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\centerline{\textbf{HW 2 CMSC 456. Morally DUE Sep 28}}

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


\begin{enumerate}
\item
(0 points)
\begin{enumerate}
\item 
What is the day and time of the midterm?
\item
What is the day and time of the final?
\item
What is the {\it dead-cat policy}?
\end{enumerate}



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\item
(20 points--4 points each)
Zan has two 4-sided dice $d_1$ and $d_2$.

$d_1$ is NOT a fair die!  

$\Pr(d_1=1)=0.1$

$\Pr(d_1=2)=0.2$

$\Pr(d_1=3)=0.3$

$\Pr(d_1=4)=0.4$

$d_2$ is NOT a fair die!

$\Pr(d_2=1)=0.4$

$\Pr(d_2=2)=0.3$

$\Pr(d_2=3)=0.2$

$\Pr(d_2=4)=0.1$

Zan will roll the two 4-sided dice and then take their sum $s$.
What are the following (Express as both an exact fraction and decimal approximations):

\begin{enumerate}
\item
$\Pr(s=3)$
\item
$\Pr(s=3 \mid d_1=1)$
\item
$\Pr(s=3 \mid d_1=2)$
\item
$\Pr(s=3 \mid s \equiv 1 \pmod 2)$
\end{enumerate}

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\item
(20 points---4 points each)
For each of the following linear functions over mod 26 either give its inverse or 
tell me correctly that there is no inverse. Show your work.

Here is an example of how we want the answer: If I ask about $f(x)=9x+8$ then do the following

$y=9x+8$

The inverse of 9 mod 26 is 3 so mult both sides by 3

$3y = 27x + 24 = x+24$

$x=3y-24=3y+2$.

So answer is $g(y)=3y+2$.

\bigskip

\begin{enumerate}
\item
$f(x)=x+1$

\item
$f(x)=2x+1$

\item
$f(x)=3x+1$


\item
$f(x)=4x+1$

\item
$f(x)=5x+1$


\end{enumerate}

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\item
(20 points--10 points each.. BASED ON SLIDES Gen Sub and Random Looking Cipher)
Let $c\ge 2$. Let $\Sigma = \{0,\ldots,c-1\}$ be the alphabet.
Alice and Eve play the following game.
\begin{itemize}
\item
Alice flips a coin to get either RP (for Random Permutation) or SH (for Shift).
\begin{itemize}
\item
If she gets RP then she generates a random perm of $\Sigma$ and sends it to Eve.
\item
If she gets SH then she generates a random number $s\in \{0,\ldots,c-1\}$ and 
finds the permutation where $x\in\Sigma$ maps to $x+s \pmod c$, and sends that perm to Eve.
\end{itemize}
\item
Eve looks at the perm she gets and yells out either RP or SH. If she is correct she
wins. If not then Alice wins.
\end{itemize}

And now finally the problem:

\begin{enumerate}
\item
Come up with a strategy for Eve that will win over half of the time (she will probability win
a lot more than that). You may assume has Eve has unlimited computing power.


\item
Given $c$ find the probability that Eve loses when using the strategy above as a function of $c$. 
Show your work. 

\end{enumerate}



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\item
(20 points. BASED ON SLIDES: Vig Cipher)
Alice and Bob are going to use the Vig cipher.
The keyword is {\it josh.}
Alice wants to send 

\emph{Bill's course on Ramsey Theory last spring was awesome!}

Format it as blocks of 5, all capital, no punctuation. (NOTE: In the future I will not remind you
of this.)

What does Alice send?
(You can either (1) do this by hand, (2) write a program to do it for you,
or (3) find software on the web to do it for you. Let us know which one
you did. If (3) then give us the website where you found it and say if the
answer leaked information.)




For the next problem

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\item
(20 points---4 points each. NO SLIDES NEEDED)
Let $\phi(n)$ be the number of numbers in $\{1,\ldots,n-1\}$ that are relatively
prime to $n$. EXAMPLES

$\phi(5)=4$ since all four elts of $\{1,2,3,4\}$ are rel prime to 5.

$\phi(p)=p-1$ for any prime $p$.

$\phi(15)=8$ since $\{1,2,4,7,8,11,13,14\}$ are the elts in $\{1,\ldots,14\}$ that are rel prime to 15.

And now finally the problem

We will determine $\phi(91)$ without having to look at all of the numbers.
We will need to factor $91 = 7\times 13$. Note that a number is rel prime to
91 if it has \emph{neither} 7 \emph{nor} 13 as a factor.
\begin{enumerate}
\item
(4 points)
How many numbers in $\{1,\ldots,90\}$ have 7 as a factor. 
DO NOT do this by listing  them all out.
Show your work.
\item 
(4 points)
How many numbers in $\{1,\ldots,90\}$ have 13 as a factor. 
DO NOT do this by listing  them all out.
Show your work.
\item
(4 points)
How many numbers in $\{1,\ldots,90\}$ have 7 AND 13 as a factor. 
DO NOT do this by listing  them all out.
Show your work.
\item
(4 points)
Using the information from the last three parts, and the law of inclusion-exclusion,
find $\phi(91)$.
\item
(4 points)
Let $p,q$ be two primes.
Give a formula for $\phi(pq)$ in terms of $p$ and $q$.
\end{enumerate}


\end{enumerate}
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