# IoT-Applicable Generalized Frameproof Combinatorial Designs

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Combinatorial RTS

#### 1.2. A Drawback and an Idea of Extension

#### Repairability Problem

#### 1.3. Frameproofness

## 2. Results

#### Organization of the Paper

## 3. Stinson and Wei’s Model [9]

- an initialization phase, in which the dealer chooses distinct, non-zero public elements ${x}_{1},{x}_{2},\dots ,{x}_{b}$ from ${\mathbb{F}}_{q}$, and gives value ${x}_{i}$ to player ${P}_{i}$;
- a share distribution phase in which the dealer chooses a secret $K={a}_{0}\in {\mathbb{F}}_{q}$, then secretly chooses ${a}_{1},\dots ,{a}_{\tau -1}\in {\mathbb{F}}_{q}$ independently and uniformly at random, and finally computes the share ${y}_{i}=a\left({x}_{i}\right)$ $\left(\mathrm{where}a\left(x\right):={\displaystyle \sum _{j=0}^{\tau -1}}{a}_{j}{x}^{j}\right)$ and gives it to player ${P}_{i}$.

**Definition 1.**

- 1.
- $\left|X\right|=v$;
- 2.
- each block $B\in \mathcal{B}$ contains exactly k points;
- 3.
- every pair of distinct points from X is contained in exactly λ blocks.

- (i)
- $bk=vr$;
- (ii)
- $\lambda (v-1)=r(k-1)$;
- (iii)
- $b\ge v$ (and hence $r>k$).

**Definition 2.**

#### Design Properties

**Theorem 1**

**.**Every point in a $(v,k,\lambda )$-BIBD occurs in exactly $r=\frac{\lambda (v-1)}{k-1}$ blocks. The value r is termed the replication number of the scheme.

**Theorem 2**

**.**A $(v,k,\lambda )$-BIBD has exactly

## 4. Tensor Design Generated by Two BIBDs

#### 4.1. Definition of the Krönecker Product

#### 4.2. Krönecker Product of Two BIBDs

1 | 2 | 3 | 2 | 4 | 6 | 3 | 6 | 9 |

2 | 3 | 5 | 4 | 6 | 10 | 6 | 9 | 15 |

3 | 5 | 1 | 6 | 10 | 2 | 9 | 15 | 3 |

5 | 1 | 2 | 10 | 2 | 4 | 15 | 3 | 6 |

2 | 4 | 6 | 1 | 2 | 3 | 4 | 8 | 12 |

4 | 6 | 10 | 2 | 3 | 5 | 8 | 12 | 20 |

6 | 10 | 2 | 3 | 5 | 1 | 12 | 20 | 4 |

10 | 2 | 4 | 5 | 1 | 2 | 20 | 4 | 8 |

3 | 6 | 9 | 4 | 8 | 12 | 2 | 4 | 6 |

6 | 9 | 15 | 8 | 12 | 20 | 4 | 6 | 10 |

9 | 15 | 3 | 12 | 20 | 4 | 6 | 10 | 2 |

15 | 3 | 6 | 20 | 4 | 8 | 10 | 2 | 4 |

4 | 8 | 12 | 3 | 6 | 9 | 1 | 2 | 3 |

8 | 12 | 20 | 6 | 9 | 15 | 2 | 3 | 5 |

12 | 20 | 4 | 9 | 15 | 3 | 3 | 5 | 1 |

20 | 4 | 8 | 15 | 3 | 6 | 5 | 1 | 2 |

#### 4.3. Some Results on the Krönecker Product of BIBDs

**Lemma 1.**

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Proof.**

**Theorem 3**

**.**Consider a $({v}_{1},{k}_{1},{\lambda}_{1},{b}_{1},{r}_{1})$-BIBD $\mathcal{A}$ and a $({v}_{2},{k}_{2},{\lambda}_{2},{b}_{2},{r}_{2})$-BIBD $\mathcal{B}$.

- 1.
- The matrix $\mathcal{A}\otimes {\mathcal{B}}_{d}$ produces a tensor design (over the integer ring $\mathbb{Z}$) for a (public) integer d such that there are no multiplicative collisions of the type ${x}_{i}({y}_{j}+d)={x}_{k}({y}_{l}+d)$ for $(i,j)\ne (k,l)$.
- 2.
- If $gcd({x}_{1},{x}_{2},\dots ,{x}_{{v}_{1}})=1$;
- if $gcd({y}_{1},{y}_{2},\dots ,{y}_{{v}_{2}})=1$;

then $\mathcal{A}$ and $\mathcal{B}$ can be reproduced from a collection of players in the new scheme $\mathcal{A}\otimes {\mathcal{B}}_{d}$, hence enabling share repair and secret reconstruction.

**Proof.**

**[I]**- A collection of players that has
- (i)
- ${\tau}_{2}$ players from one row-block ${T}_{i}$ of M;
- (ii)
- at least one player from distinct ${\tau}_{1}-1$ row-blocks ${T}_{j}\ne {T}_{i}$ of the remaining ${b}_{1}-1$ row-blocks

can reconstruct the secret. **[II]**- Let ${S}_{j}$ ($j\in \{1,2,\dots ,{b}_{2}\}$) be the collection of players $\{{P}_{{b}_{2}k+j}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}k\in \{0,1,\dots $$\dots ,{b}_{1}-1\left\}\right\}$. A collection of players that contains
- (i)
- ${\tau}_{1}$ players from one ${S}_{j}$;
- (ii)
- at least one player from ${\tau}_{2}-1$${S}_{i}$, $i\ne j$

can also reconstruct the secret.

- The share of the $j\mathrm{th}$ player ${P}_{i\xb7{b}_{2}-1+j}$ of the $i\mathrm{th}$ row-block ${T}_{i}$ is of the form$${\mathfrak{a}}_{i1}\xb7\{{\mathfrak{b}}_{j1},{\mathfrak{b}}_{j2},\dots ,{\mathfrak{b}}_{j{k}_{2}}\},{\mathfrak{a}}_{i2}\xb7\{{\mathfrak{b}}_{j1},{\mathfrak{b}}_{j2},\dots ,{\mathfrak{b}}_{j{k}_{2}}\},\dots ,{\mathfrak{a}}_{i{k}_{1}}\xb7\{{\mathfrak{b}}_{j1},{\mathfrak{b}}_{j2},\dots ,{\mathfrak{b}}_{j{k}_{2}}\}.$$Fix any $i\in \{1,2,\dots ,{b}_{1}\}$ and choose ${j}_{1},{j}_{2},\dots ,{j}_{{\tau}_{2}}$ to ensure that$gcd({\mathfrak{b}}_{{j}_{1}1},{\mathfrak{b}}_{{j}_{1}2},\dots ,{\mathfrak{b}}_{{j}_{1}{k}_{2}},{\mathfrak{b}}_{{j}_{2}1},{\mathfrak{b}}_{{j}_{2}2},\dots ,{\mathfrak{b}}_{{j}_{2}{k}_{2}},\dots ,{\mathfrak{b}}_{{j}_{{\tau}_{2}}1},{\mathfrak{b}}_{{j}_{{\tau}_{2}}2},\dots ,{\mathfrak{b}}_{{j}_{{\tau}_{2}}{k}_{2}})=1$.
- Therefore, the values of ${\mathfrak{a}}_{i1},{\mathfrak{a}}_{i2},\dots ,{\mathfrak{a}}_{i{k}_{1}}$ become known. Divide ${\mathfrak{a}}_{i\alpha}{\mathfrak{b}}_{{j}_{k}\beta}$ by ${\mathfrak{a}}_{i\alpha}$ (for $\alpha \in \{1,2,\dots ,{k}_{1}\},\phantom{\rule{0.166667em}{0ex}}\beta \in \{1,2,\dots ,{k}_{2}\}$ and $k\in \{1,2,\dots ,{\tau}_{2}\}$) to obtain ${\mathfrak{b}}_{{j}_{k}1},{\mathfrak{b}}_{{j}_{k}2},\dots ,{\mathfrak{b}}_{{j}_{k}{k}_{2}}$.
- Construct the complete matrix ${\mathcal{B}}_{d}$ using the shares of ${\tau}_{2}$ players of ${\mathcal{B}}_{d}$ that are now known. Hence construct $\mathcal{B}$.
- Using the values of the elements in ${\mathcal{B}}_{d}$, compute the values ${\mathfrak{a}}_{{i}^{\prime}1},{\mathfrak{a}}_{{i}^{\prime}2},\dots ,{\mathfrak{a}}_{{i}^{\prime}{k}_{1}}$ for ${\tau}_{1}-1$ indices ${i}^{\prime}$ that are distinct from each other as well as from i.
- Hence, construct $\mathcal{A}$ from the shares of ${\tau}_{1}$ players of $\mathcal{A}$ thus obtained.
- Finally compute the secret from $\mathcal{A}$ and $\mathcal{B}$.

#### 4.4. Proof of Existence of Secret Reconstruction

**[I]**- (i)
- $\underset{k\in \left[{b}_{1}\right]}{max}{n}_{k}\ge {\tau}_{2}$,
- (ii)
- ${n}_{k}\ge 1$ for at least ${\tau}_{1}$ indices k.

**[II]**- (i)
- $\underset{j\in \left[{b}_{2}\right]}{max}{r}_{j}\ge {\tau}_{1}$,
- (ii)
- ${r}_{j}\ge 1$ for at least ${\tau}_{2}$ indices j.

**Lemma 4.**

**Proof.**

**Lemma 5.**

**Proof.**

**Lemma 6.**

**Proof.**

**Lemma 7.**

**Proof.**

**Lemma 8.**

**Proof.**

**Lemma 9.**

**Proof.**

#### 4.5. A Generalized Share Distribution Scheme

- Dealer selects n (not necessarily distinct) BIBDs ${\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{n}$, where for $i\in \{1,2,\dots ,n\}$, ${\mathcal{A}}_{i}$ is defined over points $\{{x}_{1}^{i},{x}_{2}^{i},\dots ,{x}_{{v}_{i}}^{i}\}$.
- Dealer finds an integer ${d}_{1}$ such that $gcd({x}_{1}^{1}+{d}_{1},{x}_{2}^{1}+{d}_{1},\dots ,{x}_{{v}_{1}}^{1}+{d}_{1})=1$.
- For $i\in \{2,\dots ,n\}$:
- Dealer finds an integer ${d}_{i}$ (using Lemmas 2 and 3) such that ${d}_{i}$ breaks all pairwise multiplicative collisions and makes the gcd of all elements ${x}_{l}^{j}+{d}_{j}$ ($j\in \{1,\dots ,i-1\},\phantom{\rule{0.166667em}{0ex}}l\in \{1,\dots ,{v}_{j}\}$) and ${x}_{1}^{i}+{d}_{i},{x}_{2}^{i}+{d}_{i},\dots ,{x}_{{v}_{i}}^{i}+{d}_{i}$ is 1.

- M←${\mathcal{A}}_{1}\otimes {\mathcal{A}}_{2}\otimes \cdots \otimes {\mathcal{A}}_{n}$.
- Dealer distributes each row i of M as share to player ${P}_{i}$ and outputs $({d}_{1},{d}_{2},\dots ,{d}_{n})$ publicly.

## 5. Example

#### 5.1. Secret Reconstruction

- A collection of three players—exactly two from one of the sets ${T}_{1},{T}_{2}.{T}_{3},{T}_{4}$ and one from another—allows reconstruction of the secret. For example, consider the set of three players $\{{P}_{1},{P}_{2},{P}_{5}\}$. This set can reconstruct the secret:
- (i)
- $gcd(22,23,24,23,24,26)=1$; hence, the first row of ${M}_{\mathcal{A}}$ is $\left(1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}3\right)$ and the first two rows of ${M}_{\mathcal{B}}$ are $\left(22\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}23\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}24\right)$ and $\left(23\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}24\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}26\right)$. As ${\tau}_{2}=2$, ${M}_{\mathcal{B}}$ can be obtained from its two rows.
- (ii)
- Now, observing $5=4\xb71+1$, we readily know ${P}_{5}$ uses the first row of ${M}_{\mathcal{B}}$ and the second row of ${M}_{\mathcal{A}}$; this yields the second row of ${M}_{\mathcal{A}}$, $\left(2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}4\right)$. Since ${\tau}_{1}=2$ and we have two rows of ${M}_{\mathcal{A}}$, the whole matrix ${M}_{\mathcal{A}}$ is known.

- Any collection of three players—two from one of the sets ${S}_{1},{S}_{2},{S}_{3},{S}_{4}$ and one from another—also allows reconstruction of the secret.
- Reconstruction of the secret is ensured for a collection of five or more players.

## 6. Share Repair for a Krönecker Product-Induced Distribution Design

## 7. Frameproofness

- There exist $({b}_{2}-1)+({r}_{1}-1)\xb7{b}_{2}$ players that possess the element ${\mathfrak{a}}_{11}{\mathfrak{b}}_{ij}$ for some $i\in \{1,2,\dots ,{b}_{2}\}$ and $j\in \{1,2,\dots ,{k}_{2}\}$, since ${r}_{1}$ is the replication number of $\mathcal{A}$. Of these, $({r}_{1}-1)\xb71$ players possess the first ${k}_{2}$ elements of the share, i.e., ${\mathfrak{a}}_{11}{\mathfrak{b}}_{11}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{11}{\mathfrak{b}}_{12}\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{11}{\mathfrak{b}}_{1{k}_{2}}$. If any of these players know the ratios $\frac{{\mathfrak{a}}_{12}}{{\mathfrak{a}}_{11}},\frac{{\mathfrak{a}}_{13}}{{\mathfrak{a}}_{11}},\dots $, then they could construct the entire share of ${P}_{1}$.
- Note that for $j\ne 1$, any of the ${b}_{2}-1$ players with shares$$\begin{array}{ccc}\hfill {\mathfrak{a}}_{11}{\mathcal{B}}_{2}\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{12}{\mathcal{B}}_{2}\phantom{\rule{0.166667em}{0ex}}\right|& \dots & |\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{1{k}_{1}}{\mathcal{B}}_{2},\hfill \\ \hfill {\mathfrak{a}}_{11}{\mathcal{B}}_{3}\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{12}{\mathcal{B}}_{3}\phantom{\rule{0.166667em}{0ex}}\right|& \dots & |\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{1{k}_{1}}{\mathcal{B}}_{3},\hfill \\ & \vdots & \\ \hfill {\mathfrak{a}}_{11}{\mathcal{B}}_{{b}_{2}}\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{12}{\mathcal{B}}_{{b}_{2}}\phantom{\rule{0.166667em}{0ex}}\right|& \dots & |\phantom{\rule{0.166667em}{0ex}}{\mathfrak{a}}_{1{k}_{1}}{\mathcal{B}}_{{b}_{2}}\hfill \end{array}$$

#### 7.1. A Modified Scheme

#### 7.2. Example

#### 7.3. Secret Reconstruction for the Modified Scheme

- Choose a player ${P}_{i}^{m}$ (which is the $i\mathrm{th}$ player in the $m\mathrm{th}$ row-block of $\mathcal{A}\otimes \mathcal{B}$, or the $\left((m-1){b}_{2}+i\right)\mathrm{th}$ player from the top), for any $m\in \{1,2,\dots ,{b}_{1}\}$ and $i\in \{1,2,\dots ,{b}_{2}\}$.
- Consider elements ${a}_{mt}{b}_{ij}$ in the share of player ${P}_{i}^{m}$, i.e., ${\theta}_{mt}=l$ and ${\tilde{\pi}}^{l}\left(i\right)=j$. For such an element ${a}_{mt}{b}_{ij}$, set $y={b}_{ij}$ (note that the value $y\in \{{y}_{1},{y}_{2},\dots ,{y}_{{v}_{2}}\}$ is not known, but the positions at which the matrix $\mathcal{B}$ contains elements ${b}_{\widehat{i}\widehat{j}}=y$ is known).
- Construct set ${\mathcal{S}}_{y}\left\{\widehat{l}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\left({\tilde{\pi}}^{\widehat{l}}\left(\widehat{i}\right)=\widehat{j}\right)\wedge \left({b}_{\widehat{i}\widehat{j}}=y\right)\right\}$. By Theorem 6, for a maximal set ${\mathcal{S}}_{y}$ (if not, then another value y may be chosen by selecting a different element ${a}_{{m}^{\prime}{t}^{\prime}}{b}_{{i}^{\prime}{j}^{\prime}}$) the set$$\begin{array}{ccc}\hfill \{{a}_{\widehat{m}\widehat{t}}& :& {a}_{\widehat{m}\widehat{t}}{b}_{\widehat{i}\widehat{j}}\in \mathrm{the}\mathrm{share}\mathrm{of}\mathrm{player}{P}_{\widehat{i}}^{\widehat{m}}\mathrm{such}\mathrm{that}{b}_{\widehat{i}\widehat{j}}=y\}\hfill \\ & =& \{{x}_{1},{x}_{2},\dots ,{x}_{{v}_{1}}\}\hfill \end{array}$$
- Construct matrix $\mathcal{A}$, since the positions of all values ${x}_{1},{x}_{2},\dots ,{x}_{{v}_{1}}$ in this matrix are now known.
- Compute ${b}_{{i}^{\prime}{j}^{\prime}}$ for ${a}_{{m}^{\prime}{t}^{\prime}}{b}_{{i}^{\prime}{j}^{\prime}}$∈ share of player ${P}_{{i}^{\prime}}^{{m}^{\prime}}$ using the known values ${a}_{{m}^{\prime}{t}^{\prime}}$ until all values ${y}_{1},{y}_{2},\dots ,{y}_{{v}_{2}}$ are known.
- Construct matrix $\mathcal{B}$, since the positions of all values ${y}_{1},{y}_{2},\dots ,{y}_{{v}_{2}}$ in this matrix are now known.
- Compute $\mathcal{A}\otimes \mathcal{B}$ from the two known matrices.

## 8. Graphical Representation and Proof of Existence of Permutations

#### Matching in Bipartite Graphs

**Theorem 4**

**.**In any bipartite graph, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.

**Theorem 5**

**.**Given a bipartite graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$ with $\mathcal{V}=\mathbf{A}\bigsqcup \mathbf{B}$, $\mathcal{G}$ has a matching of size $\left|\mathbf{A}\right|$ if and only if for every $S\subseteq \mathbf{A}$ we have $\left|N\right(S\left)\right|\ge \left|S\right|$, where $N\left(S\right)=\{b\in \mathbf{B}:\exists \phantom{\rule{0.166667em}{0ex}}a\in Swith(a,b)\in \mathcal{E}\}$.

**Definition 3.**

- the vertex set $\mathcal{V}=\mathbf{P}\bigsqcup \mathbf{V}$ the disjoint union of the set of players $\mathbf{P}=\{{P}_{1},\dots ,{P}_{b}\}$ and the set of points $\mathbf{V}=\{{x}_{1},\dots ,{x}_{v}\}$ of $\mathcal{B}$;
- the edge set is the collection ${\bigcup}_{\begin{array}{c}i\in \left[b\right]\\ j\in \left[v\right]\end{array}}\{({P}_{i},{x}_{j})\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}{x}_{j}\in shareof{P}_{i}\}$.

**Theorem 6.**

- (i)
- If ${\bigcup}_{i\in \left[b\right]}\delta \left({P}_{i}\right)=\mathbf{V}$, then reconstruction of the modified scheme ${(\mathcal{A}\otimes \mathcal{B})}_{\mathrm{modified}}$ is possible.
- (ii)
- If $s\ge 1$, then (i) holds.

**Proof.**

## 9. Secret Sharing Schemes and the Internet of Things

## 10. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RTS | Repairable Threshold Scheme |

BIBD | Balanced Incomplete Block Design |

IoT | Internet of Things |

GDPR | General Data Protection Regulation |

SBIoT | Secret sharing-Based IoT |

eID | electronic IDentification |

IDA | Information Dispersal Algorithm |

BFS | Best-First Search |

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**Figure 1.**The matrix $\mathcal{A}\otimes {\mathcal{B}}_{21}$ is the Krönecker product of $\mathcal{A}$ and ${\mathcal{B}}_{21}$ as in Equation (7), and is a secret sharing scheme with reconstruction number 2. A secret reconstruction algorithm for this scheme is detailed in Section 5.1.

**Figure 2.**The matrix $\mathcal{A}\otimes \mathcal{B}$ is the Krönecker product of $\mathcal{A}$ and $\mathcal{B}$ as in Equation (12), and is a secret sharing scheme with reconstruction number 2.

**Figure 3.**The matrix N, right-operated as $\odot N$ on the tensor design $\mathcal{A}\otimes \mathcal{B}$ in Figure 2.

**Figure 4.**The matrix on the left is $\left(\mathcal{A}\otimes \mathcal{B}\right)\odot N$, and the one on the right is the share distribution scheme obtained from this operation, as described in Section 7.1.

**Figure 5.**A bipartite graph for the tensor design $\mathcal{B}$ defined in Sect. with 5 players and 5 points. Each edge $({P}_{i},{x}_{j})$ denotes the inclusion of point ${x}_{j}$ in the share of player ${P}_{i}$. The collection of red edges shows one possible maximal matching for the graph.

**Figure 6.**An application of the tensor product of repairable threshold schemes in multi-system IoT, where each system (say, a single hospital) may possess a separate RTS for sharing its own secret key, while multiple systems (say, a chain of hospitals) may share their individual secrets to non-colluding cloud storage providers through a tensor product of the individual schemes.

$\mathcal{A}$ | $\mathcal{B}$ | $\mathit{R}\mathbf{\left(}\mathit{p}\mathbf{\right)}$ | ${\mathit{R}}^{\mathbf{*}}\mathbf{\left(}\mathit{p}\mathbf{\right)}$ |
---|---|---|---|

$(3,2,1)$ | $(3,2,1)$ | ${(1-{q}^{3})}^{4}$ | $>{(1-q)}^{4}+\dots $ |

$(3,2,1)$ | $(7,3,1)$ | ${(1-{q}^{5})}^{6}$ | $>{(1-{q}^{2})}^{6}+\dots $ |

$(7,3,1)$ | $(7,3,1)$ | ${(1-{q}^{8})}^{9}$ | $>{(1-{q}^{4})}^{9}+\dots $ |

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**MDPI and ACS Style**

Roy, B.K.; Roy, A.
IoT-Applicable Generalized Frameproof Combinatorial Designs. *IoT* **2023**, *4*, 466-485.
https://doi.org/10.3390/iot4030020

**AMA Style**

Roy BK, Roy A.
IoT-Applicable Generalized Frameproof Combinatorial Designs. *IoT*. 2023; 4(3):466-485.
https://doi.org/10.3390/iot4030020

**Chicago/Turabian Style**

Roy, Bimal Kumar, and Anandarup Roy.
2023. "IoT-Applicable Generalized Frameproof Combinatorial Designs" *IoT* 4, no. 3: 466-485.
https://doi.org/10.3390/iot4030020