Scholar

[22]

D. R. Stinson, R. Wei and K. Chen, On generalized separating hash families, J. Combin. Theory Ser. A, 115 (2008), 105-120.  doi: 10.1016/j.jcta.2007.04.005.  Google Scholar

[23]

D. R. Stinson, R. Wei and L. Zhu, New constructions for perfect hash families and related structures using combinatorial designs and codes, J. Combin. Des., 8 (2000), 189-200.  doi: 10.1002/(SICI)1520-6610(2000)8:3<189::AID-JCD4>3.0.CO;2-A.  Google Scholar

[24]

G. M. Zaverucha and D. R. Stinson, Group testing and batch verification, In: Kurosawa K. (eds) Information Theoretic Security – ICITS 2009. Lecture Notes in Comput. Sci., vol 5973, Springer, Berlin, Heidelberg, 2009, 140–157. doi: 10.1007/978-3-642-14496-7_12.  Google Scholar

[25]

G. M. Zaverucha and D. R. Stinson, Short one-time signatures, Adv. Math. Commun., 5 (2011), 473-488.  doi: 10.3934/amc.2011.5.473.  Google Scholar

show all references

References:
[1]

D. Boneh, C. Gentry, B. Lynn and H. Shacham, Aggregate and verifiably encrypted signatures from bilinear maps, In: Biham E. (eds) Advances in Cryptology – EUROCRYPT 2003. Lecture Notes in Comput. Sci., vol 2656, Springer, Berlin, Heidelberg, 2003, 416–432. doi: 10.1007/3-540-39200-9_26.  Google Scholar

[2]

K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Statistics, 23 (1952), 426-434.  doi: 10.1214/aoms/1177729387.  Google Scholar

[3]

C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, 2007.  Google Scholar

[4] D. Z. Du and F. K. Hwang, Combinatorial Group Testing and Its Applications, World Scientific, 2000.   Google Scholar
[5]

P. Erdös, P. Frankl and Z. Füredi, Families of finite sets in which no set is covered by the union of r others, Israel J. Math., 51 (1985), 79-89.  doi: 10.1007/BF02772959.  Google Scholar

[6]

Z. Füredi, On r-Cover-free Families, J. Combin. Theory Ser. A, 73 (1996), 172-173.  doi: 10.1006/jcta.1996.0012.  Google Scholar

[7]

E. Gafni, J. Staddon and Y. L. Yin, Efficient Methods for Integrating Traceability and Broadcast Encryption, In: Wiener M. (eds) Advances in Cryptology – CRYPTO 1999. Lecture Notes in Comput. Sci., vol 1666, Springer, Berlin, Heidelberg, 1999, 372–387. doi: 10.1007/3-540-48405-1_24.  Google Scholar

[8]

G. Hartung, B. Kaidel, A. Koch, J. Koch and A. Rupp, Fault-tolerant aggregate signatures., In Public-Key Cryptography – PKC 2016. Lecture Notes in Comput. Sci., vol 9614, Springer, Cham, 2016, 331–356. doi: 10.1007/978-3-662-49384-7_13.  Google Scholar

[9]

T. B. Idalino and L. Moura, Efficient unbounded fault-tolerant aggregate signatures using nested cover-free families, In: Iliopoulos C., Leong H., Sung WK. (eds) Combinatorial Algorithms – IWOCA 2018. Lecture Notes in Comput. Sci., vol 10979, Springer, Cham, 2018, 52–64. doi: 10.1007/978-3-319-94667-2_5.  Google Scholar

[10]

T. B. Idalino, L. Moura, R. F. Custódio and D. Panario, Locating modifications in signed data for partial data integrity, Inform. Process. Lett., 115 (2015), 731-737.  doi: 10.1016/j.ipl.2015.02.014.  Google Scholar

[11]

K. M. Kim, Perfect Hash Families: Constructions and Applications, Master's thesis, University of Waterloo, 2003. Google Scholar

[12]

P. C. Li, G. H. J. van Rees and R. Wei, Constructions of 2-cover-free families and related separating hash families, J. Combin. Des., 14 (2006), 423-440.  doi: 10.1002/jcd.20109.  Google Scholar

[13]

S. Ling, H. Wang and C. Xing, Cover-Free Families and Their Applications, In: Security in Distributed and Networking Systems, chapter 4, 2007. Google Scholar

[14]

J. Pastuszak, J. Pieprzyk and J. Seberry, Codes identifying bad signature in batches, In: Roy B., Okamoto E. (eds) Progress in Cryptology – INDOCRYPT 2000. Lecture Notes in Comput. Sci., vol 1977, Springer, Berlin, Heidelberg, 2000, 143–154. doi: 10.1007/3-540-44495-5_13.  Google Scholar

[15]

J. Pieprzyk, H. Wang and C. Xing, Multiple-time signature schemes against adaptive chosen message attacks, In: Matsui M., Zuccherato R.J. (eds) Selected Areas in Cryptography – SAC 2003. Lecture Notes in Comput. Sci., vol 3006, Springer, Berlin, Heidelberg, 2004, 88–100. doi: 10.1007/978-3-540-24654-1_7.  Google Scholar

[16]

I. S. Reed and G. Solomon, Polynomial codes over certain finite fields, J. Soc. Indust. Appl. Math., 8 (1960), 300-304.  doi: 10.1137/0108018.  Google Scholar

[17]

R. Safavi-Naini and H. Wang, New results on multi-receiver authentication codes, In: Nyberg K. (eds) Advances in Cryptology – EUROCRYPT 1998. Lecture Notes in Comput. Sci., vol 1403, Springer, Berlin, Heidelberg, 1998, 527–541. doi: 10.1007/BFb0054151.  Google Scholar

[18]

J. N. Staddon, D. R. Stinson and R. Wei, Combinatorial properties of frameproof and traceability codes, IEEE Trans. Inform. Theory, 47 (2001), 1042-1049.  doi: 10.1109/18.915661.  Google Scholar

[19]

B. Stevens and E. Mendelsohn, Packing arrays, Theoret. Comput. Sci., 321 (2004), 25-148.  doi: 10.1016/j.tcs.2003.06.004.  Google Scholar

[20]

D. R. Stinson and R. Wei, Combinatorial properties and constructions of traceability schemes and frameproof codes, SIAM J. Discrete Math., 11 (1998), 41-53.  doi: 10.1137/S0895480196304246.  Google Scholar

[21]

D. R. Stinson and R. Wei, Generalized cover-free families, Discrete Math., 279 (2004), 463-477.  doi: 10.1016/S0012-365X(03)00287-5.  Google Scholar

[22]

D. R. Stinson, R. Wei and K. Chen, On generalized separating hash families, J. Combin. Theory Ser. A, 115 (2008), 105-120.  doi: 10.1016/j.jcta.2007.04.005.  Google Scholar

[23]

D. R. Stinson, R. Wei and L. Zhu, New constructions for perfect hash families and related structures using combinatorial designs and codes, J. Combin. Des., 8 (2000), 189-200.  doi: 10.1002/(SICI)1520-6610(2000)8:3<189::AID-JCD4>3.0.CO;2-A.  Google Scholar

[24]

G. M. Zaverucha and D. R. Stinson, Group testing and batch verification, In: Kurosawa K. (eds) Information Theoretic Security – ICITS 2009. Lecture Notes in Comput. Sci., vol 5973, Springer, Berlin, Heidelberg, 2009, 140–157. doi: 10.1007/978-3-642-14496-7_12.  Google Scholar

[25]

G. M. Zaverucha and D. R. Stinson, Short one-time signatures, Adv. Math. Commun., 5 (2011), 473-488.  doi: 10.3934/amc.2011.5.473.  Google Scholar

Figure 1.  Example of a $ 2 $-CFF($ 9,12 $) used in group testing
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Figure 2.  Example of a 0-CFF(3; 9), 1-CFF(6; 9) and a 2-CFF(9; 9)
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Figure 3.  Example of a 4-CFF(81; 729)
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Figure 4.  Compression ratio for $ q = 16,256; 1 \leq k \leq 3,; d = \log_4 n $
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Figure 5.  An SHF(2; 6; 4; {1; 2})
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Table 1.  Example of prioritizing $ d $ increases with fixed $ k = 2 $
$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 2 1 64 12 5.33
1 16 2 7 4096 240 17.06
2 256 2 127 16777216 65280 257.00
3 65536 2 32767 281474976710656 4294901760 65537.00
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$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 2 1 64 12 5.33
1 16 2 7 4096 240 17.06
2 256 2 127 16777216 65280 257.00
3 65536 2 32767 281474976710656 4294901760 65537.00
Table 2.  Example of prioritizing $ d $ increases with fixed $ k = 3 $
$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 3 1 256 16 16
1 16 3 5 65536 256 256
2 256 3 85 4294967296 65536 65536
3 65536 3 21845 $ 65536^4 $ 4294967296 4294967296
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$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 3 1 256 16 16
1 16 3 5 65536 256 256
2 256 3 85 4294967296 65536 65536
3 65536 3 21845 $ 65536^4 $ 4294967296 4294967296
Table 3.  Example of prioritizing ratio increase with fixed $ d = 2 $
$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 1 2 16 12 1.33
1 16 7 2 4294967296 240 17895697.07
2 256 127 2 $ 256^{128} $ 65280 $ 2.75 \times 10^{303} $
3 65536 32767 2 $ 65536^{32768} $ 4294901760 $ 6.04 \times 10^{157816} $
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$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 1 2 16 12 1.33
1 16 7 2 4294967296 240 17895697.07
2 256 127 2 $ 256^{128} $ 65280 $ 2.75 \times 10^{303} $
3 65536 32767 2 $ 65536^{32768} $ 4294901760 $ 6.04 \times 10^{157816} $
Table 4.  Example of prioritizing ratio increase with fixed $ d = 3 $
$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 1 3 16 16 1
1 16 5 3 16777216 256 65536
2 256 85 3 $ 256^{86} $ 65536 $ 1.95 \times 10^{202} $
3 65536 21845 3 $ 65536^{21846} $ 4294967296 $ 1.54 \times 10^{105211} $
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$ i $ $ q $ $ k $ $ d $ $ n $ $ t $ $ n/t $
0 4 1 3 16 16 1
1 16 5 3 16777216 256 65536
2 256 85 3 $ 256^{86} $ 65536 $ 1.95 \times 10^{202} $
3 65536 21845 3 $ 65536^{21846} $ 4294967296 $ 1.54 \times 10^{105211} $
Table 5.  Summary of results for $ k \geq 2 $
$ k $ $ d $ $ \rho(n) $ Feature
Corollary 1 fixed $ d \sim \frac{n^{1/(k+1)}}{k} $ $ n^{1-\frac{2}{k+1}} $ increasing $ d $
Corollary 2 increasing fixed $ \frac{n}{\log n} $ optimal ratio
Theorem 3.5 fixed fixed $ n^{1 - \frac{1}{k+1}} $ monotone
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$ k $ $ d $ $ \rho(n) $ Feature
Corollary 1 fixed $ d \sim \frac{n^{1/(k+1)}}{k} $ $ n^{1-\frac{2}{k+1}} $ increasing $ d $
Corollary 2 increasing fixed $ \frac{n}{\log n} $ optimal ratio
Theorem 3.5 fixed fixed $ n^{1 - \frac{1}{k+1}} $ monotone
Table 6.  Compression ratio for $ q = 16,256;1 \leq k \leq 3; d = \log_4 n $
$ q $ $ k $ $ d $ $ n $ $ t $ $ \rho(n)=n/t $
16 1 3 128 64 2.00
16 1 4 256 80 3.20
16 2 4 512 144 3.55
16 2 5 1024 176 5.81
16 2 5 2048 176 11.63
16 2 6 4096 208 19.69
256 2 6 8192 3328 2.46
256 2 7 16384 3840 4.26
256 2 7 32768 3840 8.53
256 2 8 65536 4352 15.05
256 2 8 131072 4352 30.11
256 2 9 262144 4864 53.89
256 2 9 524288 4864 107.78
256 2 10 1048576 5376 195.04
256 2 10 2097152 5376 390.09
256 2 11 4194304 5888 712.34
256 2 11 8388608 5888 1424.69
256 2 12 16777216 6400 2621.44
256 3 12 33554432 9472 3542.48
256 3 13 67108864 10240 6553.60
256 3 13 134217728 10240 13107.20
Table Options

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$ q $ $ k $ $ d $ $ n $ $ t $ $ \rho(n)=n/t $
16 1 3 128 64 2.00
16 1 4 256 80 3.20
16 2 4 512 144 3.55
16 2 5 1024 176 5.81
16 2 5 2048 176 11.63
16 2 6 4096 208 19.69
256 2 6 8192 3328 2.46
256 2 7 16384 3840 4.26
256 2 7 32768 3840 8.53
256 2 8 65536 4352 15.05
256 2 8 131072 4352 30.11
256 2 9 262144 4864 53.89
256 2 9 524288 4864 107.78
256 2 10 1048576 5376 195.04
256 2 10 2097152 5376 390.09
256 2 11 4194304 5888 712.34
256 2 11 8388608 5888 1424.69
256 2 12 16777216 6400 2621.44
256 3 12 33554432 9472 3542.48
256 3 13 67108864 10240 6553.60
256 3 13 134217728 10240 13107.20
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