Embedding cover-free families and cryptographical applications

Thais Bardini Idalino; Lucia Moura

doi:10.3934/amc.2019039

Article Contents

2019, Volume 13, Issue 4: 629-643. Doi: 10.3934/amc.2019039

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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

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 05B99, 94C12; Secondary: 11T71.

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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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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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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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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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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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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

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