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

Embedding cover-free families and cryptographical applications

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.

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

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

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P. ErdösP. 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

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

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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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[10]

T. B. IdalinoL. MouraR. 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

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

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P. C. LiG. 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

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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. StaddonD. 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

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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. StinsonR. 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. StinsonR. 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

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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. ErdösP. 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. IdalinoL. MouraR. 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. LiG. 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. StaddonD. 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. StinsonR. 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. StinsonR. 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
Figure 2.  Example of a 0-CFF(3; 9), 1-CFF(6; 9) and a 2-CFF(9; 9)
Figure 3.  Example of a 4-CFF(81; 729)
Figure 4.  Compression ratio for $ q = 16,256; 1 \leq k \leq 3,; d = \log_4 n $
Figure 5.  An SHF(2; 6; 4; {1; 2})
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
$ 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
$ 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} $
$ 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} $
$ 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
$ 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
$ 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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