On Secure Capacity of Multiple Unicast Traffic over Separable Networks

TL;DR

This paper determines the maximum secure data transmission rates in separable networks with multiple destinations and an eavesdropper, providing explicit capacity formulas and efficient coding schemes for certain network configurations.

Contribution

It derives the secure capacity for multiple unicast over separable networks with passive eavesdroppers, introducing a polynomial-time code construction matching the capacity bounds.

Findings

Secure capacity formulas for specific network cases.

Existence of polynomial-time secure coding schemes.

Deterministic mapping from two-layer to arbitrary networks.

Abstract

This paper studies the problem of information theoretic secure communication when a source has private messages to transmit to $m$ destinations, in the presence of a passive adversary who eavesdrops an unknown set of $k$ edges. The information theoretic secure capacity is derived over unit-edge capacity separable networks, for the cases when $k=1$ and $m$ is arbitrary, or $m=3$ and $k$ is arbitrary. This is achieved by first showing that there exists a secure polynomial-time code construction that matches an outer bound over two-layer networks, followed by a deterministic mapping between two-layer and arbitrary separable networks.

Equations

$$\displaystyle X^{n}_{e}=\left\{\begin{array}[]{ll}f_{e}\left(W_{{[m]}},\Theta\right)&\mbox{if}\ \mbox{tail}(e)=S,\\ f_{e}\left(\{X^{n}_{\ell}:\ell\in\mathcal{I}(tail(e))\}\right)&\mbox{otherwise},\end{array}\enspace\right.$$

$$M_{\mathcal{A}}$$

$$\displaystyle\left[\begin{array}[]{c}X_{1}\\ \vdots\\ X_{t}\end{array}\right]$$

$$V_i=\left[\begin{array}{c}{V}^T\\ C_i\end{array}\right],$$

$$C_1=\left[\begin{array}{cccccc}0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right],C_3=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right].$$

$$0\le {\sum }_{i\in \mathcal{A}}{R}_i\le \text{dim}\left({\sum }_{i\in \mathcal{A}}{N}_i\right),\forall \mathcal{A}\subseteq [m],$$

$${\sum }_{i\in \mathcal{A}}{R}_i\le M_{\mathcal{A}}-C_{\mathcal{A}},\forall \mathcal{A}\subseteq [m],$$

$${\sum }_{i\in \mathcal{A}}{R}_i\le M_{\mathcal{A}}-k,\forall \mathcal{A}\subseteq [m].$$

$${\sum }_{i\in \mathcal{A}}{R}_i$$

$$\stackrel{(\mathrm{b})}{=}{M}_{{\mathcal{V}}_1\cup {\mathcal{V}}_2\cup \dots \cup {\mathcal{V}}_{C_{\mathcal{A}}}}-k{C}_{\mathcal{A}}\stackrel{(\mathrm{c})}{=}{M}_{\mathcal{A}}-C_{\mathcal{A}},$$

$${\sum }_{i\in \mathcal{A}}{R}_i={\sum }_{i\in {\mathcal{V}}_1}{R}_i+{\sum }_{i\in {\mathcal{V}}_2}{R}_i\le M_{\{2,3,4\}}-2=4,$$

$$M_{\mathcal{A}}-C_{\mathcal{A}}\le \text{dim}\left({\sum }_{i\in \mathcal{A}}{N}_i\right)$$

$$=t-\text{dim}\left({\cap }_{i\in \mathcal{A}}{V}_i\right),$$

$$\forall \mathcal{A}\subseteq [m],\text{dim}\left({\cap }_{i\in \mathcal{A}}{V}_i\right)\le t-M_{\mathcal{A}}+C_{\mathcal{A}}.$$

$${\sum }_{i\in \mathcal{A}}{R}_i\le M_{\mathcal{A}}-k,\forall \mathcal{A}\subseteq [m].$$

$${\sum }_{i\in \mathcal{A}}{R}_i$$

$$\displaystyle\text{dim}\left(\textstyle{\sum}_{i\in\mathcal{A}}N_{i}\right)\!\geq\!\!\!\min_{\small\begin{array}[]{c}\mathcal{P}:\!\bigsqcup\limits_{{\mathcal{Q}}\in\mathcal{P}}\!\!{\mathcal{Q}}\!=\!\mathcal{A}\end{array}}\!\!\left\{\sum_{{\mathcal{Q}}\in\mathcal{P}}M_{\mathcal{Q}}-|\mathcal{P}|k\right\}.$$

$$\text{dim}(N_i+N_j)$$

$$=M_{\{i\}}+M_{\{j\}}-2k-\text{dim}(N_i\cap N_j),$$

$$V_{ij}^{\star }=\left[\begin{array}{c}{V}_i\\ V_j\end{array}\right]\stackrel{\text{eqref}eq:Vi}{=}\left[\begin{array}{c}{V}^T\\ C_i\\ C_j\end{array}\right]=\left[\begin{array}{c}{V}^T\\ C_{ij}\end{array}\right],$$

$$\text{dim}(N_i\cap N_j)$$

$$=\max \{0,M_{\cap \{i,j\}}-k\}=[M_{\cap \{i,j\}}-k{]}^{+},$$

$$\text{dim}(N_1+N_2+N_3)=t-\text{dim}(V_1\cap V_2\cap V_3),$$

$$T$$

$$TM=I,$$

$$\left[\begin{array}{c}{\hat{W}}_1\\ ⋮\\ {\hat{W}}_m\end{array}\right]=T\left[\begin{array}{c}{X}_1\\ ⋮\\ X_t\end{array}\right]\stackrel{\text{eqref}eq:SignRxRel}{=}\left[\begin{array}{cc}TM& TV\end{array}\right]\left[\begin{array}{c}{W}_1\\ ⋮\\ W_m\\ K\end{array}\right]$$

$$=TM\left[\begin{array}{c}{W}_1\\ ⋮\\ W_m\end{array}\right]+TVK=\left[\begin{array}{c}{W}_1\\ ⋮\\ W_m\end{array}\right].$$

$$R_{\pi (i)}=\text{dim}\left(\sum _{j=1}^i{N}_{\pi (j)}\right)-\text{dim}\left(\sum _{j=1}^{i-1}{N}_{\pi (j)}\right),i\in [m],$$

$$R_{\pi (2)}=\text{dim}(N_{\pi (1)}+N_{\pi (2)})-\text{dim}(N_{\pi (1)})$$

$$R_{\pi (3)}=\text{dim}(N_{\pi (1)}+N_{\pi (2)}+N_{\pi (3)})-\text{dim}(N_{\pi (1)}+N_{\pi (2)})$$

$$P_f:=\left\{R\in {\mathbb{R}}^{[m]}:R\ge 0,\sum _{i\in \mathcal{A}}{R}_i\le f(\mathcal{A}),\forall \mathcal{A}\subseteq [m]\right\},$$

$$f(\mathcal{A}):=\text{dim}\left(\sum _{i\in \mathcal{A}}{N}_i\right),\forall \mathcal{A}\subseteq [m]$$

$$f(\mathcal{B})$$

Full text

I Introduction

Today, a large portion of exchanged data over communication networks is inherently sensitive and private (e.g., banking, professional, health). Moreover, given the recent progress in quantum computing, we can no longer exclusively rely on computational security: we need to explore unconditionally (information theoretic) secure schemes. In this paper, we present new results for information theoretic security over networks with multiple unicast sessions.

We assume that a source has $m$ private messages to send to $m$ destinations over a network modeled as a directed graph with unit capacity edges. This communication occurs in the presence of a passive external adversary who has unbounded computational capabilities (e.g., quantum computer), but limited network presence, i.e., she can wiretap (an unknown set of) at most $k$ edges of her choice. We seek to characterize the information theoretic secure capacity for this setup.

Our results apply to the class of separable networks that, broadly speaking, are networks that can be partitioned into a number of edge disjoint subnetworks that satisfy certain properties (see Definition 3 in Section II). We establish a direct mapping between the secure capacity for separable networks, and the secure capacity for two-layer networks constructed as follows. The source is connected to a set of relays via direct edges. These relays are then connected to the $m$ destinations, such that each destination is directly connected to an (arbitrary) subset of the relays. An example of such a two-layer network with $6$ relays and $3$ destinations is shown in Fig. 1.

In [1], we characterized the secure capacity region for separable networks having $m=2$ destinations. We showed that for $m=2$ it is optimal to use different parts of the network to transmit the keys and the encoded messages. However, as we also pointed out in [1], such a scheme is not optimal when $m>2$. We proved this by constructing a joint scheme for two-layer networks that mixes the transmission of keys and encoded messages over the network, and showing that it can achieve higher secure rates than spatially separating the transmissions of the keys and the encoded messages. For instance in Fig. 1, the joint scheme achieves the rate triple $(2,2,1)$, which is not possible otherwise.

In this paper, we proved that we can leverage the polynomial-time joint scheme in [1] for two-layer networks, to prove capacity results for separable networks for the following additional cases: (i) networks where $m=3$ and $k$ is arbitrary; (ii) networks where $k=1$ and $m$ is arbitrary; (iii) networks where $k$ and $m$ are arbitrary, but the network has some special structure in terms of minimum cut. To prove optimality in these new cases, we needed new proof techniques, that include calculating the dimension of the sum of $m=3$ subspaces in a form that matches a modified outer bound. We also prove that the secure capacity region of any separable network can be characterized from the secure capacity region of the corresponding two-layer network, referred to as the child two-layer network. In particular, we provide a deterministic mapping from a secure scheme for the child two-layer network to a secure scheme for the corresponding separable network. We note that for $m=2$ every network is separable [2]; however this is no longer the case for $m\geq 3$ [1].

Related Work. Shannon [3] proved that the one-time pad can provide perfect information theoretic security with pre-shared keys. For degraded point-to-point channels, Wyner [4] showed that information theoretic security can be achieved without pre-shared keys. With feedback, Maurer [5] proved that secure communication is possible, even when the adversary has a channel of better quality than the legitimate receiver. Multicast traffic over networks of unit capacity edges was analyzed by Cai et al. in [6], and followed by several other works, such as [7, 8]. In [6], the information theoretic secure capacity was characterized for networks where a source multicasts the same information to a number of destinations in the presence of a passive external adversary eavesdropping any $k$ edges of her choice. In [9], the authors studied adaptive and active attacks and also considered multiple multicast traffic over a layered network structure, with arbitrary number of layers. However, different to this paper, every node in one layer is connected to every node in the next layer. It therefore follows that, for the case of two layers, our setting encompasses the one in [9].

Paper Organization. In Section II we define two-layer and separable networks, and formulate the problem. In Section III, we review the secure scheme proposed in [1] and in Section IV, we characterize its achieved rate region. In Section IV we also show the mapping between separable and two-layer networks. In Section V and Section VI, we prove that the scheme achieves the secure capacity when $k=1$ and $m=3$, respectively. In Section VI, we also provide sufficient conditions for the scheme to be optimal for arbitrary $k$ and $m$.

II System Model and Problem Formulation

Notation: Calligraphic letters indicate sets; $\emptyset$ is the empty set; $\mathcal{A}_{1}\sqcup\mathcal{A}_{2}$ indicates the disjoint union of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$; $\mathcal{A}_{1}\backslash\mathcal{A}_{2}$ is $\mathcal{A}_{1}\cap\mathcal{A}_{2}^{C}$; $[n]:=\{1,2,\ldots,n\}$; $[x]^{+}:=\max\{0,x\}$ for $x\in\mathbb{R}$.

A two-layer network consists of one source $S$ that wishes to communicate with $m$ destinations, by hopping information through one layer of $t$ relays. As such, a two-layer network is parameterized by: (i) the integer $t$, which denotes the number of relays in the first layer; (ii) the integer $m$, which indicates the number of destinations in the second layer; (iii) $m$ sets $\mathcal{M}_{i},\ i\in[m]$, such that $\mathcal{M}_{i}\subseteq[t]$, where $\mathcal{M}_{i}$ contains the indexes of the relays connected to destination $D_{i}$. An example of a two-layer network is shown in Fig. 1, for which $t=6$, $m=3$, $\mathcal{M}_{1}=\{1,2,4\}$, $\mathcal{M}_{2}=\{3,4,5,6\}$ and $\mathcal{M}_{3}=\{2,3\}$.

We represent a two-layer wireline network with a directed acyclic graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ is the set of edges. The edges represent orthogonal and interference-free communication links, which are discrete noiseless memoryless channels of unit capacity over a common alphabet. If an edge $e\in\mathcal{E}$ connects a node $i$ to a node $j$, we denote, $\text{tail}(e)=i$ and $\text{head}(e)=j$. $\mathcal{I}(v)$ and $\mathcal{O}(v)$ are the set of all incoming and outgoing edges of node $v$, respectively.

Source $S$ has a message $W_{i}$ for destination $D_{i},i\in[m]$. These $m$ messages are assumed to be independent. Thus, the network consists of multiple unicast traffic, where $m$ unicast sessions take place simultaneously and share the network resources. A passive external eavesdropper Eve is also present and can wiretap any $k$ edges of her choice. The symbol transmitted over $n$ channel uses on $e\in\mathcal{E}$ is denoted as $X_{e}^{n}$. In addition, for $\mathcal{E}_{t}\subseteq\mathcal{E}$ we define $X_{\mathcal{E}_{t}}^{n}=\{X_{e}^{n}:e\in\mathcal{E}_{t}\}$. We assume that $S$ has infinite sources of randomness $\Theta$, while the other nodes in the network do not have any randomness.

Over this network, we seek to reliably communicate (with zero error) the message $W_{i},i\in[m]$ to destination $D_{i}$ so that Eve receives no information about the content of the messages. In particular, we are interested in ensuring perfect information theoretic secure communication, and we aim at characterizing the secure capacity region, which is next formally defined.

Definition 1** (Secure Capacity Region).**

A rate $m$-tuple $(R_{1},R_{2},\ldots,R_{m})$ is said to be securely achievable if there exist a block length $n$ with $R_{i}=\frac{1}{n}H(W_{i}),\ \forall i\in[m]$ and encoding functions $f_{e},\forall e\in\mathcal{E}$, over a finite field $\mathbb{F}_{q}$ with

[TABLE]

such that each destination $D_{i}$ can reliably decode the message $W_{i}$ i.e., $H\left(W_{i}|\{X^{n}_{e}:e\in\mathcal{I}(D_{i})\}\right)=0,\ \forall i\in[m]$.

We also require perfect secrecy, i.e., $I\left(W_{{[m]}};X^{n}_{\mathcal{E}_{\mathcal{Z}}}\right)=0,\ \forall\ \mathcal{E}_{\mathcal{Z}}\subseteq\mathcal{E}\ \text{such that}\ |\mathcal{E}_{\mathcal{Z}}|\leq k.$ The secure capacity region is the closure of all such feasible rate $m$-tuples.

In order to prove that our designed scheme meets the perfect secrecy requirement in Definition 1, we will use the “matrix rank” condition on perfect secrecy proved in [10, Lemma 3.1].

We now provide a couple of definitions that will be used in the remaining part of the paper, and we state two remarks that highlight some properties of the networks of interest.

Definition 2** (Min-Cut).**

We denote by $M_{\mathcal{A}}$ the capacity of the min-cut between the source $S$ and the set of destinations $D_{\mathcal{A}}:=\{D_{i},\ i\in\mathcal{A}\}$, and refer to it as the min-cut capacity.

Definition 3** (Separable Graph).**

A graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with a source and $m$ destinations is said to be separable if it can be partitioned into $2^{m}-1$ edge disjoint graphs (graphs with empty edge sets are also allowed). In particular, these graphs are denoted as $\mathcal{G}_{\mathcal{J}}^{\prime}=(\mathcal{V},{\mathcal{E}_{\mathcal{J}}^{\prime}}),{{\mathcal{J}}\subseteq[m],\mathcal{J}\neq{\emptyset}}$ and are such that ${\mathcal{E}_{\mathcal{J}}^{\prime}}\subseteq\mathcal{E}$ and $\mathcal{E}_{\mathcal{J}}^{\prime}\cap\mathcal{E}_{\mathcal{L}}^{\prime}=\emptyset,\ \forall\mathcal{J}\neq\mathcal{L}\subseteq[m]$. Moreover, their min-cut capacities satisfy the following condition

[TABLE]

where, for $\mathcal{G}$, $M_{\mathcal{A}}$ is defined in Definition 2, and the graph $\mathcal{G}_{\mathcal{J}}^{\prime}$ has the following min-cut capacities: (i) $M_{\mathcal{J}}^{\prime}$ from the source $S$ to any non-empty subset of destinations in $\mathcal{J}$, and (ii) zero from the source $S$ to the set of destinations $\{D_{i}:i\in[m]\setminus\mathcal{J}\}$.

Remark 1**.**

For two-layer networks, we have $M_{\mathcal{A}}=\left|\cup_{i\in\mathcal{A}}\mathcal{M}_{i}\right|$. For notational convenience, we let $M_{\cap\{i,j\}}=|\mathcal{M}_{i}\cap\mathcal{M}_{j}|$ and $M_{\cap\{i,\mathcal{A}\}}=|\mathcal{M}_{i}\cap\left(\cup_{j\in\mathcal{A}}\mathcal{M}_{j}\right)|$. Moreover, we also assume that $M_{\{i\}}\geq K,\forall i\in[m]$ (otherwise secure communication is not possible) with $M_{\emptyset}:=K$ for consistency.

Remark 2**.**

The single unicast secure capacity [6] is $R_{i}=M_{\{i\}}-k,\forall i\in[m]$. However, when multiple unicast sessions share common network resources, in general, it is not possible to communicate at rate $R_{i}=M_{\{i\}}-k,\ i\in[m]$ to all destinations simultaneously (for any choice of $k$). This is also highlighted in the outer bound that we derived in [2], which is contained inside the region given by $R_{i}=M_{\{i\}}-k,\forall i\in[m]$.

III Secure Transmissions Scheme

We here review the secure transmission scheme for two-layer networks that we recently proposed in [1]. The source $S$ encodes the message packets with $k$ random packets and transmits these packets on its outgoing edges to the $t$ relays. We can write the received symbols at the $t$ relays as

[TABLE]

where: (i) $W_{i},i\in[m]$ is a column vector of $R_{i}$ message packets for destination $D_{i}$, (ii) $K$ is a column vector which contains the $k$ random packets, (iii) $M$ is a matrix of dimension $t\times\left(\sum_{i=1}^{m}R_{i}\right)$, and (iv) $V$ is a Vandermonde matrix of dimension $t\times k$. The matrix $V$ is chosen to guarantee security as per [10, Lemma 3.1]; hence, no matter which $k$ edges Eve wiretaps, she learns nothing about the messages $W_{[m]}$.

Each relay $i\in[t]$ forwards the received symbol $X_{i}$ in (15) to the destinations it is connected. As such, each destination will observe a subset of symbols from $\{X_{1},X_{2},\ldots,X_{t}\}$. Finally, destination $D_{i},i\in[m]$ selects a decoding vector and performs the inner product with $[X_{1},X_{2},\ldots,X_{t}]$. This decoding vector is chosen such that it has two characteristics: (1) it is in the left null space of $V$, i.e., in the right null space of $V^{T}$; this ensures that each destination is able to cancel out the random packets (encoded with the message packets); (2) it has zeros in the positions corresponding to the relays it is not connected to; this ensures that each destination uses only the symbols that it observes. In other words, all the decoding vectors that $D_{i}$ can choose belong to the null space of the matrix $V_{i}$ defined as

[TABLE]

where $C_{i}$ is a matrix of dimension $\bar{t}\times t$, with $\bar{t}$ being the number of relays to which $D_{i}$ is not connected to. In particular, each row of $C_{i}$ has all zeros except a one in the position corresponding to a relay to which $D_{i}$ is not connected to. For instance, with reference to the network in Fig. 1, we have

[TABLE]

IV Achieved Secure Rate Region

In this section, we first derive the rate region achieved by the secure scheme in Section III, and then present the mapping between separable and two-layer networks. In particular, we have the following lemma, whose proof is in Appendix A.

Lemma 1**.**

The secure rate region achieved by the proposed scheme is given by

[TABLE]

where $N_{i}$ is the right null space of the matrix $V_{i}$ in (16).

IV-A Secure Scheme for any Separable Network

We will here first show that for any separable network, a corresponding two-layer network can be created such that both networks have the same min-cut capacities $M_{\mathcal{A}}$ for all $\mathcal{A}\subseteq[m]$. We will then show that a secure scheme designed for a two-layer network can be converted to a secure scheme on the corresponding separable network.

By Definition 3, a separable network $\mathcal{G}$ with $m$ destinations, can be separated into $2^{m}-1$ networks $\mathcal{G}_{\mathcal{J}}^{\prime},\ \mathcal{J}\subseteq[m],\mathcal{J}\neq\emptyset$ where $\mathcal{G}_{\mathcal{J}}^{\prime}$ has min-cut capacity $M^{\prime}_{\mathcal{J}}$ to every subset of destinations in $\mathcal{J}$. To construct the corresponding two-layer network, we use the following iterative procedure: (1) we place the source node $S$ in layer [math] of our network, and the $m$ destination nodes $D_{i},i\in[m]$, in layer $2$ of our network; (2) for each $\mathcal{J}\subseteq[m]$, we add $M^{\prime}_{\mathcal{J}}$ relays in layer $1$ of our network; (3) for each $\mathcal{J}\subseteq[m]$, we connect: (i) the source in layer [math] with all the added $M^{\prime}_{\mathcal{J}}$ relays, and (ii) all the added $M^{\prime}_{\mathcal{J}}$ relays with the destinations $D_{i},i\in\mathcal{J}$ in layer $2$. By following the above procedure, for each $\mathcal{A}\subseteq[m]$, the min-cut capacity in the constructed two-layer network is $M_{\mathcal{A}}$ as given in (1). As such, the new constructed two-layer network has the same min-cut capacity $M_{\mathcal{A}}$ of the corresponding separable network.

In what follows, we refer to the original separable network as parent separable network, and to the corresponding two-layer network as child two-layer network. We now show that a secure scheme designed for the child two-layer network can be converted to a secure scheme for the corresponding parent separable network. Towards this end, we assume that we have a secure scheme for the child two-layer network as described in (15), and proceed as follows. On every graph $\mathcal{G}_{\mathcal{J}}^{\prime}$ in the parent separable network, we transmit (multicast) the symbols that were transmitted in the child two-layer network from the source $S$ in layer [math] to the set of $M^{\prime}_{\mathcal{J}}$ relays in layer $1$ that were added when constructing the child two-layer network for $\mathcal{G}_{\mathcal{J}}^{\prime}$. Note that this multicast towards all destinations $D_{i},i\in\mathcal{J}$, is possible since $\mathcal{G}_{J}^{\prime}$ has min-cut capacity $M^{\prime}_{\mathcal{J}}$. With such a strategy, at the end of the transmissions every destination in the parent separable graph still receives the same set of packets as it would have received in the child two-layer network. Thus, all the destinations can still decode their respective messages. In Appendix C we also prove that this scheme satisfies the security condition in [10, Lemma 3.1], and hence it is secure. Moreover, since the child two-layer and the parent separable networks have equal min-cut capacities, they have the same outer bound on the secure capacity region [2]. Thus, an optimal scheme on a child two-layer network results in an optimal scheme on the corresponding parent separable network.

V Secure Capacity for $k=1$

In this section, we consider the case when Eve wiretaps any $k=1$ edge of her choice, and characterize the secure capacity region. In particular, we prove the following theorem.

Theorem 2**.**

For the two-layer network when Eve wiretaps any $k=1$ edge of her choice, the secure capacity region is

[TABLE]

with $C_{\mathcal{A}}$ being the number of connected components in an undirected graph where: (i) there are $|\mathcal{A}|$ nodes, i.e., one for each $i\in\mathcal{A}$; (ii) an edge between node $i$ and node $j$, $\{i,j\}\in\mathcal{A}$, $i\neq j$, exists if $\mathcal{M}_{i}\cap\mathcal{M}_{j}\neq\emptyset$.

Outer Bound: The secure capacity region [2] is contained in:

[TABLE]

We now show that the outer bound in (19) can be equivalently written as in (18). Let ${\mathcal{V}_{i},}\ i\in[C_{\mathcal{A}}]$, represent the set of nodes in the $i$-th component of the graph constructed as explained in Theorem 2. Then, clearly $\mathcal{A}=\bigsqcup_{i=1}^{C_{\mathcal{A}}}\mathcal{V}_{i}$ and we can write

[TABLE]

where: (i) the inequality in $\rm{(a)}$ follows by applying (19) for each set ${\mathcal{V}_{i}},i\in[C_{\mathcal{A}}]$, (ii) the equality in $\rm{(b)}$ follows since, by construction, $\mathcal{M}_{i}\cap\mathcal{M}_{j}=\emptyset$ for all $i\in{\mathcal{V}_{x}}$ and $j\in{\mathcal{V}_{y}}$ with $x\neq y$, and (iii) the equality in $\rm{(c)}$ follows since $\mathcal{A}=\bigsqcup_{i=1}^{C_{\mathcal{A}}}\mathcal{V}_{i}$ and $k=1$. Thus, (19) implies (18). Moreover, since $C_{\mathcal{A}}\geq 1$, (18) implies (19). This shows that the rate region in Theorem 2 is an outer bound on the secure capacity region when $k=1$.

We now consider an example of a two-layer network and show how the upper bound derived above applies to it.

Example: Let $\mathcal{A}=\{2,3,4\}$, and assume that $\mathcal{M}_{1}=\{1,2\}$, $\mathcal{M}_{2}=\{3,4\}$, $\mathcal{M}_{3}=\{4,5,6\}$ and $\mathcal{M}_{4}=\{7,8\}$. Then, we construct an undirected graph such that: (i) it has $3$ nodes since $|\mathcal{A}|=3$ and (ii) has an edge between node $2$ and node $3$ since $\mathcal{M}_{2}\cap\mathcal{M}_{3}=\{4\}\neq\emptyset$. It therefore follows that this graph has $C_{\mathcal{A}}=2$ components. In particular, we have

[TABLE]

where ${\mathcal{V}_{1}}=\{2,3\}$ and ${\mathcal{V}_{2}}=\{4\}$.

Achievable Rate Region: We here show that the rate region in Theorem 2 is achieved by the scheme described in Section III. In particular, we show

[TABLE]

where recall that $\text{dim}\left(\sum_{i\in\mathcal{A}}N_{i}\right)$ is the secure rate performance of our proposed scheme in Section III (see Lemma 1). Note that the equality in ${\rm{(a)}}$ follows by using the property of the dual space and the rank nullity theorem, and $V_{i},i\in\mathcal{A}$ is defined in (16). In other words, we next show that

[TABLE]

Towards this end, we would like to count the number of linearly independent vectors $x\in\mathbb{F}_{q}^{t}$ that belong to $\left(\cap_{i\in\mathcal{A}}V_{i}\right)$.

We note that, by our construction: (i) $V^{T}$ consists of one row of $t$ ones, and (ii) $C_{i}$ has zeros in the positions indexed by $\mathcal{M}_{i}$. Hence, if a vector belongs to $V_{i}$, then all its components indexed by $\mathcal{M}_{i}$ have to be the same, i.e., either they are all zeros, or they are all equal to a multiple of one. Thus, we have $q$ choices to fill these positions indexed by $\mathcal{M}_{i}$.

Now, consider $V_{j}$ with $j\in\mathcal{A}$ and $j\neq i$. By using the same logic as above, if a vector belongs to $V_{j}$, then all its components indexed by $\mathcal{M}_{j}$ have to be the same and we have $q$ choices to fill these. We now need to count the number of such choices that are consistent with the choices made to fill the positions indexed by $\mathcal{M}_{i}$.

Towards this end, we consider two cases:

$\bullet$ Case 1: $\mathcal{M}_{i}\cap\mathcal{M}_{j}=\emptyset$. In this case, there is no overlap in the elements indexed by $\mathcal{M}_{i}$ and $\mathcal{M}_{j}$ and hence we can select all the available $q$ choices for the positions indexed by $\mathcal{M}_{j}$;

$\bullet$ Case 2: $\mathcal{M}_{i}\ \cap\mathcal{M}_{j}\neq\emptyset$. There is an overlap in the elements indexed by $\mathcal{M}_{i}$ and $\mathcal{M}_{j}$. Since we have already fixed the elements indexed by $\mathcal{M}_{i}$, there is no choice for the elements indexed by $\mathcal{M}_{j}$ (as all the elements have to be the same).

By iterating the same reasoning as above for all $i\in\mathcal{A}$, we conclude that we can fill all the positions indexed by $\cup_{i\in\mathcal{A}}\mathcal{M}_{i}$ of a vector $x\in\mathbb{F}_{q}^{t}$ and make sure that $x\in\left(\cap_{i\in\mathcal{A}}V_{i}\right)$ in $q^{C_{\mathcal{A}}}$ ways. This is because, there are $C_{\mathcal{A}}$ connected components, and for each of these components we have only $q$ choices to fill the corresponding positions in the vector $x$ (i.e., the positions that correspond to the relays to which at least one of the destinations inside that component is connected). Once we fix any position inside a component, in fact all the other positions inside that component have to be the same, and thus we have no more freedom in choosing the other positions. Moreover, the remaining $t-M_{\mathcal{A}}$ positions of $x$ can be filled with any value in $\mathbb{F}_{q}$ and for this we have $q^{t-M_{\mathcal{A}}}$ possible choices. Therefore, the number of vectors $x\in\mathbb{F}_{q}^{t}$ that belong to $\left(\cap_{i\in\mathcal{A}}V_{i}\right)$ is at most $q^{C_{\mathcal{A}}+t-M_{\mathcal{A}}}$, which implies $\forall\mathcal{A}\subseteq[m],\ \text{dim}\left(\cap_{i\in\mathcal{A}}V_{i}\right)\leq t-M_{\mathcal{A}}+C_{\mathcal{A}}$. This proves that the secure scheme in Section III achieves the rate region in Theorem 2. We now illustrate our method of identifying vectors that belong to $\cap_{i\in\mathcal{A}}V_{i}$ through an example.

Example: Let $t=8$, $m=4$, $\mathcal{M}_{1}=\{1,2\}$, $\mathcal{M}_{2}=\{3,4\}$, $\mathcal{M}_{3}=\{4,5,6\}$ and $\mathcal{M}_{4}=\{7,8\}$. Let $\mathcal{A}=\{2,3,4\}$.

We want to count the number of vectors $x\in\mathbb{F}_{q}^{8}$ such that $x\in V_{2}\cap V_{3}\cap V_{4}$. We use the following iterative procedure:

$\bullet$ For $x$ to belong to $V_{2}$ its elements in the $3$rd and $4$th positions have to be the same since $\mathcal{M}_{2}=\{3,4\}$. Thus, we have $q$ choices to fill the $3$rd and $4$th positions.

$\bullet$ For $x$ to belong to $V_{3}$, its elements in the $4$th, $5$th and $6$th positions have to be equal since $\mathcal{M}_{3}=\{4,5,6\}$. However, the element in the $4$th position has already been fixed in selecting vectors that belong to $V_{2}$. Thus, there is no further choice in filling the $5$th and $6$th position.

$\bullet$ For $x$ to belong to $V_{4}$, its elements in the $7$th and $8$th positions have to be the same since $\mathcal{M}_{4}=\{7,8\}$. Since in the previous two steps, we have not filled yet the elements in these positions, then we have $q$ possible ways to fill the elements in the $7$th and $8$th positions.

$\bullet$ Moreover, we can fill the elements in the $1$st and $2$nd positions of $x$ in $q^{2}$ possible ways.

With the above procedure we get that $\text{dim}\left(\cap_{i\in\{2,3,4\}}V_{i}\right)=4$, which is equal to the upper bound that we computed in (20) for the same example.

VI Secure Capacity For $m=3$

In this section, we consider the case $m=3$, and we characterize the secure capacity region through the theorem below.

Theorem 3**.**

For a two-layer network with $m=3$ destinations, the secure capacity region is given by

[TABLE]

Clearly the rate region in (22) is an outer bound on the secure capacity region [2] and can be equivalently written as

[TABLE]

where $\mathcal{P}$ is a partition of $\mathcal{A}$. We will show that $\forall\mathcal{A}\subseteq[m]$,

[TABLE]

We prove (24) by considering three different cases.

Case 1: $|\mathcal{A}|=1$, i.e., $\mathcal{A}=\{i\}$. For this case, $V_{i}$ in (16) has $k+t-M_{\{i\}}$ rows. All these rows are linearly independent since: (i) the rows of $V^{T}$ are linearly independent as $V$ is a Vandermonde matrix, (ii) $C_{i}$ is full row rank by construction, and (iii) any linear combination of the rows of $V^{T}$ will have a weight of at least $t-k+1$ (from the Vandermonde property), whereas any linear combination of the rows of $C_{i}$ will have a weight of at most $t-M_{\{i\}}\leq t-k$. It therefore follows that, $\forall i\in[3]$, we have that $\text{dim}(N_{i})=t-{\text{dim}(V_{i})}=t-(k+t-M_{\{i\}})=M_{\{i\}}-k$, where the first equality follows by using the rank-nullity theorem. Thus, (24) is satisfied.

Case 2: $|\mathcal{A}|=2$, i.e., $\mathcal{A}=\{i,j\}$. $\forall(i,j)\in[3]^{2},i\neq j$,

[TABLE]

where the second equality follows by using $\text{dim}(N_{i})$ derived in Case 1. Thus, we need to compute $\text{dim}(N_{i}\cap N_{j})$. Note that, by definition, $N_{i}\cap N_{j}$ is the right null space of

[TABLE]

where in the last equality, $C_{ij}$ is a matrix of dimension $(t-M_{\cap\{i,j\}})\times t$, with all unique rows. Using a similar argument as in Case 1 the number of linearly independent rows of $V^{\star}_{ij}$ is $\min\{t,t-M_{\cap\{i,j\}}+k\}$. Thus,

[TABLE]

where the first equality follows from the rank-nullity theorem. We can now write $\text{dim}(N_{i}+N_{j})$ from (25) as ${\text{dim}(N_{i}+N_{j})=\min\left\{M_{\{i\}}+M_{\{i\}}-2k,M_{\{i,j\}}-k\right\}}$, and the condition in (24) is satisfied.

Case 3: $\mathcal{A}=\{1,2,3\}$. We will compute

[TABLE]

that is, the number of linearly independent vectors $x\in\mathbb{F}_{q}^{t}$ that belong to $V_{1}\cap V_{2}\cap V_{3}$. Similar to the case $k=1$, we have $t-M_{\{1,2,3\}}$ degrees of freedom to fill the positions of $x$ corresponding to $[t]\setminus\cup_{i\in[3]}\mathcal{M}_{i}$. We now select a permutation $(i,j,\ell)$ of $(1,2,3)$. In order for $x$ to belong to $V_{i}$, the positions of $x$ corresponding to $\mathcal{M}_{i}$ can be filled with $k$ degrees of freedom. This is because: (i) $C_{i}$ in (16) has zeros in the positions specified by $\mathcal{M}_{i}$, and (ii) $V^{T}$ has $k$ rows. Then, to fill the positions of $x$ specified by $\mathcal{M}_{j}$ so that $x\in V_{j}$, we have at most $[k-M_{\cap\{i,j\}}]^{+}$ degrees of freedom. This is because the positions of $x$ corresponding to $\mathcal{M}_{i}\cap\mathcal{M}_{j}$ are already fixed. Finally, to fill the positions of $x$ corresponding to $\mathcal{M}_{\ell}$ so that $x\in V_{\ell}$, we have at most $[k-M_{\cap\{\ell,\{i,j\}\}}]^{+}$ degrees of freedom. This is because the positions of $x$ corresponding to $\mathcal{M}_{\ell}\cap(\mathcal{M}_{i}\cup\mathcal{M}_{j})$ are already fixed. Thus, we obtain $\text{dim}(V_{1}\cap V_{2}\cap V_{3})\leq k+{[k-M_{\cap\{i,j\}}]^{+}}+{[k-M_{\cap\{\ell,\{i,j\}\}}]^{+}}+t-{M_{\{1,2,3\}}}$, which when substituted in (26), satisfies (24) (see Appendix B). This proves Theorem 3.

We now conclude this section with the following lemma.

Lemma 4**.**

The scheme in Section III achieves the secure capacity region of a two-layer network with arbitrary values of $k$ and $m$ whenever $\mathcal{M}_{\cap\{i,j\}}\geq k$ for all $(i,j)\in[m]^{2},i\neq j$.

Proof.

We can compute $\text{dim}(\cap_{i=1}^{m}V_{i})$ as follows:

$\begin{aligned} \text{dim}(\cap_{i\in\mathcal{A}}V_{i})&\stackrel{{\scriptstyle{\rm{(a)}}}}{{\leq}}t-M_{\mathcal{A}}+k+{[k-\mathcal{M}_{\cap\{i_{1},i_{2}\}}]^{+}}\\ &\quad+\textstyle\sum_{j=3}^{m}{[k-\mathcal{M}_{\cap\{i_{j},\{i_{1},i_{2},\ldots,i_{j-1}\}\}}]^{+}}\\ &{\stackrel{{\scriptstyle{\rm{(b)}}}}{{\leq}}}k+t-M_{\mathcal{A}},\end{aligned}$ where: $\rm{(a)}$ follows by extending to arbitrary $m$ the iterative algorithm for Case 3 above to select $x\in\cap_{i=1}^{m}V_{m}$, and $\rm{(b)}$ follows since $\mathcal{M}_{\cap\{i_{j},\{i_{1},i_{2},\ldots,i_{j-1}\}\}}\geq\mathcal{M}_{\cap\{i_{j},i_{j-1}\}}\geq k$. By using the property of the dual space and the rank-nullity theorem, we obtain $\text{dim}(\sum_{i\in\mathcal{A}}N_{i})\geq M_{\mathcal{A}}-k$, which satisfies (24) $\forall\mathcal{A}\subseteq[m]$. This proves Lemma 4. ∎

Appendix A Proof of Lemma 1

We let $T$ be a matrix of dimension $\left(\sum_{i=1}^{m}R_{i}\right)\times t$ that, for each destination $D_{i},i\in[m],$ contains $R_{i}$ decoding vectors that belong to $N_{i}$. Mathematically, we have

[TABLE]

where $d^{(i)}_{j}$ denotes the $j$-th decoding vector (of length $t$) selected from the null space $N_{i}$, with $i\in[m],j\in[R_{i}]$. Note that, if for all $i\in[m]$, we can select $R_{i}$ decoding vectors from $N_{i}$ such that all the $d_{j}^{(i)}$ in (34) are linearly independent (i.e., such that $T$ has a full row rank), then it is possible to construct the matrix $M$ in (15) such that

[TABLE]

which ensures that all the destinations are able to correctly decode their intended message as

[TABLE]

We propose an iterative algorithm to select $R_{i},i\in[m]$ decoding vectors from $N_{i}$ such that $T$ in (34) has indeed a full row rank. The performance of the proposed algorithm is provided in the following lemma.

Lemma 5**.**

For any given permutation $\pi=\left\{\pi(1),\ldots,\pi(m)\right\}$ of $[m]$, it is possible to select

[TABLE]

vectors from $N_{\pi(i)}$ so that all the $\sum_{i=1}^{m}R_{i}$ selected vectors are linearly independent.

Proof.

We use an iterative algorithm that, for any permutation $\pi=\{\pi(1),\ldots,\pi(m)\}$ of $[m]$, allows to select $R_{\pi(i)}$ vectors from $N_{\pi(i)}$ (with $R_{\pi(i)}$ being defined in (35)) so that all the selected $\sum_{i=1}^{m}R_{i}$ vectors are linearly independent. We next illustrate the main steps of the proposed algorithm.

We select $R_{\pi(1)}=\text{dim}(N_{\pi(1)})$ independent vectors from $N_{\pi(1)}$. Note that one possible choice for this consists of selecting the basis of the subspace $N_{\pi(1)}$. 2. 2.

Next we would like to select independent vectors from $N_{\pi(2)}$ that are also independent of the $R_{\pi(1)}$ vectors that we selected in the previous step. Towards this end, we note that a basis of the subspace $N_{\pi(1)}+N_{\pi(2)}$ is a subset of the union between a basis of $N_{\pi(1)}$ and a basis of $N_{\pi(2)}$. Therefore, we can keep selecting vectors from a basis of $N_{\pi(2)}$ as long as we select an independent vector. Since there are $\text{dim}(N_{\pi(1)}+N_{\pi(2)})$ independent vectors in a basis of $N_{\pi(1)}+N_{\pi(2)}$, then we can select

[TABLE]

independent vectors from $N_{\pi(2)}$ that are also independent of the $R_{\pi(1)}$ vectors that we selected in the previous step. 3. 3.

Similar to the above step, we now would like to select independent vectors from $N_{\pi(3)}$ that are also independent of the $R_{\pi(1)}+R_{\pi(2)}$ vectors that we selected in the previous two steps. Towards this end, we note that a basis of the subspace $N_{\pi(1)}+N_{\pi(2)}+N_{\pi(3)}$ is a subset of the union between a basis of $N_{\pi(1)}+N_{\pi(2)}$ and a basis of $N_{\pi(3)}$. Therefore, we can keep selecting vectors from a basis of $N_{\pi(3)}$ as long as we select an independent vector. Since there are $\text{dim}(N_{\pi(1)}+N_{\pi(2)}+N_{\pi(3)})$ independent vectors in a basis of $N_{\pi(1)}+N_{\pi(2)}+N_{\pi(3)}$, then we can select

[TABLE]

independent vectors from $N_{\pi(3)}$ that are also independent of the $R_{\pi(1)}+R_{\pi(2)}$ vectors that we selected in the previous two steps. 4. 4.

We keep using the iterative procedure above for all the elements in $\pi$, and we end up with $\sum_{i=1}^{m}R_{i}$ vectors that are linearly independent.

This concludes the proof of Lemma 5. ∎

Remark 3**.**

Note that, since there are $m!$ possible permutations of $[m]$, then Lemma 5 offers $m!$ possible choices for selecting $R_{i},i\in[m]$ vectors from $N_{i}$ so that all the $\sum_{i=1}^{m}R_{i}$ selected vectors are linearly independent.

Remark 4**.**

The result in Lemma 5 implies that rate $m$-tuple $(R_{1},R_{2},\ldots,R_{m})$, with $R_{i},i\in[m]$ being defined in (35), can be securely achieved by our proposed scheme.

We now leverage the result in Lemma 5 to prove Lemma 1. We start by noting that the rate region in (17) can be expressed as the following polyhedron:

[TABLE]

where $f(\mathcal{A}):=\text{dim}\left(\sum_{i\in\mathcal{A}}N_{i}\right)$. We now prove the following lemma, which states that this function $f(\cdot)$ is a non-decreasing and submodular function over subsets of $[m]$.

Lemma 6**.**

The set function

[TABLE]

is a non-decreasing and submodular function.

Proof.

Let $\mathcal{A}\subset\mathcal{B}\subseteq[m]$, then

[TABLE]

which proves that the function $f(\cdot)$ is non-decreasing. For proving submodularity, consider two subsets $\mathcal{C},\mathcal{D}\subseteq[m]$. Then, we have

[TABLE]

which proves that the function $f(\cdot)$ is submodular. ∎

Since $f(\cdot)$ is a submodular function, then the polyhedron defined in (36) is the polymatroid associated with $f(\cdot)$. Moreover, since $f(\cdot)$ is also non-decreasing, then the corner points of the polymatroid in (36) can be found as follows [11, Corollary 44.3a]. Consider a permutation $\pi=\{\pi(1),\ldots,\pi(m)\}$ of $[m]$. Then, by letting ${\mathcal{S}}_{\ell}=\{\pi(1),\ldots,\pi(\ell)\}$ for $1\leq\ell\leq m$, we get that the corner points of the polymatroid in (36) can be written as

[TABLE]

Note that by using $f(\mathcal{A})=\text{dim}\left(\sum_{i\in\mathcal{A}}N_{i}\right)$, the above corner points are precisely those in (35) in Lemma 5. Since each rate $m$-tuple $(R_{1},R_{2},\ldots,R_{m})$, with $R_{i},i\in[m]$ being defined in (35), can be securely achieved by our proposed scheme, it follows that the secure rate region in (17) can also be achieved by our scheme. This concludes the proof of Lemma 1.

Appendix B Analysis of the dimension of $(V_{1}\cap V_{2}\cap V_{3})$

From our analysis, we have obtained

[TABLE]

We now further consider two cases.

Case 3A: There exist $(i,j)\in[3]^{2},i\neq j,$ such that $M_{\cap\{i,j\}}\geq k$. In this case, with the permutation $(i,j,\ell)$, the expression in (37) becomes

[TABLE]

From (26), this implies that

[TABLE]

where the last equality follows since $M_{\{1,2,3\}}=M_{\{i,j\}}+{M_{\{\ell\}}}-M_{\cap\{\ell,\{i,j\}\}}$. With this, the condition in (24) is satisfied.

Case 3B: We have $M_{\cap\{i,j\}}<K,\forall{(i,j)\in[3]^{2},i\neq j}$. In this case, we compute $\text{dim}(V_{1}\cap V_{2}\cap V_{3})$ as follows: we first fill the positions of $x$ indexed by $\mathcal{M}_{1}$ with $k$ degrees of freedom, and then fill the positions of $x$ indexed by $\mathcal{M}_{2}$ with $(k-M_{\cap\{1,2\}})$ degrees of freedom as before. Now, we may have fixed more than $k$ positions of $x$ corresponding to indexes in $\mathcal{M}_{3}$, which is not feasible. If that is the case, we backtrack (i.e., remove excess degrees of freedom) that we have used for filling positions of $x$ indexed by $\mathcal{M}_{2}$. Thus,

If $M_{\cap\{3,\{1,2\}\}}\leq k$, then

[TABLE]

This, from (26), implies

[TABLE]

which satisfies the condition in (24). 2. 2.

If $M_{\cap\{3,\{1,2\}\}}>k$, then

[TABLE]

This, from (26), implies

[TABLE]

which satisfies the condition in (24).

Appendix C Proof of Security: Separable Networks

In this section we will show that the scheme described for any separable network is also secure. Towards this end, we start by noting that the only property of $V$ that we have used in our scheme for two-layer networks is the Maximum Distance Separable (MDS) property (i.e., any $k$ rows of $V$ are linearly independent). This implies that, even if in (15) we select a random matrix $\tilde{V}$ instead of $V$, with high probability (close to $1$ for large field size) we will have a secure scheme for the child two-layer network.

Let $Y$ be the collection of the symbols transmitted (multicast) on the parent separable network, as described above. Since multicasting involves network coding, we have

[TABLE]

where $G$ is an encoding matrix of dimension $|\mathcal{E}|\times M_{[m]}$ Thus,

[TABLE]

From the security condition in [10, Lemma 3.1], it follows that the scheme above is secure if we can show that for any choice of $G$, there exists a $\tilde{V}$ such that $\tilde{V}$ is an MDS matrix (i.e., any $k$ rows of $\tilde{V}$ are linearly independent) and

[TABLE]

We will show that for any choice of $G$ of size $|\mathcal{E}|\times M_{[m]}$ with $M_{[m]}\geq k$, there exists a $\tilde{V}$ such that $\tilde{V}$ is an MDS matrix (i.e., any $k$ rows of $\tilde{V}$ are linearly independent) and

[TABLE]

We start by noting that

[TABLE]

Thus, if we prove that, for all $|\mathcal{Z}|\leq k$,

[TABLE]

then we also show that (42) holds. In what follows, we formally prove that a $\tilde{V}$ such that $\tilde{V}$ is an MDS matrix that satisfies the condition in (43) for all $|\mathcal{Z}|\leq k$ can be constructed with a non-zero probability. Towards this end, we let $\hat{k}=rk\left(\left.G\right|_{\mathcal{Z}}\right)$, where $\hat{k}\leq k$ since $|\mathcal{Z}|\leq k$. We define the event

[TABLE]

We have

[TABLE]

where: (i) the equality in $\rm{(a)}$ follows by using the De Morgan’s laws, and (ii) the inequality in $\rm{(b)}$ follows since for two events $A$ and $B$, we have $\Pr(A\cup B)\leq\Pr(A)+\Pr(B)$. We now further upper bound the two probability terms $P_{1}$ and $P_{2}$. For $P_{1}$, we obtain

[TABLE]

where: (i) the equality in $\rm{(c)}$ follows by defining, for a given $\mathcal{Z}$ such that $|\mathcal{Z}|\leq k$, the event

[TABLE]

(ii) the equality in $\rm{(d)}$ follows by using the De Morgan’s laws; (iii) the inequality in $\rm{(e)}$ follows by using the union bound; (iv) the inequality in $\rm{(f)}$ follows since ${{n}\choose{t}}\leq\left(\frac{{\rm{e}}n}{t}\right)^{t}$; (v) the inequality in $\rm{(g)}$ follows by defining the event $\hat{A}_{\mathcal{Z}}$ as

[TABLE]

where $\hat{G}$ is the matrix formed by the \hat{k}=rk\left(\left.\left[\begin{array}[]{c}G\end{array}\right]\right|_{\mathcal{Z}}\right) independent rows of \left.\left[\begin{array}[]{c}G\end{array}\right]\right|_{\mathcal{Z}}, and $\hat{V}$ is formed by the first $\hat{k}$ columns of $\tilde{V}$. Thus, the inequality in $\rm{(g)}$ then follows since $\hat{A}_{\mathcal{Z}}\subseteq A_{\mathcal{Z}}$; (vi) the equality in $\rm{(h)}$ follows due to the following computation. We write

[TABLE]

Note that the matrix $\hat{G}\hat{V}$ is of full rank (equal to $\hat{k}$) if the only solution to $\sum_{i=1}^{\hat{k}}c_{i}\hat{G}{v}_{i}=0$ is $c_{i}=0,\forall i\in[\hat{k}]$. Let $\hat{N}$ be the null space of $\hat{G}$, and $\hat{N}^{\perp}$ be the space such that $\hat{N}^{\perp}\cap\hat{N}={\emptyset}$ and $\hat{N}^{\perp}\cup\hat{N}=\mathbb{F}_{q}^{M_{[m]}}$. Then, we can write each $v_{i},i\in[\hat{k}],$ as the sum of its projection on $\hat{N}$ (say $v_{i}^{(a)}$) and the residual in $\hat{N}^{\perp}$ (say $v_{i}^{(b)}$). This implies that $\hat{G}\hat{V}$ is of full rank if the only solution to $\sum_{i=1}^{\hat{k}}c_{i}\hat{G}{v}_{i}^{(b)}=0$ is $c_{i}=0,\forall i\in[\hat{k}]$ (because $\hat{G}v_{i}^{(a)}=0$). Since a random choice of $v_{i}$ results in a random choice on $v_{i}^{(b)}$, then the probability of $\hat{G}\hat{V}$ being of full rank is equal to the probability that all the vectors $v_{i}^{(b)},\ i\in[\hat{k}]$ are mutually independent in $\hat{N}^{\perp}$. This probability, since $\text{dim}(\hat{N}^{\perp})=\hat{k}$, is equal to $\prod_{i=0}^{\hat{k}-1}\left(1-\frac{q^{i}}{q^{\hat{k}}}\right)$; finally, (vii) the inequality in $\rm{(i)}$ follows since $i-\hat{k}\leq-1$ for all $i\in[0:\hat{k}-1]$ and $\hat{k}\leq k$.

For $P_{2}$, we obtain

[TABLE]

where: (i) the equality in $\rm{(j)}$ follows by defining, for a given $\mathcal{S}$ such that $|\mathcal{S}|=k$, the event

[TABLE]

(ii) the equality in $\rm{(k)}$ follows by using the De Morgan’s laws; (iii) the equality in $\rm{(\ell)}$ follows by selecting uniformly at random all the subsets of $k$ rows out of the $M_{[m]}$ rows, (iv) the equality in $\rm{(m)}$ follows by counting arguments to ensure that the $k$ selected rows are all independent, and (v) the inequality in $\rm{(n)}$ follows since $i-k\leq-1$ for all $i\in[0:k-1]$.

Thus, we obtain

[TABLE]

where the last inequality holds for sufficiently large $\mathbb{F}_{q}$.