\documentclass[12pt,ifthen]{article}
\usepackage{url}

\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
\newcommand{\lc}{\left\lceil}
\newcommand{\rc}{\right\rceil}
\newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\bit }{\{0,1\}}
\newcommand{\abit }{\hat{a}}
\newcommand{\bbit }{\hat{b}}
\newcommand{\bits}[1]{\{0,1\}^{{#1}}}

\newcommand{\nth}{n^{th}}

\newif{\ifshowsoln}
\showsolntrue
\newcommand{\und}{\_\_\_\_\_\_\_\_\_}
\newcommand{\Z}{\mathbb{Z}}
\usepackage{amsmath}
\usepackage{amssymb}  % for \nmid

\begin{document}

\centerline{\textbf{HW 12 CMSC 456. MORALLY DUE Dec 10}}

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

{\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 of the final?
READ the slides and notes on Perfect and Comp Secrecy.

\centerline{\bf GOTO NEXT PAGE}

\newpage

\item
(40 points) 

All of the arithmetic in this problem is mod 2.

Write a program do do the following.

Input $c_0,c_1,c_2,c_3\in \bit$ and do the following:
\begin{enumerate}
\item
Let 
$$f(s_3,s_2,s_1,s_0) = (c_3s_3+c_2s_2+c_1s_1+c_0s_1,s_3,s_2,s_1)$$
\item
For all $b_0,b_1,b_2,b_3 \in \{0,1\}$ compute

$v_0=(b_3,b_2,b_1,b_0)$

$v_1=f(v_0)$

$v_2=f(v_1)$

$\vdots$

UNTIL you find $i<j$ such that $f(v_i)=f(v_j)$.
Keep track of the $j$'s seen (see next step).
(So you will need to store all of the $v_1,v_2,\ldots$. 
Since there are only 16 possibilities you can do this in an inelegant
but easy way. And DO NOT WORRY- I am NOT going to ask you to later to it
for 10 bits or 100 bits or something that would require a clever way to store it.)
\item
For each $(c_3,c_2,c_1,c_0)$ note which $(b_3,b_2,b_1,b_0)$ lead to the LARGEST
sequence without a repeat- so the largest $j$.
\end{enumerate}

You final output should look like this (I made up the numbers and only give the
first two rows. Yours should have 16 rows).

\begin{tabular}{|c|c|c|}
\hline
$c$-vector & best $b$-vector & length of sequence \cr
\hline
0000       &  1100           & 2 \cr
0001       & 1010            & 13 \cr 
\hline
\end{tabular}

\centerline{\bf GOTO NEXT PAGE}

\newpage


\item
(30 points)
All of the arithmetic in this problem is mod 2.

Let

$$f(s_3,s_2,s_1,s_0) = (s_3s_2+s_1+s_0,s_3,s_2,s_1)$$

Write a program do do the following.

\begin{enumerate}
\item
For all $b_0,b_1,b_2,b_3$ compute

$v_0=(b_3,b_2,b_1,b_0)$

$v_1=f(v_0)$

$v_2=f(v_1)$

$\vdots$

UNTIL you find $i<j$ such that $f(v_i)=f(v_j)$.
Keep track of the $j$'s seen (see next step).
(So you will need to store all of the $v_1,v_2,\ldots$. 
Since there are only 16 possibilities you can do this in an inelegant
but easy way. And DO NOT WORRY- I am NOT going to ask you to later to it
for 10 bits or 100 bits or something that would require a clever way to store it.)
\item
For each $(b_3,b_2,b_1,b_0)$ output the length of the sequence before a repeat.
sequence without a repeat.
\end{enumerate}

You final output should look like this (I made up the numbers and only gave the
first two rows. Yours should have 16 rows.)

\begin{tabular}{|c|c|}
\hline
$b$-vector & length of sequence \cr
\hline
0000       &  5 \cr
0001       & 19 \cr
\hline
\end{tabular}


\centerline{\bf GOTO NEXT PAGE}

\newpage

\item
(30 points)
Give a rigorous definition of a psuedorandom FUNCTION that uses a game.
It should begin with $F_k(x)$ where $k$ is unif in $\bits n$.
$F_k$ goes from $\bits n$ to $\bits n$.

\end{document}