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\begin{document}

\centerline{\textbf{HW 7 CMSC 456. DUE Nov 5}}
\centerline{\textbf{Will Be Graded By Nov 12}}
\centerline{\textbf{Regrade Requests Due by Nov 19 NO DEAD CAT POLICY}}

\ifshowsoln
\centerline{\textbf{SOLUTIONS}}
\fi

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


\begin{enumerate}
\item
(0 points)
READ the syllabus- Content and Policy.
What is your name?  Write it clearly.


\item
(30 points)
Look up the Euclidean Algorithm to find the Greatest Common Divisor of two numbers.
\begin{enumerate}
\item
(5 points)
Write a program to implement the algorithm.
\item
(5 points)
Modify it (actually add to it) so that if it is given $(a,b)$ which are relatively
prime it finds the inverse of $a$ mod $b$.
\item
(10 points)
Use your program to find the GCD of all $(a,b)$ such that $50\le a < b \le 60$, and
\item
(10 points) 
Use your program to find, for all $1\le x\le 100$, the inverse of $x$ mod 101.
\end{enumerate}

\item
(30 points)
Dr. Batz is trying to factor a large number $N$ using the method that 
worked on the Jevon's Number. So he wants to find $x,y$ such that
$x^2-y^2=N$. Instead he finds $x,y$ such that

$$x^2-y^2 \equiv 0 \pmod N$$

\begin{enumerate}
\item
Tell her how she might be able to use this.
\item
Will there be scenarios where $x,y$ do not help?
\end{enumerate}



\centerline{\bf THERE IS ANOTHER PAGE. GOTO IT!!!!!!!!!}

\newpage

\item
(20 points) Read the slides for Misc Crypto where I talk about making the Vig Cipher better.
We will be using that method throughout this question.
FOR THIS PROBLEM WE USE $A=1$, $B=2$, etc, $Z=26$.

\begin{enumerate}
\item
Alice says to Bob {\it NSF Good}.
Give the resulting key.
(It will be a sequence of numbers)
  \item
Alice says to Bob {\it Problems with a Point}.
How long is the resulting key? 
(You DO NOT need to find the key.)
\item
Give two phrases $p_1$ and $p_2$ (in English) where
$p_1$ has LESS letters than $p_2$, but using $p_1$ results in a LONGER key.
\end{enumerate}


\item
(20 points)
Alice and Bob are bridge partners.
And they cheat! Here is their scheme:
\begin{itemize}
\item
If the first card is placed horizontally then the person placing it
has 0 or 1 Ace.
\item
If the first card is placed Vertically then the person placing it
has 2 or 3  or 4 Aces.
\end{itemize}
In this problem we will both Alice and Bob and also help
the bridge community.
\begin{enumerate}
\item
Alice and Bob will be playing 20 games and are worried that their
cheating may be discovered. Show how they can use a 1-time pad
to make their cheating harder to discover.
\item
Change something about how Bridge is played so that Alice and
Bob cannot use their method to cheat.
\end{enumerate}
\end{enumerate}






\end{document}