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\centerline{\textbf{HW 5 CMSC 456. DUE Oct 16}}

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{\textbf{NOTE- THE HW IS FOUR PAGES LONG}}


\begin{enumerate}
\item
(0 points)
READ the syllabus- Content and Policy.
What is your name?  Write it clearly.
What is the day and time of the first midterm?
Read slides on Dr. Mazurek's lecture.

\item
(25 points)
Write a simple program which does the following:
\begin{enumerate}
\item INPUT: A key K, a nonce N, and a text string M
\item OUTPUT: Ciphertext corresponding to M encrypted under AES256-GCM (i.e. the AES algorithm with key length 256 in GCM mode) with K as the key and N as the IV.
\end{enumerate}
Do this two ways and WRITE IN ENGLISH
the contrast of experience:
Include your code, an input of your choice, and the corresponding output.
You have TWO choices:

\noindent
I) Do both in PYTHON:

\begin{enumerate}
\item
Crytography library on the hw website, and
\item
PyCrypto on the hw website
\end{enumerate}

\noindent
II) Do both in C (which would be harder)

\begin{enumerate}
\item
C via OpenSSL on the hw website, and
\item
libsodium on the hw website
\end{enumerate}

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\item
(20 points) 
Let $N=pq$ where $p,q$ are primes.
Let $m \in \{2,\ldots,N-1\}$.
\begin{enumerate}
\item
(4 points)
Exactly how many multiplications do you need to compute $m^{2^{16}+1}$
using repeated squaring.
\item
(4 points)
Exactly how many multiplications do you need to compute $m^{2^{16}-1}$
using repeated squaring.
\item
(0 points, this is just here for information)
If you did the last two problems right then 
$m^{2^{16}+1}$ took MUCH LESS mults then 
$m^{2^{16}-1}$. This is one reason why $e=m^{2^{16}+1}$ is so popular in RSA.
\item
(4 points) $2^{16}+1$ is prime. Is $2^{32}+1$ prime? If not then give its factors.
(HINT- look up Fermat Primes on the web)
\item
(4 points) Why is choosing $e$ to be prime a good thing to do?
\item
(4 points) I had said in class that we do not want to pick $e$ too low.
Roughly how big does $N$ have to be before picking 
$e=2^{16}+1$ is a bad thing to do.
How does this $N$ compare to the number of protons in the universe?
(Look up Eddington's Number on the web)
\end{enumerate}


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\item
(25 points) 
(HINT --- look up the Chinese Remainder Theorem.)
Give an algorithm (psuedocode but more descriptive) for the following:

{\bf Input:} $N_1,\ldots,N_L$, $x_1,\ldots,x_L$ where $N_1,\ldots,N_L$ are rel prime.

{\bf Output:} An $x$ such that

$x\equiv x_1 \pmod {N_1}$

$x\equiv x_2 \pmod {N_2}$

$\vdots$

$x\equiv x_L \pmod {N_L}$

AND $0\le x < N_1\cdots N_L$.

You can assume you have a program that finds inverses of numbers in mods if
they exist.
% Note that since all of the $N_i$ are rel prime, for all $i$ there exists
% a number which you can denote $N_i^{-1}$ which is the inverse of $N_i$
% mod $N_1N_2\cdots N_{i-1}N_{i+1}\cdots N_L$.
% (In the solution you should denote 
% $N_1N_2\cdots N_{i-1}N_{i+1}\cdots N_L$ by $M_i$.)

Note that since all of the $N_{i}$ are rel prime, for all $i$ there
exists a number which you can denote $M_{i}^{-1}$ which is the inverse
of $M_{i} \mod N_{i}$, where
$M_{i}=N_{1}N_{2}\dots N_{i-1}N_{i+1}\dots N_{L}$.


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\item
(30 points)
(Read the slides on low-exponent attacks on RSA.)
Before getting to the specs of the psuedocode you are to write,
here is the setting.
\begin{itemize}
\item
Zelda will do RSA with $L$ people $A_1,\ldots,A_L$.
\item
Zelda is using RSA as follows: For person $A_i$ she uses $(e,N_i)$.
\item
The $N_i$ are all relatively prime.
\item
$N_1<\cdots<N_L$.
\item
The parameter $e$ -- we think of it as being small but the algorithm should
run even if $e$ is not small. It may report back NO could not crack.
\item
We assume that Zelda sent the same message to everyone.
The message is $m$. So she send $A_i$ the number $m^e \mod {N_i}$.
\item
You are Eve. You already have a program that will do the Chinese Remainder Theorem.
That is, you have a program that will, on input
$x_1,\ldots,x_L$, $N_1,\ldots,N_L$ where the $N_i$'s are rel prime, output
$x$ such that, for all $1\le i\le L$, $x\equiv x_i \pmod {N_i}$.
\end{itemize}

NOW YOUR ASSIGNMENT:

Write a program such that
\begin{enumerate}
\item
{\bf Input:} $e$, $N_1<\ldots<N_L$ and $c_1,\ldots,c_L$.
The $N_i$ are all rel prime.
There is an $m$ such that,
for all $1\le i\le L$, $c_i=m^e \pmod {N_i}$.
\item
{\bf Output:} Either find $m$ as in the example in class OR say that you can't find $m$
Prove that if $e\le L$ then your algorithm does find $m$.
\end{enumerate}


\end{enumerate}

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