The PKG performs the following steps
- Choose $p,q \in \mathbb{P}$.
- Calculate $N=pq$.
- Calculate $\phi (n)=(p-1)(q-1)$.
- Choose $e$ with $gcd(e,\phi(n))=1$ and $1 < e < \phi(n)$.
- Let it be $e = {p^{e_1}_1} \cdot {p^{e_2}_2} \cdot \ldots
{p^{e_k}_k}$ the prime factorization of $e$ for $i \in k:p_i \in \mathbb{P},e_i \in \mathbb{N}$.
Choose an injective mapping $H$ with
\begin{align*}
H &: \begin{cases}
\{0,1\}^i \rightarrow \mathbb{Z} / N \mathbb{Z} & \\
ID \mapsto m = {p^{e_{m_1}}_1} \cdot {p^{e_{m_2}}_2} \cdot \ldots {p^{e_{m_k}}_k} & (i \in k:p_i \in \mathbb{P},e_{m_i} \in
\mathbb{N})
\end{cases}
\end{align*}
and $eH(ID)<\phi(n)$ for $i \in \mathbb{n}$. The publicly available parameters are $\texttt{params} = \langle e, N, H \rangle$ and the $\texttt{master-key}$ is $\phi(n) \in \mathbb{Z} / N \mathbb{Z}$.
The PKG takes then an $ID \in \{0,1\}^{*}$ (from Alice) and calculates the corresponding Secret Key $d_{ID}$ with
\begin{align*}
(e H(ID)) d_{ID} \equiv 1 \text{ mod } \phi(n)
\end{align*}
When Bob wants to encrypt a message $m \in \mathbb{Z} / N \mathbb{Z}$, he takes $\texttt{params}$ and calculates
\begin{align*}
c \equiv m^{e H(ID)} \text{ mod } N
\end{align*}
Alice decrypts this ciphertext $c$ with
\begin{align*}
m \equiv c^{d_{ID}} \text{ mod } N
\end{align*}
EXAMPLE
$p = 1010231362240711373894507355467 \in \mathbb{P}$ and
$q = 793738224882014450642935586909 \in \mathbb{P}$.
$N=pq=801859248185081566400631735533731882269717325788593134781503$
$\phi(N) = 2^3 \cdot 31 \cdot 283 \cdot 29347 \cdot 39547129
\cdot 422250739 \cdot 1354514929 \cdot 17211833615713895353775639$.
$e = 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23
\cdot 29$.
It apllies $ID \in \{0,1\}^8$ with $ID=\langle b_1,b_2,\ldots,b_8
\rangle$ for $i \in 8:b_i \in \{0,1\}$. Choose $H$ as:
\begin{align*}
H &: \begin{cases}
\{0,1\}^8 \rightarrow \mathbb{Z} / N \mathbb{Z} & \\
ID \mapsto m = {5^{{b_1}}} \cdot {7^{{b_2}}} \cdot \ldots \cdot {29^{{b_8}}} &
\end{cases}
\end{align*}
The publicly available parameters are
\begin{align*}
\texttt{params} &= \langle 1078282205, 801859248185081566400631735533731882269717325788593134781503, H \rangle
\end{align*}
The $\texttt{master-key}$ is
\begin{align*}
\phi(N) &= 801859248185081566400631735531927912682594599964055691839128
\end{align*}
The PKG takes then $ID = 01101111$ as the ID of user "o". Then $H(ID) = 5^0 \cdot 7^1 \cdot 11^1 \cdot 13^0 \cdot 17^1 \cdot 19^1 \cdot 23^1 \cdot 29^1 = 16588957$, $eH(ID)=17887577132610185$ and $d_{ID}=308315206989333722335381678529602981822693965290742774973561$.
User "i" wants now to encrypt the message 3463463463463424234234234. He calculates
\begin{align*}
c &\equiv 3463463463463424234234234^{17887577132610185} \text{ mod N} \\
&\equiv 353097511425650359803351296367609508451542189692844760010085 \text{ mod N}
\end{align*}
User "o" decrypt the ciphertext with:
\begin{align*}
m &\equiv 353097511425650359803351296367609508451542189692844760010085^{D_{ID}} \text{ mod N} \\
&\equiv 3463463463463424234234234 \text{ mod N}
\end{align*}
