Nt1330 Unit 1 Test Paper

\section{Building Blocks}

\subsection{Access Structures}

\textbf{Definition 3.8.}(Access Structure\citeup{beimel1996secure}) \emph{Let $\{P_1, P_2,...,P_n\}$ be a set of parties. A collection $\mathbb{A}\subseteq 2^{\{P_1,P_2,...,P_n\}}$ is monotone if $B\in\mathbb{A}$ and $B\subseteq C$ implies $C\in\mathbb{A}$. An access structure is a monotone collection $\mathbb{A}$ of non-empty subsets of $\{P_1,P_2,...,P_n\}$, i.e., $\mathbb{A}\subseteq 2^{\{P_1,P_2,...,P_n\}} \setminus\{\emptyset\}$. The sets in $\mathbb{A}$ are called the authorized sets, and the sets not in $\mathbb{A}$ are called the unauthorized sets}.

In our settings, attributes will play the role of the parties such that the access structure $\mathbb{A}$ will contain the authorized

\end{align}

Observe that $p(0)=s$. Each party $P_i$ is given by the share $p(i)$ which is a linear combination of the random inputs and the secret. Therefore, any group of $t$ parties can reconstruct the secret $s$ by computing the Lagrange interpolation formula (described below) in which $x$ is substituted by $0$.

\subsection{Lagrange Interpolation}

\emph{Let $i\in \mathbb{Z}$ and $S\subseteq\mathbb{Z}$, the Lagrange basis polynomial is defined as $\bigtriangleup_{i,S}(x)=\prod\limits_{j\in S,j\neq i}(\frac{x-j}{i-j})$. Let $f(x)\in\mathbb{Z}[x]$ be a $d^{th}$ degree polynomial. If $|S|=d+1$, from a set of $d+1$ points ${(i,f(i))}_{i\in S}$, one reconstruct $f(x)$ as follows:\\}

\begin{align}\label{equ:3-1}

f(x)=\sum\limits_{i\in S}f(i)\cdot{\bigtriangleup}_{i,S}(x).

\end{align}

\subsection{Access

More precisely, an access structure is expressed by an access tree ${\cal T}$ where every non-leaf node $x$ has assigned a threshold gate and every leaf $x$ is assigned a party $P_i \in \{P_1, P_2,...,P_n\}$ . A threshold gate is described by its children $n_x$ and a threshold value $t_x$, where $0iteup{di2007over} is a technique that makes use of a two-layer encryption to enforce selective encryption without requesting the data owner to re-encrypt the data every time there is a change in the AC policy. In the over-encryption technique, two layers of encryption are imposed on data: the inner layer is applied by the data owner to provide initial protection, and the outer layer is done by the server to reflect access control policy changes. Intuitively, this technique allows data owners outsource, besides their data, authorization and revocation tasks on their data to a semi-trusted server without revealing the underlying data to the