
\chapter{Miscellaneous problems}

%=====================================================================
\section{Fun problems}
\label{sec:funproblems}
\sectionWithAnswers{Fun problems}{sec:funproblems}

In this section we present a number of problems for the interested
reader to pursue.

\begin{Exercise}\label{ex:misc:riffle}
  A riffle shuffle of a deck of even-numbered cards splits the deck
  into two equal parts, and then alternates the cards. That is, if the
  cards were originally numbered $1$ \ldots $10$, they will end up $1,
  6, 2, 7, 3, 8, 4, 9, 5, 10$. Write a function {\tt riffle} that
  takes an even number of elements and returns the elements in the new
  order. The signature of your function should be:
\begin{code}
   riffle : {a b} (fin a) => [2*a]b -> [2*a]b;
\end{code}
You might find the following Cryptol primitive functions useful:
\begin{Verbatim}
   width : {a b c} (c >= width a) => [a]b -> [c]
   take  : {a b c} (fin a,b >= 0) => (a,[a+b]c) -> [a]c
   drop  : {a b c} (fin a,a >= 0) => (a,[a+b]c) -> [b]c
   join  : {a b c} [a][b]c -> [a*b]c
\end{Verbatim}
Use the interpreter to pass arguments to these functions to
familiarize yourself with their operation first. What happens when you
call {\tt riffle} with a sequence that has an odd number of elements?
\end{Exercise}

\begin{Answer}\ansref{ex:misc:riffle}
\begin{code}
  riffle xs = join [| [x y] || x <- fh || y <- sh |]
    where [fh sh] = split xs;
\end{code}
The Cryptol type-checker will {\em not} allow passing an odd number of
elements to {\tt riffle}. To see the error message, issue: {\tt riffle
  [1..5]}.
\end{Answer}

\begin{Exercise}\label{ex:misc:riffle2}
  Given a deck of 52 cards, 8 riffles returns the deck to its original
  position. Write a theorem stating this fact and prove it.
\end{Exercise}
\begin{Answer}\ansref{ex:misc:riffle2}
  Note that the actual contents of the input sequence is immaterial;
  we will just use 8-bit numbers.
\begin{code}
   riffle8 : [52][8] -> Bit;
   theorem riffle8: {deck}. decks @ 8 == deck
     where decks = [deck] # [| riffle d || d <- decks |];
\end{code}
\end{Answer}

\begin{Exercise}\label{ex:misc:sort}
  Define the merge-sort function in Cryptol, that works over arbitrary
  finite sequences of words. Prove its correctness. You might want to
  split the proof in two parts. First prove that the output is a
  permutation of the input, and then prove that the output is always
  in increasing order.
\end{Exercise}
\begin{Answer}\ansref{ex:misc:sort}
  A solution to the merge-sort problem can be found in the {\tt
    Examples} directory of the Cryptol distribution.
\end{Answer}

\begin{Exercise}\label{ex:misc:legato}
  Legato's challenge refers to the verification of an 8-bit
  multiplication algorithm encoded in Mostek assembly code. More
  information on Legato's challenge can be found at:
\begin{center}
 \url{http://www.cs.utexas.edu/~moore/acl2/workshop-2004/contrib/legato/Weakest-Preconditions-Report.pdf}
\end{center}
Write a Cryptol program to model Legato's multiplication algorithm
following the machine model. Prove that the multiplier is correct.
\end{Exercise}
\begin{Answer}\ansref{ex:misc:legato}
A solution can be found at:
\begin{center} \url{http://www.galois.com/blog/2009/07/08/legatosmultiplierincryptol}
\end{center}
\end{Answer}

\begin{Exercise}\label{ex:misc:skein}
  The Cryptol distribution comes with a variety of crypto algorithm
  implementations, including AES and the SHA-3 candidate Skein.  Study
  these definitions and experiment with them using the Cryptol
  interpreter.
\end{Exercise}
