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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 %%% ### ### %%%
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%%%%%% Ensure that you do not write the questions before each of the solutions because it is not necessary. %%%%%%
\documentclass[12pt,a4paper]{article}
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\usepackage{amsmath}
\usepackage{fancyhdr}
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\usepackage{hyperref}
\usepackage{caption}
\usepackage{subcaption}
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\newcommand{\student}{\textbf{Azamuke Denish}}
\newcommand{\course}{\textbf{Demo Template for Msc. CS}}
\newcommand{\assignment}{\textbf{1}}
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\usepackage{lipsum}%% a garbage package you don't need except to create examples.
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%\lhead{Azamuke Denish}
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%\cfoot{\textbf{AIMS Rwanda Academic Year 2020 - 2021}}
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\begin{document}
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\begin{center}
\includegraphics[scale = 0.6]{cs4.png}
%\textbf{AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES \\[0.5cm]
%(AIMS RWANDA, KIGALI)}
\vspace{0.5cm}
\end{center}
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\noindent
\rule{17cm}{0.2cm}\\[0.3cm]
Name: \student \hfill Assignment Number: \assignment\\[0.1cm]
Course: \course \hfill Date: \today\\
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\vspace{1.0cm}
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\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}