## 3 Mathematical Preliminaries
### Access Structure for Privacy Preservation  
Here we first define normal access structure. **Definition 2**. : _Let us name the universe of attributes \(A\)U. An access structure \(\varphi\) on \(A\)U is a collection of non-empty attribute sets, i.e., \(\varphi\)\(\subseteq\)\(2^{A\)U\(}\backslash\)\(\varnothing\). The collection of attribute sets, which is present in \(\varphi\), is called the authorized set; all the other attributes lie in an unauthorized set. In addition, an access structure is called monotonic if \(\forall\)C, \(D\): if \(C\in\varphi\) and \(C\subseteq D\), then \(D\in\varphi\)._  
Now let us define linear secret sharing scheme (LSSS). **Definition 3**. : _Let \(\Pi\) be a secret-sharing scheme in which:_  
1. _The generated share for each participant is a vector over_ \(Z_{p}\)_._
2. \(\exists\) _A matrix_ \(W\) _of_ \(m\) _rows and_ \(n\) _columns, where_ \(\forall\)_rows_ \(\in\) _W, the_ \(j^{th}\) _row is marked with the function_ \(\rho(j)\)_; then it is called a linear scheme. Secret_ \(s\) _is randomly chosen such that_ \(s\) \(\in\)__\(Z_{p}\) _and a vector is formed so that_ \(v\) _=_ \((s,\ v_{2}\ \ldots\ v_{n})\)__\(\in\)__\(Z_{p}^{n}\)_. Now let us take_ \(\lambda\)_=_ \(W\)__\(v^{T}\) _such that share_ \(\lambda_{j}\) _is for participant_ \(\rho(j)\) _so we can write_ \(\lambda_{j}\) _=_ \(W_{j}\)_._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multirow{2}{*}{**CP/KP**} & \multirow{2}{*}{
\begin{tabular}{c} **Multi-** \\ **Authority** \\ \end{tabular} } & \multirow{2}{*}{**Privacy-Aware**} & \multirow{2}{*}{**Expressiveness**} & \multirow{2}{*}{**Security**} & \multirow{2}{*}{**Group**} & \multirow{2}{*}{**Order**} & \multirow{2}{*}{**Security Model**} \\ \hline  & & & & & & & \\ \hline \multicolumn{7}{c}{**Multi-Authority Schemes**} \\ \hline
[16] & CP & \(\surd\) & \(\times\) & LSSS & Adaptive & Composite & Random oracle \\ \hline
[23] & CP & \(\surd\) & \(\times\) & LSSS & Adaptive & Composite & Standard \\ \hline
[26] & CP & \(\surd\) & \(\times\) & LSSS & Adaptive & Composite & Standard \\ \hline
[25] & KP & \(\surd\) & \(\times\) & LSSS & Adaptive & Composite & Standard \\ \hline \multicolumn{7}{c}{**Multi-Authority Privacy Preserving Schemes**} \\ \hline
[35] & CP & \(\surd\) & \(\surd\) (Fully hidden policy) & LSSS & Selective & Prime & Random oracle \\ \hline
[39] & CP & \(\surd\) & \(\surd\) (Partially hidden policy) & LSSS & Selective & Prime & Standard \\ \hline \multicolumn{7}{c}{**Multi-Authority Privacy Preserving (Partially Hidden policy) Scheme with Adaptive Security in Standard Model**} \\ \hline
**Ours** & CP & \(\surd\) & \(\surd\) (Partially hidden policy) & LSSS & Adaptive & Composite & Standard \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont.__Linear reconstruction property: Let us denote \(S\) as an authorized set and take \(I=\{\)j: \(\rho(j)\)\(\in\)\(S\}\). For an LSSS scheme, there exists a constant set \(\left(\mu_{j}\in Z_{p}\right)_{j\in I^{\prime}}\), used to compute the secret \(s\): \(\sum_{j\in I}\mu_{j}\lambda_{j}=s\). Access Structure for privacy preservation:_  
**Definition 4**. : _Let us take an access structure \(\varphi\) = (\(W\), \(\rho\), \(Z\)) for describing an access policy. \(W\) is a share-generating matrix with the dimensions \(l\) by \(n\) that is connected with a secret sharing scheme, \(\rho\) is a map function and \(Z=\left(Z_{\rho(j)}\right)_{1\leq j\leq I}\) is a set of corresponding possible values of the respective attribute. Function \(\rho\) maps each row of \(W\) to the name of the attribute present in the access policy. We are keeping \(Z\) hidden in our scheme and the share-generating matrix \(W\) and function \(\rho\) are attached to the ciphertext._  
**Definition 5**. : _In discrete mathematics, a canonical normal form of a Boolean formula can be written as OR of ANDs. It is termed the sum of products (SOP). This normal form is called a disjunctive normal form (DNF).