```
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%%% Template for AIMS Rwanda Assignments %%% %%%
%%% Author: AIMS Rwanda tutors %%% ### %%%
%%% Email: tutors2017-18@aims.ac.rw %%% ### %%%
%%% Copyright: This template was designed to be used for %%% ####### %%%
%%% the assignments at AIMS Rwanda during the academic year %%% ### %%%
%%% 2017-2018. %%% ######### %%%
%%% You are free to alter any part of this document for %%% ### ### %%%
%%% yourself and for distribution. %%% ### ### %%%
%%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Ensure that you do not write the questions before each of the solutions because it is not necessary. %%%%%%
\documentclass[12pt,a4paper]{article}
%%%%%%%%%%%%%%%%%%%%%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{graphicx}
\usepackage{tabulary}
\usepackage{amsmath}
\usepackage{fancyhdr}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{placeins}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{verbatim}
\usepackage[left=2cm,right=2cm,top=3cm,bottom=2.5cm]{geometry}
\usepackage{hyperref}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{multirow}
\usepackage{psfrag}
%%%%%%%%%%%%%%%%%%%%% students data %%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\student}{\textbf{Azamuke Denish}}
\newcommand{\course}{\textbf{Demo Template for Msc. CS}}
\newcommand{\assignment}{\textbf{1}}
%%%%%%%%%%%%%%%%%%% using theorem style %%%%%%%%%%%%%%%%%%%%
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{defn}[thm]{Definition}
\newtheorem{exa}[thm]{Example}
\newtheorem{rem}[thm]{Remark}
\newtheorem{coro}[thm]{Corollary}
\newtheorem{quest}{Question}[section]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{lipsum}%% a garbage package you don't need except to create examples.
\usepackage{fancyhdr}
\pagestyle{fancy}
%\lhead{Azamuke Denish}
\rhead{ \thepage}
%\cfoot{\textbf{AIMS Rwanda Academic Year 2020 - 2021}}
\renewcommand{\headrulewidth}{0.4pt}
\renewcommand{\footrulewidth}{0.4pt}
%%%%%%%%%%%%%% Shortcut for usual set of numbers %%%%%%%%%%%
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%555
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%% title page %%%%%%%%%%%%%%%%%%%%%%%%%%
\thispagestyle{empty}
\begin{center}
\includegraphics[scale = 0.6]{cs4.png}
%\textbf{AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES \\[0.5cm]
%(AIMS RWANDA, KIGALI)}
\vspace{0.5cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%% assignment information %%%%%%%%%%%%%%%%
\noindent
\rule{17cm}{0.2cm}\\[0.3cm]
Name: \student \hfill Assignment Number: \assignment\\[0.1cm]
Course: \course \hfill Date: \today\\
\rule{17cm}{0.05cm}
\vspace{1.0cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Problem 1}
Given that \textbf{Airqo}, \textit{Africa's leading air quality monitoring, research and analytics network company} has used its public RSA key $(n, e)$ for years. After a security check they had to change it to $(n, e')$, with the same $n$ but with a different number $e'$ which is relatively prime to $e$. A customer had previously sent his message $\overline{a}$ which was encoded with the old key. After he got the news of the security check he encodes this same message $\overline{a}$ with the new public key. \\How can an attacker get
$\overline{a}$ from the knowledge of the old and new encrypted message $\overline{c_1}$ and $\overline{c_2}$ respectively
using only the public keys? You are required to evaluate this for the example where $n = 247, e = 11, e' = 17, c_1 = 24, c_2 = 93.$\\\\
\newpage
\begin{thebibliography}{99}
\bibitem{am} Bitter, R., Mohiuddin, T., \& Nawrocki, M. (2017). LabVIEW: {\em Advanced programming techniques.\em} CRC press.
\end{thebibliography}
\end{document}
```