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@ -55,7 +55,6 @@
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}
% TODO: Fix the remaining descriptions, such that they all use the environment!
\date{\today}
\begin{document}
@ -63,16 +62,11 @@
\tableofcontents
\newpage
\section{Introduction}
By definition, \emph{Encryption} is the process of converting information into a \emph{code} with the purpose of preventing unauthorized access \cite{oxford}. Traditionally, the way this was accomplished was via some a priori established secret key $k$, which could then be used both for \emph{encryption}, but also for \emph{decryption}. This concept was then challenged by the concept of \emph{Public Key Cryptography}, which allows two parties to communicate with each other in a secure and private fashion, without having already shared the aforementioned secret key. This allowed each party to have a \emph{Public Key} and a \emph{Secret Key}, which could then be used to encrypt and decrypt, respectively. This works well and is used in many applications, such as \emph{SSH} and \emph{SSL}. It does however have one caveat. \emph{Public Key Encryption} is notoriusly slow, compared to the \emph{Symmetrical}-scheme with only a single key, but if you wish to send to several people you will need to encrypt whatever you wish to send several times, once for each party and furthermore, public key encryption is very \emph{all-or-nothing}. Either a single party can decrypt what you send and see everything, or she will not be able to decrypt and thus see nothing.
By definition, \emph{Encryption} is the process of converting information into a \emph{code} with the purpose of preventing unauthorized access \cite{oxford}. Traditionally, the way this was accomplished was via some a priori established secret key $k$, which could then be used both for \emph{encryption}, but also for \emph{decryption}. This concept was then challenged by the concept of \emph{Public Key Cryptography}, \texttt{PKE}, which allows two parties to communicate with each other in a secure and private fashion, without having already shared the aforementioned secret key. This allowed each party to have a \emph{Public Key} and a \emph{Secret Key}, which could then be used to encrypt and decrypt, respectively. This works well and is used in many applications, such as \emph{SSH} and \emph{SSL}. It does however have one caveat. \emph{Public Key Encryption} is notoriusly slow, compared to the \emph{Symmetrical}-scheme with only a single key. Thus, we introduce the concept of \emph{Key Encapsulation Mechanisms}, or \texttt{KEM}, in which a decryption key is now what is sent, rather than a message. This naturally solves one of the underlying issues of \texttt{PKE}, as we can now encrypt a symmetrical key and change our way of communication after the first message.
% TODO: Change slightly
However, there are cases where public-key encryption is insufficient. There is often a need to specify a decryption policy in the ciphertext and only individuals who satisfy the policy can decrypt. More generally, we may want to only give access to a function of the plaintext, depending on the decryptor’s authorization. Consider a cloud service storing encrypted images. Law enforcement may require the cloud to search for images containing a particular face. Thus, the cloud needs a restricted secret key that decrypts images that contain the target face, but reveals nothing about other images. More generally, the secret key may only reveal a function of the plaintext image, for example an image that is blurred everywhere except for the target face. Traditional public-key cryptography cannot help with such tasks.
Now, imagine a scenario where a user has an entire network of people in which he only wishes to send to a subset of these at a time. In the world of \texttt{PKE}, this user will have to fetch each users public key and encrypt the same symmetrical key for each user, resulting in a ciphertext for the same key for each user. This is highly inefficient, and becomes even worse if the \emph{authorised set} of receivers changes, at which point a new symmetrical key must be created and sent to each user. To this end, \emph{Broadcast Encryption}, \texttt{BE}, can be adopted as a solution. In such a scheme, a user will pick his set of recievers, $S$, fetch their public keys, but only encrypt the key once and broadcast this. Hence, there is only a single message and this can be broadcasted to all users of the system, but only those users within the set $S$ can decrypt it.
To this end, \emph{Functional Encryption} can be adopted. Essentially, in functional encryption systems, the decryption key allows the user to learn a specific, pre-defined \emph{function} of the encrypted data, rather than necessarily the actual encrypted data, note that if this function is the \emph{Identity function}, then the system will be equivalent to regular public key encryption. In a functional encryption system for some functionality $F(\cdot, \cdot)$, an authority holding a master secret key $MSK$ can generate a secret key $sk_k$ which enables the computation of the function $F(k,\cdot)$. So given some plaintext $x$, one can compute $F(k,x)$, given only the ciphertext $c$ of $x$.
Full blown functional encryption is however quite a mouthful to implement in an efficient manner and as such there are some subclasses of it. This paper will focus on \emph{Identity Based Encryption (IBE)} and \emph{Broadcast Encryption (BE)}.
% TODO: Explain IBE intuitively
In this paper we first cover the concept of \emph{Identity Based Encryption}, \texttt{IBE}, as an introduction to some of the mathmatical concepts used within the paper, such as that of \emph{bilinear maps} and the mathmatical assumption that is the \emph{Bilinear Diffie Hellman}-problem. These two concepts can be extended and applied to many branches of encryption, such as \texttt{BE} or \emph{Threshold Public Key Encryption}, \texttt{TPKE}, both of which will also be introduced in this paper. Thus, we wish to introduce three different types of encryption schemes, \texttt{IBE, BE} and \texttt{TE}, give a notion of what security is defined to be for these as well as an actual construction.
\section{Syntax and Preliminaries}
@ -86,6 +80,8 @@ Let $p$ be a large prime number. Let $\mathbb{G}_1, \mathbb{G}_2$ be two groups
\end{itemize}
A bilinear map satisfying all the above three properties is said to be \emph{admissible}.
Intuitively, a bilinear map is simply a linear map, or function, taking two arguments, such that if either is lifted to an exponent within the transformation, the transformation can be applied first and the result lifted to the same exponent instead.
\subsection{Mathmatical Assumptions}
All of the following mathmatical assumptions will be derived from the \emph{Diffie-Hellman} assumption. The reader is assumed to be familiar with this. We note that the \emph{Decisional Diffie-Hellman} problem is easy within the setting of bilinear maps, as, given some generator $g$ and $g^a$, $g^b$ and $g^c$ where the question is if $c = ab$, it is straightforward to check if $e(g,g^c) = e(g^a,g^b)$, which holds for the case where $c = ab$. As such, new assumptions which are difficult within this setting, are required.
@ -119,28 +115,30 @@ and finally a $Z \in \Gm_T$. The deciding part is then to decide whether $Z$ is
We want to note that the paper does not define either $\alpha$ or $\gamma$, which essentially means this problem is not well defined. We assume however, that both $\alpha$ and $\gamma$ have to come from $\mathbb{Z}^*_p$, whenever this problem is referenced.
\subsubsection{Notation}
We would like to note that whenever something is prefixed with a capital B, it is in the context of \emph{Broadcast Encryption}. Furthermore we would like to note that for all setup algorithms mentioned, the security parameter will be implicit, but naturally all schemes relies on one in some shape or form, to either denote a bit length, graph size of whatever makes sense within the context.
% TODO: Write up all of the mathematical assumptions
\section{Identity Based Encryption}
We will cover a basic identity based encryption scheme which illustrates a basic usage of bilinear maps as well as one way to extend the \emph{Diffie-Hellman Assumption} known from Public Key Encryption. This scheme is not secure against an adaptive chosen ciphertext attack (\texttt{IND-ID-CCA}). Note that it can be extended to cover this, but this is out of the scope of this paper. We note that although this scheme is not awefully relevant for the rest of this paper, it still is something we covered throughout the semester and it offers a delicate and simple introduction to some of the mathematical concepts and encryption schemes which will be used throughout this paper, specifically that of bilinear maps and public key cryptography.
We will cover an identity based encryption scheme which illustrates a basic usage of bilinear maps as well as one way to extend the \emph{Diffie-Hellman Assumption} known from Public Key Encryption. This scheme is not secure against an adaptive chosen ciphertext attack (\texttt{IND-ID-CCA}). The scheme can however be extended to cover this, but this is out of the scope of this paper. We note that although this scheme is not awfully relevant for the rest of the paper, it still is something we covered throughout the semester and it offers a delicate and simple introduction to some of the mathematical concepts and encryption schemes which will be used throughout this paper, specifically that of bilinear maps and public key cryptography.
\subsection{The structure}
\subsection{Modelling Identity Based Encryption}
\label{sec:IBEStruct}
\textbf{Identity-Based Encryption.} \quad An Identity-Based encryption scheme is specified by four different algorithms, all containing some sort of randomness: \texttt{Setup, Extract, Encrypt, Decrypt}:
\begin{description}
\item[Setup] Takes some security parameter $k$ and returns the system parameters and a master-key. These system parameters include a description of a finite message space $\mathcal{M}$ as well as a description of a finite ciphertext space $\mathcal{C}$. These parameters are known publicly, wherre the master-key is known only to the trusted authority, the so called Private Key Generator (\texttt{PKF}).
\item[Extract] Takes the system parameters, the master-key and an arbitrary \texttt{ID} $\in \{0,1\}^*$ and returns a private key $d$. \texttt{ID} is essentially any arbitrary string which will be used a public key and $d$ is the corresponding decryption key, which can be used by the owner of the \texttt{ID}. Thus, the \texttt{extract} algorithm extracts a private key from the given public key.
\item[Encrypt] Takes the system parameters, \texttt{ID}, and $M \in \mathcal{M}$. Returns some ciphertext $C \in \mathcal{C}$.
\item[Decrypt] Takes the system parameters, some private key $d$ and $C \in \mathcal{C}$. Returns the plaintext $M \in \mathcal{M}$.
\end{description}
Naturally, these algorithms must satisfy that:
\item[Setup$()$] Uses some security parameter $k$ and returns the system parameters, $\text{params}$, and a master-key, $MK$. These system parameters include a description of a finite message space $\mathcal{M}$ as well as a description of a finite ciphertext space $\mathcal{C}$. These parameters are known publicly, wherre the master-key is known only to the trusted authority, the so called Private Key Generator (\texttt{PKF}).
\item[Extract$(\text{params}, MK, \mathtt{ID})$] Takes the system parameters, the master-key and an arbitrary \texttt{ID} $\in \{0,1\}^*$ and returns a private key $d$. \texttt{ID} is essentially any arbitrary string which will be used a public key and $d$ is the corresponding decryption key, which can be used by the owner of the \texttt{ID}. Thus, the \texttt{extract} algorithm extracts a private key from the given public key.
\item[Encrypt$(\text{params}, \mathtt{ID}, M)$] Takes the system parameters, \texttt{ID}, and $M \in \mathcal{M}$. Returns some ciphertext $C \in \mathcal{C}$.
\item[Decrypt$(\text{params}, C, d)$] Takes the system parameters, some private key $d$ and $C \in \mathcal{C}$. Returns the plaintext $M \in \mathcal{M}$.
\item[Correctness] Naturally, these algorithms must satisfy that:
$$ \forall M \in M\ :\ \text{Decrypt}(\text{params}, C, d) = M\quad \text{where}\quad C = \text{Encrypt}(\text{params}, ID, M)$$
\end{description}
\subsubsection{Security}
\subsection{Security Definition}
\textbf{Chosen Ciphertext Security.} \quad To this end, we will focus on Chosen Ciphertext Security (\texttt{IND-CPA}), as this is the standard acceptable notion of security for a public key encryption scheme \cite{security_notion}. The standard definition however, is not strong enough, as we must also require that the adversary might already know of several \texttt{ID}s and decryption keys, given by the \texttt{PKG} and these should not aid the adversary in breaking the security. We define an \emph{extraction query} to be a query which yields the decryption key for a given \ID. Furthermore, the adversary is given the choice of which \ID to be challenged on, rather than it being a random public key. \cite{WeilIBE}
An Identity-Based Encryption scheme is semantically secure against an adaptive chosen ciphertext attack (\texttt{IND-ID-CPA}) if no polynomially bounded adversary $\mathcal{A}$ has non-negligible advantage against the Challenger in the following game \adv{E}:
\begin{description}
@ -165,7 +163,8 @@ These queries may be run adaptively, as in Phase 1.
This definition closely resembles the standard definition of \texttt{IND-CPA} but extended with the addition of extraction queries and that the challenger is now challenged on an \ID picked by the adversary. The addition of the extraction queries is supported by \cite{ExtractionDef}, when the scheme is to support multiple users, which is likely the case for any IBE scheme. Furthermore, the weaker notion of security known as \emph{Semantic Security} (\texttt{IND-ID-CPA}) can be defined based on \texttt{IND-ID-CCA}, except now the adversary is not allowed to issue any decryption queries, i.e. he is only allowed extraction queries.
\subsection{A Scheme}
\subsection{A Instantiation of \texttt{IBE}}
\label{sec:IBEConst}
The scheme we will focus on is that of Boneh and Franklin as described in \cite{WeilIBE}. The structure will be as defined in Section \ref{sec:IBEStruct}. We let $\lambda$ be the given security parameter given implicitly to the setup algorithm. We let $\mathcal{G}$ be a BDH parameter generator.
\begin{description}
\item[Setup] Given $k$;
@ -192,43 +191,140 @@ $$e(d_{\mathtt{ID}}, u) = e(Q_{\mathtt{ID}}^s, g^r) = e(Q_{\mathtt{ID}}, g)^{sr}
\end{description}
\subsection{Security}
The scheme can be shown to be semantically secure (\texttt{IND-ID-CPA}), assuming that the BDH problem is hard in the groups generated by $\mathcal{G}$.
% TODO: Consider finishing this security proof. Is it really that important? It's the last from this paper.
\subsection{Security of the Instantiation}
The scheme can be shown to be semantically secure (\texttt{IND-ID-CPA}), assuming that the BDH problem is hard in the groups generated by $\mathcal{G}$. It is proven by reducing an adversary who can break the \texttt{IBE} construction from Section \ref{sec:IBEConst}, to breaking an instantiation of the mathmatical problem defined in Section \ref{sec:BDHProb}. The actual proof is quite long and is left within the Appendix \ref{app:IBE-Sec} in its entirety.
\section{Broadcast Encryption}
\label{sec:BE}
Broadcast Encryption systems \cite{BEDef} in a nutshell, allows one sender to send to a subset $S \subseteq [1,n]$ of users with a single message. Traditionally, the user would have to encrypt this message once per user in a horribly inefficient manner. This is fixed, by defining the encryption key in such a way to allow for any user within the $S$ to decrypt the message, while not allowing anyone outside of $S$ to do so. It is preferable for this kind of schem to be \emph{public key based}, rather than symmetric. This allows any user to encrypt. It should allow \emph{stateless receivers} s.t. users won't need to keep any state such as updating a private key, and the system should be \emph{fully collusion resistant}, i.e. not allow decryption even if everybody outside of the set $S$ cooperated.
\subsection{Modelling Broadcast Encryption}
Broadcast Encryption systems can be defined as a \emph{Key Encapsulation Mechanism}, \texttt{KEM}, where what is actually encrypted is a symmetrical key, which can then be used for encryption and decryption at a later stage.
% TODO: BSetup takes n and ell, where ell defines the maximum size of the set. Ergo is the security parameter lambda implicit. Change this at later stages!
\begin{description}
\item[BSetup$(n, \ell)$] Where $n$ defines the number of receivers and $\ell$ is the maximum size of the receiver set, $n \leq \ell$. Outputs a public/secret key pair, $(PK, SK)$ of the BE scheme.
\item[KeyGen$(i, SK)$] $i$ defines a specific index for a user, $i \in [1,n]$, as well as the secret key $SK$. Outputs a decryption key $d_i$ specific for this user.
\item[BEnc$(S, PK)$] Takes as input some subset $S \subseteq [1,n]$ where $S$ will henceforth be known as the receiver or authorised set, as well as the public key of the BE scheme. Given that $|S| \leq \ell$, outputs a pair $(\hdr, K)$, where \hdr is called the \emph{header} and $K \in \mathcal{K}$ is the message encryption key. \hdr can be seen as the ciphertext for the key $K$, which in this case is a symmetrical key which can be used after having been extracted from the \hdr.
\item[BDec$(S, i, d_i, \hdr, PK)$] Takes the receiver set $S$, an index $i \in [1,n]$, a corresponding decryption key, a \hdr and the public key PK. Given $|S| \leq \ell$ and $i \in S$, this extracts the key $K \in \mathcal{K}$ from \hdr.
\item[Correctness] For all $S \subseteq [1,n]$ and all $i \in S$, if $(PK, SK) \xleftarrow{R} BSetup(n, \ell)$, $d_i \xleftarrow{R} KeyGen(i, SK)$ and $(\hdr, K) \xleftarrow{R} BEnc(S, PK)$, then $BDec(S, i, d_i, \hdr, PK) = K$, i.e. if a key $K \in \mathcal{K}$ is encrypted for a receiver set, then a user from within this receiver set can also extract the same $K$ from the \hdr.
\end{description}
\subsection{Security Defintions}
\label{sec:BESec}
We define three levels of security, \emph{Static, Semi-Static} and \emph{Adaptive}. For the sake of simplicity, we will explain Semi-static and then emphasise the differences. Note that Semi-Static security is stronger than Static security, but weaker than Adaptive. The definition of Semi-Static is due to Gentry and Waters \cite{BESecDef, GentryWaters}.
\begin{description}
\item[Initialisation] The adversary \adv{A} first commits to a \emph{potential} set of receivers which he wishes to attack, $\tilde{S}$, and outputs this.
\item[Setup] The challenger \CH runs the $\mathbf{BSetup}(n, \ell)$ algorithm of the BE scheme, obtaining a public key PK. \CH gives this PK to \adv{A}.
\item[Key Extraction Phase] The adversary \adv{A} is allowed to issue private key queries for indices $i \in [1,n] \setminus \tilde{S}$, i.e. he is allowed to ask for the private keys of any user not in the set of potential receivers.
\item[Challenge] Once the adversary \adv{A} has extracted all desired keys, he specifies an attack set $S^* \subseteq \tilde{S}$, on which he wants to be challenged. The challenger \CH then sets $(\hdr^*, k_0) \leftarrow BEnc(S^*, PK)$ and $k_1 \in_R \mathcal{K}$. Then $b \in_R \{0,1\}$ and \CH sends $(\hdr^*, k_b)$ to \adv{A}.
\item[Guess] Adversary \adv{A} outputs a guess $b' \in \{0,1\}$ and he wins if $b' = b$.
The advantage of \adv{A} is then defined as: $$Adv_{SS,BE,n,\ell}(\lambda) = |Pr(b'=b) - \frac{1}{2}|$$
Static security is the least strongest type and it requires the adversary to commit to the set of receivers of which he wants to be challenged on, in the initialisation phase, rather than the potential set the Semi-Static adversary has to commit to. Adaptive security is arguably the most desired and correct type, as it enforces nothing in regards to the attack set $S^*$. The adversary is allowed to see the public key PK and ask for several private keys, before choosing which set he wishes to be challenged on. We note here, that due to Gentry and Waters \cite{GentryWaters}, we can transform a Semi-Statically secure BE scheme to an Adaptively secure BE scheme.
\end{description}
% TODO: Consider using description environment or https://tex.stackexchange.com/questions/436977/how-to-insert-multiple-hspace-into-one-row-line
% TODO, maybe new page this
\subsection{Their construction}
\label{sec:GentryWatersConst}
Let $GroupGen(\lambda,n)$ be an algorithm which generates a group \G and \Gp{_T} of prime order $p = poly(\lambda, n) > n$ with a bilinear map $e : \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$, based on a security parameter $\lambda$.
\begin{description}
\item[BSetup$(n, \ell)$] Run $(\mathbb{G}, \mathbb{G}_T, e) \xleftarrow{R} GroupGen(\lambda, n)$. Set $\alpha \in_R \Z_p$ and $g,h_1,\dots,h_n \in_R \mathbb{G}^{n+1}$. Finally, set $PK = (\mathbb{G}, \mathbb{G}_T, e), g, e(g,g)^\alpha, h_1, \dots, h_n$. The secret key is $SK = g^\alpha$. The result is the pair $(PK, SK)$.
\item[BKeyGen$(i, SK)$] Set $r_i \in_R \Z_p$ and output; $$d_i \leftarrow (d_{i,0},\dots,d_{i,n}) \quad \text{ where } \quad d_{i,0} = g^{-r_i}, \quad d_{i,i} = g^\alpha h^{r_i}_i, \quad d_{i,j \text{ for } i\neq j} h^{r_i}_j$$
\item[BEncrypt$(S, PK)$] Set $t \in_R \Z_p$ and $$Hdr = (C_1,C_2), \quad \text{ where }\quad C_1 = g^t, \quad C_2 = (\prod_{i \in S}h_i)^t $$ Finally, set $K = e(g,g)^{t\cdot \alpha}$. Output $(\hdr, K)$.
\item[BDecrypt$(S,i,d_i,\text{Hdr}, PK)$] Check if $i \in S$, if so; let $d_i = (d_{i,0},\dots,d_{i,n})$, Hdr$=(C_1,C_2)$, output $$k =e(d_{i,i} \cdot \prod_{j \in S \setminus \{i\}} d_{i,j}, C_1) \cdot e(d_{i,0}, C_2)$$
\item[Correctness] Correctness is given by;
\begin{align*}
K &= e(d_{i,i} \cdot \prod_{j \in S \setminus \{i\}} d_{i,j}, C_1) \cdot e(d_{i,0}, C_2) \\
&= e(g^{\alpha}h^{r_i}_i \cdot (\prod_{j \in S \setminus \{i\}} h_j)^{r_i}, g^t) \cdot e(g^{-r_i}, (\prod_{j \in S}h_j)^t) \\
&= e(g^{\alpha} \cdot (\prod_{j \in S} h_j)^{r_i}, g^t) \cdot e(g^{-r_i}, (\prod_{j \in S}h_j)^t) \\
&= e(g,g)^{t \cdot \alpha}
\end{align*}
\end{description}
\subsection{Proof of Security}
The proof is a reduction from their construction to the \emph{BDHE}-problem. The scheme is proven secure in the semi-static model. We note that the proof in the original paper does not hold, likely due to a typo, but we'll emphasise the fix.
We wish to build an algorithm \adv{B}, which will use an adversary \adv{A} of the system described in \ref{sec:GentryWatersConst}, to break the \emph{BDHE} problem. \\ \\
\adv{B} receives a problem instance which contains $g^s, Z, \{g^{a^i}: i \in [0,m] \cup [m+2, 2m]\}$.
\begin{description}
\item[Init] \adv{A} commits to a set $\tilde{S} \subseteq [1,n]$.
\item[Setup] \adv{B} generates $y_0,\dots,y_n \in_R \Z_p$. \adv{B} sets:
$$
h_i =
\begin{cases}
g^{y_i} & \text{ for } i \in \tilde{S} \\
g^{y_i + a^{i}} & \text{ for } i \in [1,n] \setminus \tilde{S}
\end{cases}
$$
\adv{B} then sets $\alpha = y_0 \cdot a^{n+1}$. $PK$ is then defined as the scheme dictates where the only oddity is $e(g,g)^\alpha$, which can be computed as $e(g^a,g^{a^{n}})^{y_0}$ due to the definition of $\alpha$. $PK$ is sent to \adv{A}.
\item[Key Extraction Phase] \adv{A} is allowed to query private keys for indices $i \in [1,n] \setminus \tilde{S}$. Intuitively, you should not be allowed to query the indices of which you wish to be challenged. To answer a query, \adv{B} will generate a $z_i \in_R \Z_p$ and set $r_i = z_i - y_0 \cdot a^{n+1-i}$. \adv{B} then outputs
$$ d_i = (d_{i,0},\dots,d_{i,n})\quad \text{ where } \quad d_{i,0} = g^{-r_i},\quad d_{i,i} = g^\alpha h^{r_i}_i, \quad d_{i,j \text{ where } i\neq j}h^{r_i}_j $$
\item[Challenge] \adv{A} will then choose a subset $S^* \subseteq \tilde{S}$ to which \adv{B} sets:
$$\text{Hdr} = (C_1, C_2) \quad \text{ where } C_1 = g^s, \quad C_2 = (\prod_{j \in S^*}h_j)^s$$
Note that $g^s$ comes from the original challenge and due to the construction of the $h_j$ values, $C_2$ is computable, as \adv{B} knows the discrete log of each of them, specifically $h_j = g^{y_j}$, as long as $j \in \tilde{S}$.
\adv{B} sets $K = Z^{y_0}$ (The original; $K = Z$) and sends $(\text{Hdr},K)$ to \adv{A}.
\item[Guess] \adv{A} will output a guess $b'$. \adv{B} forwards this bit to the Challenger.
\item[Correctness] This simulation intuitively works, as if \adv{A} returns $b' = 0$ then the pair $(\text{Hdr}, K)$ is generated according to the same distribution as in the real world, according to \adv{A}. This is also true for \adv{B}'s simulation, as for $b=0$, $K = e(g,g)^{\alpha \cdot s} = e(g,g)^{(a^{n+1} \cdot s) \cdot y_0} = Z^{y_0}$, so it's a valid ciphertext under randomness $s$. When $b=1$, the $K$ is however picked randomly from $\mathcal{K}$, resulting in a correctly header Hdr with randomness $s$, but the ciphertext is random.
\end{description}
This construction we'll be the foundation of the \emph{Ad-Hoc Broadcast Encryption} which we will explore shortly and likewise will this proof be brought up when exploring possible proofs of security of said \emph{Ad-Hoc Broadcast Encryption} scheme.
\section{Dynamic Threshold Public-Key Encryption}
In a Threshold Public-Key Encryption (\texttt{TPKE}) scheme, the decryption key corresponding to a public key is shared among a set of $n$ users \cite{TPKE}. Specifically for \texttt{TPKE} is that for any ciphertext to be correctly decrypted, $t$ receivers has to participate and cooperate. Thus, if any number of users less than $t$ try to decrypt, they will gain nothing, hence the threshold part of \texttt{TPKE}. A limitation of existing \texttt{TPKE} schemes however, is that the threshold value of $t$ is tightly connected to the public key of the system, as such, one has to fix the threshold for good, when setting up the system. Many applications would benefit from a flexibility to choose $t$ whenever broadcasting. As such Dynamic Threshold Public-Key Encryption (\texttt{DTPKE}) is proposed \cite{DTPKE}.
In a sense, Broadcast Encryption Systems can be related to notion of \emph{Threshold Public Key Encryption Systems} (\texttt{TPKE}) if we define the authorized set of the \texttt{TPKE} system to be equal to $S$ and the threshold parameter $t$ is set to be $1$. This is only true however, for the specific value of $t=1$, thus, specialized systems can be designed for the purpose of being broadcast encryption systems and \texttt{TPKE} can be seen as a general case of \texttt{BE}.
\subsection{Modelling Dynamic Threshold Public-Key Encryption}
A \texttt{DTPKE}-scheme consist of $7$ algorithms: \texttt{DTPKE} $= ($\texttt{Setup}, \texttt{Join}, \texttt{Encrypt}, \texttt{ValidateCT}, \texttt{ShareDecrypt}, \texttt{ShareVerify}, \texttt{Combine}$)$.
\begin{description}
\item[Setup$(\lambda)$] Takes security parameter $\lambda$. Outputs a set of system parameters: $$\mathtt{params} = (MK,EK,DK,VK,CK).$$ $MK$ is a Master Secret Key, $EK$ is the Encryption Key, $DK$ is the Decryption Key, $VK$ is the Validation Key and $CK$ is the Combination Key. $MK$ is kept secret by the issuer, but the other four are public parameters.
\item[Join$(MK, \mathtt{ID})$] Takes the $MK$ and an identity \ID of a user. Outputs the user's keys $(usk, upk, uvk)$, where $usk$ is the secret key used for decryption, $upk$ is the public key used for encrypting and $uvk$ is the verification key. $upk, uvk$ are both public, whereas $usk$ is given privately to the user.
\item[Encrypt$(EK, S, t, M)$] Takes the Encryption Key, the public keys of the users within the receiver set $S$, a threshold $t$ and a message to be encrypted, $M$. Outputs a ciphertext.
\item[ValidateCT$(EK, S, t, C)$] Takes the encryption key, the public keys of the receiver set, a threshold and a ciphertext. Checks whether $C$ is a valid ciphertext with respect to $EK, S$ and $t$.
\item[ShareDecrypt$(DK, \mathtt{ID}, usk, C)$] Takes the decryption key, a user id \ID and his private key $usk$, as well as a ciphertext $C$. Outputs a decryption share $\sigma$ or $\perp$.
\item[ShareVerify$(VK, \mathtt{ID}, uvk, C, \sigma)$] Takes the verification key $VK$, a user id \ID and his verification key $uvk$ plus a ciphertext $C$ and decryption share $\sigma$. Checks whether $\sigma$ is a valid decryption share with respect to $uvk$.
\item[Combine$(CK, S, t, C, T, \Sigma)$] Takes the combination key $CK$, a ciphertext $C$, some subset $T \subseteq S$ of $t$ authorised users and $\Sigma = (\sigma_1, \dots, \sigma_t)$ which is a list of $t$ decryption share. Outputs the plaintext $M$ or $\perp$.
\end{description}
\subsection{Security Model}
\-\hspace{5mm} \textbf{Setup:}\quad The challenger runs Setup$(\lambda)$ of the \texttt{DTPKE} scheme, obtaining the $$\mathtt{params} = (MK,EK,DK,VK,CK)$$. All the public parameters (all except for $MK$) are given to the adversary \adv{A}. \vsp{3mm}
\-\hspace{5mm} \textbf{Phase 1:}\quad The adversary is allowed to adaptively issue queries where query $q_i$ is one of three queries;
\begin{description}
\item[Setup] The challenger runs Setup$(\lambda)$ of the \texttt{DTPKE} scheme, obtaining the $$\mathtt{params} = (MK,EK,DK,VK,CK)$$. All the public parameters (all except for $MK$) are given to the adversary \adv{A}.
\item[Phase 1] The adversary is allowed to adaptively issue queries where query $q_i$ is one of three queries;
\begin{itemize}
\item A \texttt{Join} query on an id \texttt{ID}; The challenger runs the \texttt{Join} algorithm on input $(MK,\mathtt{ID})$, to create a new user in the system. Note that the challenger has $MK$ from the setup step.
\item A \texttt{Corrupt} query on an id \texttt{ID}: The challenger forwards the corresponding private key to the adversary.
\item A \texttt{ShareDecrypt} query on an id \texttt{ID} and a header \texttt{Hdr}: The challenger runs the \texttt{ShareDecrypt} algorithm of the \texttt{DTPKE} scheme on \texttt{Hdr}, using the corresponding private key, and forwards the partial decryption to the adversary.
\end{itemize}
\hsp{5mm} \textbf{Challenge:}\quad The adversary \adv{A} outputs a target set of users $S^*$ as well as a threshold $t^*$. The challenger selects $b \in_R \set{0, 1}$ and then runs \texttt{Encrypt} to obtain $\mathtt{Hdr}^*, k_0) \la \mathtt{Encrypt}(EK, S^*, t^*)$. Furthermore, he picks another key $k_1 \in_R \mathcal{K}$. The challenger outsputs $(\mathtt{Hdr}^*, k_b)$ to \adv{A}. \vsp{3mm}
\hsp{5mm} \textbf{Phase 2:}\quad The adversary \adv{A} is allowed to continue adaptively issuing \texttt{Join, Corrupt} and \texttt{ShareDecrypt} queries, with the only constraint that he asks less than or equal to $t^*-1$.\vsp{3mm}
\hsp{5mm} \textbf{Guess:} The adversary outputs a guess bit $b' \in \{0,1\}$ and he will win the game if $b' = b$. \vsp{5mm}
From this basic description, we can define three sub definitions:
\end{itemize}
\item[Challenge] The adversary \adv{A} outputs a target set of users $S^*$ as well as a threshold $t^*$. The challenger selects $b \in_R \set{0, 1}$ and then runs \texttt{Encrypt} to obtain $\mathtt{Hdr}^*, k_0) \la \mathtt{Encrypt}(EK, S^*, t^*)$. Furthermore, he picks another key $k_1 \in_R \mathcal{K}$. The challenger outsputs $(\mathtt{Hdr}^*, k_b)$ to \adv{A}.
\item[Phase 2] The adversary \adv{A} is allowed to continue adaptively issuing \texttt{Join, Corrupt} and \texttt{ShareDecrypt} queries, with the only constraint that he asks less than or equal to $t^*-1$.
\item[Guess] The adversary outputs a guess bit $b' \in \{0,1\}$ and he will win the game if $b' = b$.
From this basic description, we can define three sub definitions:
\begin{itemize}
\item \emph{Non-Adaptive Adversary} (\texttt{NAA}): We restrict the adversary to decide upon the challenge set $S^*$ as well as the threshold $t^*$ before the \texttt{Setup} step is run.
\item \emph{Non-Adaptive Corruption} (\texttt{NAC}): We restrict the adversary to decide before the setup is run, which identities will be corrupted.
\item \emph{Chosen-Plaintext Adversary} (\texttt{CPA}): We restrict the adversary from issuing share decryption queries.
\end{itemize}
\end{description}
% TODO: Consider having parameters for all modelling or none!
\subsection{Modelling \texttt{DTPKE}}
A \texttt{DTPKE}-scheme consist of $7$ algorithms: \texttt{DTPKE} $= ($\texttt{Setup}, \texttt{Join}, \texttt{Encrypt}, \texttt{ValidateCT}, \texttt{ShareDecrypt}, \texttt{ShareVerify}, \texttt{Combine}$)$. \vsp{4mm}
\hsp{5mm}\textbf{Setup$(\lambda)$:}\quad Takes security parameter $\lambda$. Outputs a set of system parameters: $$\mathtt{params} = (MK,EK,DK,VK,CK).$$ $MK$ is a Master Secret Key, $EK$ is the Encryption Key, $DK$ is the Decryption Key, $VK$ is the Validation Key and $CK$ is the Combination Key. $MK$ is kept secret by the issuer, but the other four are public parameters. \vsp{3mm}
\hsp{5mm}\textbf{Join$(MK, \mathtt{ID})$:}\quad Takes the $MK$ and an identity \ID of a user. Outputs the user's keys $(usk, upk, uvk)$, where $usk$ is the secret key used for decryption, $upk$ is the public key used for encrypting and $uvk$ is the verification key. $upk, uvk$ are both public, whereas $usk$ is given privately to the user.\vsp{3mm}
\hsp{5mm}\textbf{Encryptp$(EK, S, t, M)$:}\quad Takes the Encryption Key, the public keys of the users within the receiver set $S$, a threshold $t$ and a message to be encrypted, $M$. Outputs a ciphertext.\vsp{3mm}
\hsp{5mm}\textbf{ValidateCT$(EK, S, t, C)$:}\quad Takes the encryption key, the public keys of the receiver set, a threshold and a ciphertext. Checks whether $C$ is a valid ciphertext with respect to $EK, S$ and $t$. \vsp{3mm}
\hsp{5mm}\textbf{ShareDecrypt$(DK, \mathtt{ID}, usk, C)$:}\quad Takes the decryption key, a user id \ID and his private key $usk$, as well as a ciphertext $C$. Outputs a decryption share $\sigma$ or $\perp$. \vsp{3mm}
\hsp{5mm}\textbf{ShareVerify$(VK, \mathtt{ID}, uvk, C, \sigma)$:}\quad Takes the verification key $VK$, a user id \ID and his verification key $uvk$ plus a ciphertext $C$ and decryption share $\sigma$. Checks whether $\sigma$ is a valid decryption share with respect to $uvk$. \vsp{3mm}
\hsp{5mm}\textbf{Combine$(CK, S, t, C, T, \Sigma)$:}\quad Takes the combination key $CK$, a ciphertext $C$, some subset $T \subseteq S$ of $t$ authorised users and $\Sigma = (\sigma_1, \dots, \sigma_t)$ which is a list of $t$ decryption share. Outputs the plaintext $M$ or $\perp$.\vsp{3mm}
\subsection{A scheme and the Security Thereof}
It should be noted that this scheme is very long and as such will be left out of the report, but it will be included in the appendix, completely as the original authors wrote it. We will instead list their security proof, which contains an error worth of noting. Their proof is a reduction to the \texttt{MSE-DDH} problem, as defined in Section \ref{sec:MSE-DDH}. Regardless, their security proof states that the \texttt{DTPKE} scheme has \texttt{IND-NAA-NAC-CPA} security (Non-adaptive adversary, non-adaptive corruption, chosen-plaintext attack).
@ -237,7 +333,7 @@ It should be noted that this scheme is very long and as such will be left out of
For any $l,m,t,$ $\mathbf{Adv}^{ind}_{\mathtt{DTPKE}}(l,m,t) \leq 2 \cdot \mathbf{Adv}^{\text{MSE-DDH}}(l,m,t)$. Where $l$ denotes the total number of \textbf{Join} queries that can be issued by the adversary, $m$ is the maximal size the authorised set of receivers is allowed to be, $t$ is the threshold.
\end{theorem}
\begin{proof}
Let \texttt{DTPKE} denote the construction as described in Appendix A. Now, to establish the semantic security, the \texttt{IND-NAA-NAC-CPA} security, for static adversaries of the \texttt{DTPKE} scheme, we describe a reduction to the \texttt{MSE-DDH} problem. To this end, we assume an adversary \adv{A} who can break the scheme under an $(l,m,t)$-collusion. This adversary \adv{A} will be used to build an algorithm \adv{B} who can then distinguish the two distributions of the $(l,m,t)$-\texttt{MSE-DDH} problem.
Let \texttt{DTPKE} denote the construction as described in Appendix \ref{app:DTPKE-Scheme}. Now, to establish the semantic security, the \texttt{IND-NAA-NAC-CPA} security, for static adversaries of the \texttt{DTPKE} scheme, we describe a reduction to the \texttt{MSE-DDH} problem. To this end, we assume an adversary \adv{A} who can break the scheme under an $(l,m,t)$-collusion. This adversary \adv{A} will be used to build an algorithm \adv{B} who can then distinguish the two distributions of the $(l,m,t)$-\texttt{MSE-DDH} problem.
The algorithm \adv{B} is given as input some group system $Pub = (p, \Gm_1, \Gm_2, \Gm_T, e)$ as described in \ref{sec:MSE-DDH} as well as an $(l,m,t)$-\texttt{MSE-DDH} instance in $Pub$. The \texttt{MSE-DDH} instances gives us, \adv{B}, two coprime polynomials $f_{poly}$ and $g_{poly}$ of orders $l$ and $m$ with pairwise distinct roots $(x_1, \dots, x_l)$ and $(x_{l+t}, \dots, x_{l+t+m-1})$ respectively. Finally, \adv{B} has all the exponents;
\begin{align*}
@ -299,100 +395,45 @@ It should be noted that this scheme is very long and as such will be left out of
Now, as the distribution of $b$ is independent from the adversarys view; $$Pr(b'=1 | b=1 \wedge \text{random}) = Pr(b'=1 | b=0 \wedge \text{random})$$ Thus, the left side cancels out. In the real case however, the distribution of all variables which are defined by \adv{B} comply with the definition of the semantic security game, as all simulations are perfect. Thus, to conclude:
$$\mathbf{Adv}_{\mathtt{DTPKE}}^{\text{ind}}(\mathcal{A}) = Pr(b'=1 | b=1 \wedge \text{real}) - Pr(b'=1 | b=0 \wedge \text{real})$$
Is exactly equal to $$2 \cdot (\frac{1}{2} \times (Pr(b'=1 | b=1 \wedge \text{real}) + Pr(b'=1 | b=0 \wedge \text{real})).$$
\end{proof}
\end{proof}
% TODO: Consider giving some outro to this
% TODO: Explain this scheme and their security proof which doesn't work. Yikes.
% TOOD: Add the DTPKE scheme to the appendix A.
% TODO: Consider making all upk and usk bold
\section{Broadcast Encryption}
\label{sec:BE}
Broadcast Encryption systems \cite{BEDef} in a nutshell, allows one sender to send to a subset $S \subseteq [1,n]$ of users with a single message. Traditionally, the user would have to encrypt this message once per user in a horribly inefficient manner. This is fixed, by defining the encryption key in such a way to allow for any user within the $S$ to decrypt the message, while not allowing anyone outside of $S$ to do so. It is preferable for this kind of schem to be \emph{public key based}, rather than symmetric. This allows any user to encrypt. It should allow \emph{stateless receivers} s.t. users won't need to keep any state such as updating a private key, and the system should be \emph{fully collusion resistant}, i.e. not allow decryption even if everybody outside of the set $S$ cooperated.
In a sense, Broadcast Encryption Systems can be related to notion of \emph{Threshold Public Key Encryption Systems} (\texttt{TPKE}) if we define the authorized set of the \texttt{TPKE} system to be equal to $S$ and the threshold parameter $t$ is set to be $1$. This is only true however, for the specific value of $t=1$, thus, specialized systems can be designed for the purpose of being broadcast encryption systems. In this paper we will focus on a scheme due to Gentry and Waters \cite{GentryWaters}.
\subsection{Security Defintions}
\label{sec:BESec}
We define three levels of security, \emph{Static, Semi-Static} and \emph{Adaptive}. For the sake of simplicity, we will explain Semi-static and then emphasise the differences. Note that Semi-Static security is stronger than Static security, but weaker than Adaptive. The definition of Semi-Static is due to Gentry and Waters \cite{BESecDef, GentryWaters}. \vsp{4mm}
\hsp{5mm}\textbf{Initialisation:}\quad The adversary \adv{A} first commits to a \emph{potential} set of receivers which he wishes to attack, $\tilde{S}$, and outputs this. \vsp{3mm}
\hsp{5mm}\textbf{Setup:}\quad The challenger \CH runs the $\mathbf{BSetup}(n, \ell)$ algorithm of the BE scheme, obtaining a public key PK. \CH gives this PK to \adv{A}. \vsp{3mm}
\hsp{5mm}\textbf{Key Extraction Phase:}\quad The adversary \adv{A} is allowed to issue private key queries for indices $i \in [1,n] \setminus \tilde{S}$, i.e. he is allowed to ask for the private keys of any user not in the set of potential receivers. \vsp{3mm}
\hsp{5mm}\textbf{Challenge:}\quad Once the adversary \adv{A} has extracted all desired keys, he specifies an attack set $S^* \subseteq \tilde{S}$, on which he wants to be challenged. The challenger \CH then sets $(\hdr^*, k_0) \leftarrow BEnc(S^*, PK)$ and $k_1 \in_R \mathcal{K}$. Then $b \in_R \{0,1\}$ and \CH sends $(\hdr^*, k_b)$ to \adv{A}. \vsp{3mm}
\hsp{5mm}\textbf{Guess:}\quad Adversary \adv{A} outputs a guess $b' \in \{0,1\}$ and he wins if $b' = b$. \\ \\
\noindent
The advantage of \adv{A} is then defined as: $$Adv_{SS,BE,n,\ell}(\lambda) = |Pr(b'=b) - \frac{1}{2}|$$
Static security is the least strongest type and it requires the adversary to commit to the set of receivers of which he wants to be challenged on, in the initialisation phase, rather than the potential set the Semi-Static adversary has to commit to. Adaptive security is arguably the most desired and correct type, as it enforces nothing in regards to the attack set $S^*$. The adversary is allowed to see the public key PK and ask for several private keys, before choosing which set he wishes to be challenged on. We note here, that due to Gentry and Waters \cite{GentryWaters}, we can transform a Semi-Statically secure BE scheme to an Adaptively secure BE scheme.
% TODO: Consider using description environment or https://tex.stackexchange.com/questions/436977/how-to-insert-multiple-hspace-into-one-row-line
% TODO, maybe new page this
\subsection{Their construction}
\label{sec:GentryWatersConst}
Let $GroupGen(\lambda,n)$ be an algorithm which generates a group \G and \Gp{_T} of prime order $p = poly(\lambda, n) > n$ with a bilinear map $e : \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$, based on a security parameter $\lambda$. \vsp{5mm}
\-\hspace{5mm}\textbf{BSetup$(\lambda,n)$:}\quad Run $(\mathbb{G}, \mathbb{G}_T, e) \xleftarrow{R} GroupGen(\lambda, n)$. Set $\alpha \in_R \Z_p$ and $g,h_1,\dots,h_n \in_R \mathbb{G}^{n+1}$. Finally, set $PK = (\mathbb{G}, \mathbb{G}_T, e), g, e(g,g)^\alpha, h_1, \dots, h_n$. The secret key is $SK = g^\alpha$. The result is the pair $(PK, SK)$. \vspace{3mm} \\
\-\hspace{5mm}\textbf{BKeyGen$(i, SK)$:}\quad Set $r_i \in_R \Z_p$ and output; $$d_i \leftarrow (d_{i,0},\dots,d_{i,n}) \quad \text{ where } \quad d_{i,0} = g^{-r_i}, \quad d_{i,i} = g^\alpha h^{r_i}_i, \quad d_{i,j \text{ for } i\neq j} h^{r_i}_j$$ \vspace{3mm} \\
\-\hspace{5mm}\textbf{BEncrypt$(S, PK)$:}\quad Set $t \in_R \Z_p$ and $$Hdr = (C_1,C_2), \quad \text{ where }\quad C_1 = g^t, \quad C_2 = (\prod_{i \in S}h_i)^t $$ Finally, set $K = e(g,g)^{t\cdot \alpha}$. Output $(\hdr, K)$. \vspace{3mm} \\
\-\hspace{5mm}\textbf{BDecrypt}$(S,i,d_i,\text{Hdr}, PK)$\textbf{:}\quad Check if $i \in S$, if so; let $d_i = (d_{i,0},\dots,d_{i,n})$, Hdr$=(C_1,C_2)$, output $$k =e(d_{i,i} \cdot \prod_{j \in S \setminus \{i\}} d_{i,j}, C_1) \cdot e(d_{i,0}, C_2)$$ \vsp{3mm}
\hsp{5mm} \textbf{Correctness:}\quad Correctness is given by;
\begin{align*}
K &= e(d_{i,i} \cdot \prod_{j \in S \setminus \{i\}} d_{i,j}, C_1) \cdot e(d_{i,0}, C_2) \\
&= e(g^{\alpha}h^{r_i}_i \cdot (\prod_{j \in S \setminus \{i\}} h_j)^{r_i}, g^t) \cdot e(g^{-r_i}, (\prod_{j \in S}h_j)^t) \\
&= e(g^{\alpha} \cdot (\prod_{j \in S} h_j)^{r_i}, g^t) \cdot e(g^{-r_i}, (\prod_{j \in S}h_j)^t) \\
&= e(g,g)^{t \cdot \alpha}
\end{align*}
\subsection{Proof of Security}
The proof is a reduction from their construction to the \emph{BDHE}-problem. The scheme is proven secure in the semi-static model. We note that the proof in the original paper does not hold, likely due to a typo, but we'll emphasize the fix.
We wish to build an algorithm \adv{B}, which will use an adversary \adv{A} of the system described in \ref{sec:GentryWatersConst}, to break the \emph{BDHE} problem. \vsp{4mm}
\hsp{5mm} \adv{B} receives a problem instance which contains $g^s, Z, \{g^{a^i}: i \in [0,m] \cup [m+2, 2m]\}$. \vsp{3mm}
\hsp{5mm} \textbf{Init:}\quad \adv{A} commits to a set $\tilde{S} \subseteq [1,n]$. \vsp{3mm}
\hsp{5mm} \textbf{Setup:}\quad \adv{B} generates $y_0,\dots,y_n \in_R \Z_p$. \adv{B} sets:
$$
h_i =
\begin{cases}
g^{y_i} & \text{ for } i \in \tilde{S} \\
g^{y_i + a^{i}} & \text{ for } i \in [1,n] \setminus \tilde{S}
\end{cases}
$$
\adv{B} then sets $\alpha = y_0 \cdot a^{n+1}$. $PK$ is then defined as the scheme dictates where the only oddity is $e(g,g)^\alpha$, which can be computed as $e(g^a,g^{a^{n}})^{y_0}$ due to the definition of $\alpha$. $PK$ is sent to \adv{A}. \vsp{3mm}
\hsp{5mm} \textbf{Private Key Queries:}\quad \adv{A} is allowed to query private keys for indices $i \in [1,n] \setminus \tilde{S}$. Intuitively, you should not be allowed to query the indices of which you wish to be challenged. To answer a query, \adv{B} will generate a $z_i \in_R \Z_p$ and set $r_i = z_i - y_0 \cdot a^{n+1-i}$. \adv{B} then outputs
$$ d_i = (d_{i,0},\dots,d_{i,n})\quad \text{ where } \quad d_{i,0} = g^{-r_i},\quad d_{i,i} = g^\alpha h^{r_i}_i, \quad d_{i,j \text{ where } i\neq j}h^{r_i}_j $$
\hsp{5mm} \textbf{Challenge:}\quad \adv{A} will then choose a subset $S^* \subseteq \tilde{S}$ to which \adv{B} sets:
$$\text{Hdr} = (C_1, C_2) \quad \text{ where } C_1 = g^s, \quad C_2 = (\prod_{j \in S^*}h_j)^s$$
Note that $g^s$ comes from the original challenge and due to the construction of the $h_j$ values, $C_2$ is computable, as \adv{B} knows the discrete log of each of them, specifically $h_j = g^{y_j}$, as long as $j \in \tilde{S}$.
\adv{B} sets $K = Z^{y_0}$ (The original; $K = Z$) and sends $(\text{Hdr},K)$ to \adv{A}. \vsp{3mm}
\hsp{5mm} \textbf{Guess:}\quad \adv{A} will output a guess $b'$. \adv{B} forwards this bit to the Challenger. \vsp{3mm}
\hsp{5mm} \textbf{Security:}\quad This simulation intuitively works, as if \adv{A} returns $b' = 0$ then the pair $(\text{Hdr}, K)$ is generated according to the same distribution as in the real world, according to \adv{A}. This is also true for \adv{B}'s simulation, as for $b=0$, $K = e(g,g)^{\alpha \cdot s} = e(g,g)^{(a^{n+1} \cdot s) \cdot y_0} = Z^{y_0}$, so it's a valid ciphertext under randomness $s$. When $b=1$, the $K$ is however picked randomly from $\mathcal{K}$, resulting in a correctly header Hdr with randomness $s$, but the ciphertext is random. \\ \\
\noindent
This construction we'll be the foundation of the \emph{Ad-Hoc Broadcast Encryption} which we will explore shortly and likewise will this proof be brought up when exploring possible proofs of security of said \emph{Ad-Hoc Broadcast Encryption} scheme.
\section{Ad-Hoc Broadcast Encryption}
The scheme presented in \ref{sec:BE} requires a \emph{trusted dealer} to perform its \emph{setup} and \emph{keygen}. It goes for a lot of \emph{Broadcast Encryption} systems, that they require a trusted entity to generate and distribute secret keys to all users. This tends to make the system very rigid and not applicable to ad hoc networks or peer-to-peer networks. A \emph{potential} solution to this is presented by \cite{AHBE}. They present a solution to the fully dynamic case of broadcast encryption. This has significant ties to the \emph{Dynamic Threshold Encryption} scheme in which users could freely join and leave, however they did not quite get rid of the trusted dealer. This is accomplished here. Keep in mind that broadcast encryption is simply threshold encryption for the threshold of $t=1$.
In an Ad-Hoc Broadcast Encryption (\texttt{AHBE}) scheme all users possess a public key and by only seeing the public keys of users, a sender can securely broadcast to \emph{any} subset of the users. Only users within the picked subset can decrypt the message. To accomplish this, the authors create a generic transformation from any \emph{key homomorphic} BE scheme to an \texttt{AHBE} scheme. It turns out that the scheme of Gentry and Waters presented in \ref{sec:BE} is just this and the transformation will be performed on this.
\subsection{Modelling Ad-Hoc Broadcast Encryption}
As an \texttt{AHBE} system eliminate the trusted dealer, the \emph{setup} and \emph{keygen} step morph together, as there is no global \emph{setup} step required, but merely something each user should locally run. As all other schemes defined in this paper, this too is defined to be a \emph{Key Encapsulation Method} (\texttt{KEM}).
\subsection{Security Definition of Adaptive Security in AHBE}
\begin{description}
\item[KeyGen$(i,n,N)$] Let $N$ be defined as the number of potential receivers of the scheme and let $n \leq N$ be defined as the maximum number of receivers of an ad-hoc broadcast recipient group. The \emph{KeyGen} (this) algorithm is run by each user $i \in [1,N]$ to create her own public/secret key pair. A user takes $n, N$ as well as her own index $i \in [1,N]$. It's not mentioned how the user receives this index in practice, without simply having a central authority giving them, but one could imagine the users being aware of how many recipients there are in total and simply increment this to get their own index, if one disregards the issues of people joining the peer-to-peer network at the same time. The \emph{KeyGen} algorithm outputs the users public/secret key pair $(PK_i,SK_i)$. We define a shorthand for several users key pairs; $\{(PK_i, SK_i) | i \in S \subseteq [1,N] $ as $(PK_i,SK_i)_{S}$ and likewise only for the public keys; $(PK_i)_{S}$. All of this depends on a security parameter $\lambda$, which is implicitly given to the algorithm.
\item[AHBEnc$(\mathbb{S}, (PK_i)_{S})$] This is run by any sender who may or may not be in $[1,N]$, as long as the sender knows the public keys of the receivers. It takes the recipient set $S \subseteq [1,N]$ and the public keys for $i \in S$; $(PK_i)_{S}$. Given that $|S| \leq n$, the algorithm returns a pair $(\text{Hdr}, K)$ where Hdr is the header, the encapsulated key, and $K$ is the message encryption key.
\item[AHBDec$(\mathbb{S}, j, sk_j, \text{Hdr}, (PK_i)_{S})$] This allows each recipient $i \in S$ to decrypt the message encryption key which is hidden in the header. If $|S| \leq n, j \in S$, then the algorithm returns the message encryption key $k$.
\end{description}
\subsection{Definition of Adaptive Security in AHBE}
An \emph{Ad-Hoc Broadcast Encryption} system is defined to be \textbf{correct} if any user within the receiver set $S$ can decrypt a valid header. In an adaptively secure ad-hoc broadcast encryption system, the adversary is allowed access to all the public keys of the receivers and to ask for several secret keys before choosing the set of indices that the adversary wishes to attack.
Both the Challenger and an adversary \adv{A} are given the security parameter $\lambda$. \vsp{3mm}
\hsp{5mm}\textbf{Setup:}\quad The Challenger runs $KeyGen(i, n, N)$ to obtain the users' public key. These public keys and the public parameters are given to the adversary \adv{A}. \vsp{3mm}
\hsp{5mm}\textbf{Corruption:}\quad Adversary \adv{A} is allowed to adaptively issue private key queries for \emph{some} indices $i \in [1,N]$. \vsp{3mm}
\hsp{5mm}\textbf{Challenge:}\quad \adv{A} specifies some challenge set $S^* \subseteq [1,N]$ s.t. \adv{A} has corrupted none of the users $i$ within $S^*$. The challenger sets $(\text{Hdr}^*, k_0) \leftarrow \mathtt{AHBEnc}(S^*, (pk_i)_{S^*})$ and $k_1 \in_R \mathbb{K}$. The challenger sets $b \in_R \{0,1\}$. It gives $(\text{Hdr}^*, k_b)$ to the adversary \adv{A}. \vsp{3mm}
\hsp{5mm}\textbf{Guess:}\quad The adversary \adv{A} will output a bit $b' \in \{0,1\}$ as an attempt to guess the bit $b$. \adv{A} wins if $b' = b$. \\ \\
\noindent
The advantage of \adv{A} is as expected; $Adv^{\texttt{AHBE}}_{\mathcal{A},n,N}(1^\lambda) = |Pr(b = b') - \frac{1}{2}|$.
% Both the Challenger and an adversary \adv{A} are given the security parameter $\lambda$. \\
\begin{description}
\item[Setup] The Challenger runs $KeyGen(i, n, N)$ to obtain the users' public key. These public keys and the public parameters are given to the adversary \adv{A}.
\item[Key Extraction Phase] The Challenger runs $KeyGen(i, n, N)$ to obtain the users' public key. These public keys and the public parameters are given to the adversary \adv{A}.
\item[Challenge] \adv{A} specifies some challenge set $S^* \subseteq [1,N]$ s.t. \adv{A} has corrupted none of the users $i$ within $S^*$. The challenger sets $(\text{Hdr}^*, k_0) \leftarrow \mathtt{AHBEnc}(S^*, (PK_i)_{S^*})$ and $k_1 \in_R \mathbb{K}$. The challenger sets $b \in_R \{0,1\}$. It gives $(\text{Hdr}^*, k_b)$ to the adversary \adv{A}.
\item[Guess] The adversary \adv{A} will output a bit $b' \in \{0,1\}$ as an attempt to guess the bit $b$. \adv{A} wins if $b' = b$.
The advantage of \adv{A} is as expected; $Adv^{\texttt{AHBE}}_{\mathcal{A},n,N}(1^\lambda) = |Pr(b = b') - \frac{1}{2}|$.
\end{description}
\subsection{Modelling \AHBE Systems}
As an \texttt{AHBE} system eliminate the trusted dealer, the \emph{setup} and \emph{keygen} step morph together, as there is no global \emph{setup} step required, but merely something each user should locally run. As all other schemes defined in this paper, this too is defined to be a \emph{Key Encapsulation Method} (\texttt{KEM}). \vsp{4mm}
\hsp{5mm}\textbf{KeyGen$(i,n,N)$:}\quad Let $N$ be defined as the number of potential receivers of the scheme and let $n \leq N$ be defined as the maximum number of receivers of an ad-hoc broadcast recipient group. The \emph{KeyGen} (this) algorithm is run by each user $i \in [1,N]$ to create her own public/secret key pair. A user takes $n, N$ as well as her own index $i \in [1,N]$. It's not mentioned how the user receives this index in practice, without simply having a central authority giving them, but one could imagine the users being aware of how many recipients there are in total and simply increment this to get their own index, if one disregards the issues of people joining the peer-to-peer network at the same time. The \emph{KeyGen} algorithm outputs the users public/secret key pair $(pk_i,sk_i)$. We define a shorthand for several users key pairs; $\{(pk_i, sk_i) | i \in S \subseteq [1,N] $ as $(pk_i,sk_i)_{S}$ and likewise only for the public keys; $(pk_i)_{S}$. All of this depends on a security parameter $\lambda$, which is implicitly given to the algorithm. \vsp{3mm}
\hsp{5mm}\textbf{AHBEnc$(\mathbb{S}, (pk_i)_{S})$:}\quad This is run by any sender who may or may not be in $[1,N]$, as long as the sender knows the public keys of the receivers. It takes the recipient set $S \subseteq [1,N]$ and the public keys for $i \in S$; $(pk_i)_{S}$. Given that $|S| \leq n$, the algorithm returns a pair $(\text{Hdr}, K)$ where Hdr is the header, the encapsulated key, and $K$ is the message encryption key. \vsp{3mm}
\hsp{5mm}\textbf{AHBDec$(\mathbb{S}, j, sk_j,$}$ \text{Hdr}, (pk_i)_{S})$\textbf{:}\quad This allows each recipient $i \in S$ to decrypt the message encryption key which is hidden in the header. If $|S| \leq n, j \in S$, then the algorithm returns the message encryption key $k$.
\subsection{Key Homomorphism}
As mentioned, the authors present a transformation for any key homomorphic BE scheme. As such, we'll quickly define this.
@ -401,7 +442,7 @@ As mentioned, the authors present a transformation for any key homomorphic BE sc
\begin{definition}[Key Homomorphism]
\normalfont Let $\oplus : \Gamma \times \Gamma \rightarrow \Gamma$, $\odot : \Omega \times \Omega \rightarrow \Omega$ and $\ocircle : \mathbb{K} \times \mathbb{K} \rightarrow \mathbb{K}$ be efficient operations in the public key space $\Gamma$, the decryption key space $\Omega$ and the message encryption key space $\mathbb{K}$, respectively. A BE scheme is then said to be homomorpic if the following conditions hold for all $S \subseteq [1,N]$ for $|S| \leq n$ and all $i \in S$:
\begin{enumerate}
\item If $(PK_1, SK_1) \leftarrow $\texttt{BSetup}$(n,N)$, where BSetup is the setup algorithm for the BE scheme, \vsp{2mm}
\item If $(PK_1, SK_1) \leftarrow $\texttt{BSetup}$(n,N)$, where BSetup is the setup algorithm for the BE scheme, $n$ is the size of the receiver set and it is allowed to be of size $N$, \vsp{2mm}
$(PK_2, SK_2) \leftarrow $\texttt{BSetup}$(n,N)$, \vsp{2mm}
$(d_1(i) \la $ \texttt{BKeyGen}$(i, SK_1)$, \vsp{2mm}
$(d_2(i) \la $ \texttt{BKeyGen}$(i, SK_2)$, \vsp{2mm}
@ -435,16 +476,17 @@ Within Figure \ref{fig:KHBEMatrix}, the $PK_i$ is the public key of the BE insta
% TODO: Be very consistent in what you call the public keys of the AHBE scheme!
\subsubsection{Formal Conversion from KHBE to AHBE}
% TODO: Perhaps mention that anything with B infront of it, is something belonging to a broadcast scheme
As discussed, an AHBE scheme consist of three algorithms; \texttt{KeyGen, AHBEnc, AHBDec}. \vsp{4mm}
\hsp{5mm}\textbf{KeyGen:}\quad Let the potential receivers be a set $\{1,\dots,N\}$. Let $n \leq N$ be the maximum number of recipients within a single broadcast. For simplicity, we assume that $n = N$. Generate an instance $\pi$ of a KHBE scheme and let this be a system parameter. The KeyGen algorithm then does the following:
As discussed, an AHBE scheme consist of three algorithms; \texttt{KeyGen, AHBEnc, AHBDec}.
\begin{description}
\item[KeyGen] Let the potential receivers be a set $\{1,\dots,N\}$. Let $n \leq N$ be the maximum number of recipients within a single broadcast. For simplicity, we assume that $n = N$. Generate an instance $\pi$ of a KHBE scheme and let this be a system parameter. The KeyGen algorithm then does the following:
\begin{itemize}
\item For receiver $i \in [1,n]$, invoke the setup algorithm of the BE Scheme used by the underlying KHBE scheme; \texttt{BSetup}, to generate a public/private key pair $(PK_i, SK_i)$ for the KHBE scheme.
\item Receiver $i$ runs \texttt{BKeyGen} and obtains $d_i(j) \leftarrow \text{BKeyGen}(j,SK_i)$ for $j = 1,\dots,n$. The public key of the specific receiver $i$ in the AHBE scheme is then:
$$PK_{AHBE} = \{d_i(j) | 1 \leq i \neq j \leq n\} \cup \{PK_i\}$$ Where $PK_i$ came from the BSetup call.
\item The private key of receiver $i$ is then set to be the \emph{unpublished} $d_i(i)$.
\end{itemize} \vspace{3mm}
\hsp{5mm}\textbf{AHBEnc:}\quad Computes the header and key for a receiver set $S$ in the following way:
\end{itemize}
\item[AHBEnc] Computes the header and key for a receiver set $S$ in the following way:
\begin{itemize}
\item Pick receiver set $S \subseteq [1,n]$
\item Compute the public key of the broadcast:
@ -452,11 +494,13 @@ As discussed, an AHBE scheme consist of three algorithms; \texttt{KeyGen, AHBEnc
\item Invoke the underlying KHBE encryption algorithm BEnc$(\cdot)$ in order to compute the header of the key:
$$(Hdr, k) \la BEnc(S, PK_{\mathtt{AHBE}})$$
and send $(S, Hdr)$ to the receiver set.
\end{itemize} \vspace{3mm}
\hsp{5mm}\textbf{AHBDec:}\quad Due to the underlying KHBE scheme, the receiver $i \in S$ can compute a decryption key for the \texttt{AHBE} public key $PK_{AHBE}$ by computing:
\end{itemize}
\item[AHBDec] Due to the underlying KHBE scheme, the receiver $i \in S$ can compute a decryption key for the \texttt{AHBE} public key $PK_{AHBE}$ by computing:
$$d(i) = d_i(i) \odot\{\odot_{j \in S}^{j \neq i} d_j(i)\} = \odot_{j \in S} d_j(i)$$
As only user $\U_i$ knows $d_i(i)$ only she can compute $d(i)$. Due to the homomorphism of the KHBE scheme, $d(i)$ is a valid decryption key for the public key $PK_{AHBE}$, as long as $i \in S$. To perform this decryption, each user $\U_i$ for $i \in S$, invokes the KHBE decryption algorithm BDec$(\cdot)$;
$$k = BDec(S, i, d(i), Hdr, K) $$
\end{description}
\subsection{Proof of Security}
The security of the AHBE scheme is proven by a reduction to the underlying KHBE scheme. As such, if the underlying KHBE scheme is presumed to be secure, so should the AHBE scheme. Furthermore, the AHBE scheme has semi-static security, if the KHBE scheme has adaptive security.
@ -511,44 +555,53 @@ This transformation would have to be both randomised and OTP, as otherwise if we
Something something game where the algorithm can use this adversary to transform the header and thus the key into something else which he can then use to distinguish if the original underlying key was random or was constructed properly regarding the rest of the receiver set.
\subsection{An AHBE Implementation}
To end up with a Semi-statically secure AHBE scheme, we first need to produce an adaptively secure BE scheme which is key homomorphic. To this end, we use the scheme defined in \ref{sec:BE} coupled with the generic transformation from Semi-static to Adaptive by Gentry and Waters \cite{GentryWaters}. Note that $g, h_{i,s} \text{ for } i \in [1,n], s \in \{0,1\}$ be independent generators of a group $\mathbb{G}$ of prime order $p$, with a bilinear map $e : \Gm \times Gm \ra \Gm_{T}$. \vsp{5mm}
\-\hspace{5mm}\textbf{BSetup$(\lambda,n)$:}\quad Let $\alpha \in_R \mathbb{Z}_p$ and compute $g^\alpha, e(g,g)^\alpha$. The BE public key PK is then; $PK = e(g,g)^\alpha$ and the private key is $SK = g^\alpha$. \vspace{3mm} \\
\-\hspace{5mm}\textbf{BKeyGen$(i, SK)$:}\quad Set $r_i \in_R \mathbb{Z}_p$, $s_i \in_R \{0,1\}$. Output decryption key for user $i$; $d_i = (d_{i,0},\dots,d_{i,n})$:
$$d_i \leftarrow (d_{i,0},\dots,d_{i,n}) \quad \text{ where } \quad d_{i,0} = g^{-r_i}, \quad d_{i,i} = g^\alpha h^{r_i}_{i,s_i}, \quad d_{i,j \text{ for } i\neq j} h^{r_i}_{j,s_i}$$ \vspace{3mm} \\
\-\hspace{5mm}\textbf{BEnc$(S, PK)$:}\quad Set $t \in_R \Z_p$ and $$Hdr = (C_1,C_2, C_3), \quad \text{ where }\quad C_1 = g^t, \quad C_2 = (\prod_{i \in S}h_{i,0})^t,\quad C_3 = (\prod_{i \in S}h_{i,1})^t $$ Finally, set $K = e(g,g)^{t\cdot \alpha}$. Output $(\hdr, K)$. Send $(S, \hdr)$ to the receivers. \vspace{3mm} \\
\-\hspace{5mm}\textbf{BDec}$(S,i,d_i,\text{Hdr}, PK)$\textbf{:}\quad Check if $i \in S$, if so; let $d_i = (d_{i,0},\dots,d_{i,n})$, Hdr$=(C_1,C_2,C_3)$, output $$k =e(d_{i,i} \cdot \prod_{j \in S \setminus \{i\}} d_{i,j}, C_1) \cdot e(d_{i,0}, C_2)$$ \vsp{3mm}
The correctness is the exact same as defined in Section \ref{sec:GentryWatersConst}.
To end up with a Semi-statically secure AHBE scheme, we first need to produce an adaptively secure BE scheme which is key homomorphic. To this end, we use the scheme defined in \ref{sec:BE} coupled with the generic transformation from Semi-static to Adaptive by Gentry and Waters \cite{GentryWaters}. Note that $g, h_{i,s} \text{ for } i \in [1,n], s \in \{0,1\}$ be independent generators of a group $\mathbb{G}$ of prime order $p$, with a bilinear map $e : \Gm \times Gm \ra \Gm_{T}$.
As we desire a key homomorphic scheme, we define the aggregations like so; $PK_1 \oplus PK_2 = PK_1PK_2$, $d_{1_i} \odot d_{2_i} = (d_{1_{i,0}}, d_{2_{i,0}}, \dots, d_{1_{i,n}}, d_{2_{i,n}})$ and $k_1 \ocircle k_2 = k_1k_2$. Finally we instantiate the AHBE scheme: \vsp{4mm}
\hsp{5mm}\textbf{KeyGen:}\quad Let the potential receivers be a set $\{1,\dots,N\}$. Let $n \leq N$ be the maximum number of recipients within a single broadcast. For simplicity, we assume that $n = N$. Generate an instance $\pi$ of a KHBE scheme and let this be a system parameter. The KeyGen algorithm then does the following:
We define all algoritms prefixed by $AB$ to be an \emph{adaptively secure} \texttt{BE} algorithm.
\begin{description}
\item[ABSetup$(n, \ell)$] Let $\alpha \in_R \mathbb{Z}_p$ and compute $g^\alpha, e(g,g)^\alpha$. The BE public key PK is then; $PK = e(g,g)^\alpha$ and the private key is $SK = g^\alpha$.
\item[ABKeyGen$(i, SK)$] Set $r_i \in_R \mathbb{Z}_p$, $s_i \in_R \{0,1\}$. Output decryption key for user $i$; $d_i = (d_{i,0},\dots,d_{i,n})$:
$$d_i \leftarrow (d_{i,0},\dots,d_{i,n}) \quad \text{ where } \quad d_{i,0} = g^{-r_i}, \quad d_{i,i} = g^\alpha h^{r_i}_{i,s_i}, \quad d_{i,j \text{ for } i\neq j} h^{r_i}_{j,s_i}$$
\item[ABEnc$(S, PK)$] Set $t \in_R \Z_p$ and $$Hdr = (C_1,C_2, C_3), \quad \text{ where }\quad C_1 = g^t, \quad C_2 = (\prod_{i \in S}h_{i,0})^t,\quad C_3 = (\prod_{i \in S}h_{i,1})^t $$ Finally, set $K = e(g,g)^{t\cdot \alpha}$. Output $(\hdr, K)$. Send $(S, \hdr)$ to the receivers.
\item[ABDec$(S,i,d_i,\text{Hdr}, PK)$] Check if $i \in S$, if so; let $d_i = (d_{i,0},\dots,d_{i,n})$, Hdr$=(C_1,C_2,C_3)$, output $$k =e(d_{i,i} \cdot \prod_{j \in S \setminus \{i\}} d_{i,j}, C_1) \cdot e(d_{i,0}, C_2)$$
The correctness is the exact same as defined in Section \ref{sec:GentryWatersConst}.
\end{description}
As we desire a key homomorphic scheme, we define the aggregations like so; $PK_1 \oplus PK_2 = PK_1PK_2$, $d_{1_i} \odot d_{2_i} = (d_{1_{i,0}}, d_{2_{i,0}}, \dots, d_{1_{i,n}}, d_{2_{i,n}})$ and $k_1 \ocircle k_2 = k_1k_2$. Finally we instantiate the AHBE scheme:
% TODO: Fix it so that we are consistent with key of AHBE and user and BE
\begin{description}
\item[KeyGen$(i, n, N )$] Let the potential receivers be a set $\{1,\dots,N\}$. Let $n \leq N$ be the maximum number of recipients within a single broadcast. For simplicity, we assume that $n = N$. Generate an instance $\pi$ of a KHBE scheme and let this be a system parameter. The KeyGen algorithm then does the following:
\begin{itemize}
\item For receiver $i \in [1,n]$, invoke the \texttt{BSetup}, to generate a public/private key pair $(PK_i, SK_i) = e(g,g)^{\alpha_i}, g^{\alpha_i}$ for the KHBE scheme..
\item Receiver $i$ runs \texttt{BKeyGen} and obtains $d_i(j) \leftarrow \text{BKeyGen}(j,SK_i)$ for $i,l,j = 1,\dots,n$ where $d_i(j) = (d_{i,0,j}, \dots, d_{i,n,j})$ such that: \\
\item For receiver $i \in [1,n]$, invoke the \texttt{ABSetup}, to generate a public/private key pair $(PK_i, SK_i) = e(g,g)^{\alpha_i}, g^{\alpha_i}$ for the KHBE scheme..
\item Receiver $i$ runs \texttt{ABKeyGen} and obtains $d_i(j) \leftarrow \mathtt{ABKeyGen}(j,SK_i)$ for $i,l,j = 1,\dots,n$ where $d_i(j) = (d_{i,0,j}, \dots, d_{i,n,j})$ such that: \\
$$d_{i,0,j} = g^{-r_{i,j}},\quad d_{i,j,j} = g^{\alpha_i}h^{r_{i,j}}_{j,s_i}, \quad d_{i,l,j} = h^{r_{i,j}}_{l,s_i},$$ \\
For $r_{i,j} \in_R \mathbb{Z}_p$, $s_i \in_R \{0,1\}$. Receiver $i$'s private key is then $d_i(i)$. \\
\item The public key of the specific receiver $i$ in the AHBE scheme is then: \\
$$PK_{AHBE_i} = \{d_i(j) | 1 \leq i \neq j \leq n\} \cup \{PK_i\}$$ Where $PK_i$ came from the BSetup call.
\end{itemize} \vspace{3mm}
\hsp{5mm}\textbf{AHBEnc:}\quad Computes the header and key for a receiver set $S$ in the following way:
\end{itemize}
\item[AHBEnc$(S, (PK_i)_S)$] Computes the header and key for a receiver set $S$ in the following way:
\begin{itemize}
\item Pick receiver set $S \subseteq [1,n]$
\item Compute the public key of the broadcast:
$$PK_{AHBE} = \oplus_{i \in S} PK_i = \prod_{i \in S} PK_i = e(g,g)^{\sum_{i \in S} \alpha_i}$$
Note that the $PK_i$'s used here are in fact the ones from the original \texttt{BSetup} call, so it is contained within $PK_{AHBE_i}$.
\item Invoke the underlying KHBE encryption algorithm BEnc$(\cdot)$ in order to compute the header of the key $\hdr = BEnc(S, PK_{AHBE}) = (C_1,C_2,C_3)$ for:
Note that the $PK_i$'s used here are in fact the ones from the original \texttt{ABSetup} call, so it is contained within $PK_{AHBE_i}$.
\item Invoke the underlying KHBE encryption algorithm BEnc$(\cdot)$ in order to compute the header of the key $\hdr = \mathtt{ABEnc}(S, PK_{AHBE}) = (C_1,C_2,C_3)$ for:
$$C_1 = g^t, \quad C_2 = (\prod_{i \in S}h_{i,0})^t,\quad C_3 = (\prod_{i \in S}h_{i,1})^t$$
and for the secret key:
$$k = PK_{AHBE} = e(g,g)^{t \cdot \sum_{i \in S} \alpha_i}$$
$$k = PK_{AHBE}^t = e(g,g)^{t \cdot \sum_{i \in S} \alpha_i}$$
for $t \in_R \mathbb{Z}_p$
and send $(S, \hdr)$ to the receiver set.
\end{itemize} \vspace{3mm}
\hsp{5mm}\textbf{AHBDec:}\quad Due to the underlying KHBE scheme, the receiver $i \in S$ can compute a decryption key for the \AHBE public key $PK_{AHBE}$ by computing:
\end{itemize}
\item[AHBDec$(S, j, sk_j, \hdr, (PK_i)_S)$] Due to the underlying KHBE scheme, the receiver $i \in S$ can compute a decryption key for the \AHBE public key $PK_{AHBE}$ by computing:
\begin{align*}
d(i) &= d_i(i) \odot\{\odot_{j \in S}^{j \neq i} d_j(i)\} = \odot_{j \in S} d_j(i) \\
&= (\prod_{j \in S} d_{j,0,i}, \dots, \prod_{j \in S} d_{j,n,i})
\end{align*}
As only user $\U_i$ knows $d_i(i)$ only she can compute $d(i)$. Due to the homomorphism of the KHBE scheme, $d(i)$ is a valid decryption key for the public key $PK_{AHBE}$, as long as $i \in S$. To perform this decryption, each user $\U_i$ for $i \in S$, invokes the KHBE decryption algorithm BDec$(\cdot)$;
$$k = BDec(S, i, d(i), Hdr, K) $$
As only user $\U_i$ knows $d_i(i)$ only she can compute $d(i)$. Due to the homomorphism of the KHBE scheme, $d(i)$ is a valid decryption key for the public key $PK_{AHBE}$, as long as $i \in S$. To perform this decryption, each user $\U_i$ for $i \in S$, invokes the KHBE decryption algorithm \texttt{ABDec}$(\cdot)$;
$$k = \mathtt{ABDec}(S, i, d(i), Hdr, PK_{AHBE}) $$
\end{description}
\subsection{Attempt at Reducing the \AHBE Instantion to BDHE-Problem}
Seeing that the reduction had some non-salveable issues regarding the decryption keys of the target set $S^*$, we attempted to reduce their instantiation directly to the BDHE problem, which the original scheme due to Gentry and Waters was originally reduced to, to prove its Semi-static security. We recall why the original reduction worked: The values $h_1, \dots, h_n$ are originally picked completely at random from the target group of the bilinear map, $\Gm_T$, which allowed the original reduction to sample $y_1, \dots, y_n$ and lift the generator of the group $\Gm$, $g$, to specific values of $y_i$, whenever we needed to know the discrete log of $h_i$, specifically when $i \in \tilde{S}$, i.e. the set of potential receivers, $h_i = g^{y_i}$. Furthermore, for the rest of the users, $i \not\in \tilde{S}$, they generated the values of $h_i = g^{y_i + a^i}$ meaning that the adversary \adv{B} could in fact not compute the discrete log and would thus not have a chance of computing the header information, if the adversary \adv{A} decided to attack this user. Due to the semi-static nature however, this is not something they have to worry of, as \adv{A} has already commited to $\tilde{S}$. The definition of the $h_i$ for $i \not\in \tilde{S}$, means that \adv{B} can properly answer the extraction queries for these users, as \adv{B} defines the values $r_i$ in such a way, that the exponents cancels out in $d_{i,i} = g^{\alpha}h^{r_i}_i$ and we do not have to bother trying to compute the discrete log of $g^\alpha$, specically the $a^{n+1}$ part of $\alpha = y_0 \cdot a^{n+1}$. The issues then arise, as all the $h_i$ values are required for the \AHBE scheme, essentially meaning we can not fake some and define some in a very specific way, as they are \emph{all} used for the different keys, regardless of the user $i$ being in the attack set $i \in \tilde{S}$, as all the users are using the same underlying KHBE scheme. This results in the algorthim \adv{B} not being capable of answering extraction queries for any user i outside of the attack set, $i \not\in \tilde{S}$, as \adv{B} also has to generate all the $h$ values in such a way that he can compute the discrete log.
@ -563,11 +616,24 @@ As such, we conclude that, if there is a reduction to be found from the \AHBE in
\nocite{*}
\bibliography{refs}
\newpage
% https://tex.stackexchange.com/questions/49643/making-appendix-for-thesis
\begin{appendices}
\includepdf[pages=1,pagecommand={\section{\texttt{IBE} Security Proof} \label{app:IBE-Sec} },width=\textwidth]{papers/IBESecProof.pdf}
\includepdf[pages=2-,pagecommand={},width=\textwidth]{papers/IBESecProof.pdf}
% TODO: Properly crop the construction
\chapter{\texttt{DTPKE}-scheme}
\includepdf[pages=-,pagecommand={},width=\textwidth]{papers/DTPKE-Const.pdf}
\includepdf[pages=1,pagecommand={\section{\texttt{DTPKE} Scheme} \label{app:DTPKE-Scheme}},width=\textwidth]{papers/DTPKE-Const.pdf}
\includepdf[pages=2-,pagecommand={},width=\textwidth]{papers/DTPKE-Const.pdf}
\end{appendices}
\end{document}