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\begin{flushright}
  Your Name Goes Here
\end{flushright}


\begin{center}
  \textbf{Math 25}\\ Homework 5
\end{center}

\begin{enumerate}
\item Below is a screen shot of the web page
  \url{http://releases.ubuntu.com/17.10/} which happens to be a
  release page for the Linux distribution Ubuntu.  Any other vendor
  that releases software has a similar page, see
  \url{https://ftp.mozilla.org/pub/firefox/releases/55.0.3/} for
  example.


  \includegraphics*[width=9in]{releases-ubuntu}

  The file listed at the bottom of the image
  (\texttt{ubuntu-17.10-beta2-desktop-amd64.iso}) is a 1.4G file (in
  iso format meaning could be burned to a DVD).  Mounting the file, or
  burning it to a DVD and inserting in a computer would offer to
  install a new operating system on your computer.  \textbf{I am not
    suggesting you do this!}

  Visit this site, and look at the files MD5SUMS, SHA1SUMS,
  SHA256SUMS.  Each contain hash values for the file
  \texttt{ubuntu-17.10-beta2-desktop-amd64.iso}.

  \begin{enumerate}
  \item Go online, poke around, and tell me a little about the
    differences about MD5SUMS, SHA1SUMS, and SHA256SUMS.  Detail at
    least the length of the hash, and the relative security of each
    algorithm.

  \item Explain the purpose of the presence of the hash values on the
    download page.  What is a downloader supposed to do with them?

  \item More interestingly, there are three other files also listed on
    that page: MD5SUMS.gpg, SHA1SUMS.gpg and SHA256SUMS.gpg.  They are
    Ubnutu's official digital signatures of the files MD5SUMS, etc.,
    analogous to the signatures we talked about in class.

    Give an explanation of why it is important for such files to be on
    a download site, given that you as the downloader will almost
    never access them.
  \end{enumerate}

\item Set up public and private keys for an RSA system using the
  primes 17 and 23.

\item Show that for each $k \ge 1$, there are at most a finite number
  of positive integers $n$ for which $\phi(n) = k$.

\item Prove that for all $n \ge 1$, $\phi(n^2) = n\phi(n)$.

\item Let $\sigma(n) = \sum_{d\mid n} d$ be the usual sum of the
  positive divisors of $n$.  We have shown that $\sigma$ is
  multiplicative, but not completely multiplicative.  Verify, that for
  a prime $p$ and integer $k \ge 1$,
  \[\sigma(p^{k+1}) = \sigma(p)\sigma(p^k) - p \sigma(p^{k-1}).\]

\item For a prime $p$, consider the power series
  \[ \sum_{k=0}^\infty \sigma(p^k) x^k.\]
  \begin{enumerate}
  \item Find the radius of convergence of the series.  Eeek!
    Calculus!
  \item For $x$ in the interval of convergence, verify the identity
    \[
      \sum_{k=0}^\infty \sigma(p^k) x^k = \frac{1}{1 - \sigma(p)x +
        px^2}.\]

  \end{enumerate}



\end{enumerate}

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