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\assigntitle{27}{Number Theory, Part II: Modular Arithmetic and Cryptography}
%% This week: modular arithmetic. Foundational subject in number
%% theory, and also has interesting practical
%% applications. One particular application: cryptography.
This week you will study \term{modular arithmetic}---arithmetic where
we make the natural numbers ``wrap around'' by only considering
their remainder when divided by some particular number. Modular
arithmetic is a foundational subject in number theory, but as we will
see, it also has interesting practical applications---for example, it
finds many uses in cryptography (the study and design of secret
codes).
\section{Wrap-around numbers}
\label{sec:wrap}
%% All about *remainder*. "modulo n" -- only care about remainder
%% when dividing by n. E.g. 12, 7, 2, 102 all equivalent modulo 5.
%% notation for "equivalent modulo". Give LaTeX notation.
%% I.e. when working modulo 5, there are only five numbers in the
%% whole world: 0, 1, 2, 3, 4.
%% should already be familiar: clocks!
Modular arithmetic is all about \emph{remainders}. When using modular
arithmetic, we pick some particular number $n$ (often, but not always,
a prime) called the \term{modulus}, and say that we are working
``modulo $n$''---this means that we only care about \emph{remainder
when dividing by $n$}.
For example, $12$ and $17$ are \term{equivalent modulo $5$} since they
have the same remainder (namely, $2$) when divided by $5$. In this
case we write \[ 12 \equiv 17 \pmod{5}. \] In other words, when
working modulo $5$, we put on our ``modulo $5$ glasses'' and $12$ and
$17$ look the same to us. Modulo $5$, there are really only five
numbers we care about: $0$, $1$, $2$, $3$, and $4$. After that, the
naturals ``wrap around'' and the pattern repeats: $5$ has a remainder
of $0$ when divided by $5$; $6$ has a remainder of $1$, and so on.
Every natural number is equivalent, modulo $5$, to some number from
$0$ to $4$.
(You can create modular equivalences in \LaTeX\ with \verb|\equiv| and
\verb|\pmod|. For example, the equation above was typeset with
\verb|12 \equiv 17 \pmod{5}|.)
\begin{problem}
State whether each modular equivalence is true or false. For those
which are false, give the largest possible modulus which makes the
equivalence true. For example, $4 \equiv 7 \pmod{5}$ is false, but
$4 \equiv 7 \pmod{3}$ is true. ($4 \equiv 7 \pmod{1}$ is also true,
but $1$ is not the \emph{largest possible} modulus that works.)
\begin{subproblems}
\item $19 \equiv 23 \pmod{4}$
\item $222 \equiv 23 \pmod{10}$
\item $30 \equiv 280 \pmod{25}$
\item $9 \equiv 400 \pmod{2}$
\end{subproblems}
\end{problem}
\begin{problem}
Write down four natural numbers that are all equivalent modulo
$17$.
\end{problem}
\begin{problem}
Can you find five distinct natural numbers so that no two of them
are equivalent modulo $4$? If so, write down the five numbers; if
not, explain why.
\end{problem}
\begin{problem}
Explain how you can tell, just by looking at two numbers, whether
they are equivalent modulo $10$.
\end{problem}
\section{Modular arithmetic}
%% modular arithmetic.
When working modulo $n$, it is as if we have taken the usual number
line, like in \pref{fig:natural-line}, and wrapped it around to make a
circle, like the one in \pref{fig:circ-number-line}.
\diagram{natural-line}{The natural number line}
\diagram{circ-number-line}{The number line modulo $8$}
So we can count things, do arithmetic, and so on with the circular
number line just like we would with the normal number line---the only
difference is that everything above $n-1$ wraps back around (if we are
working modulo $n$).
\begin{problem}
Compute each of the following.
\begin{subproblems}
\item $(3 + 5) \bmod 7$
\item $(4 \times 7 + 6) \bmod 19$
\item $(1 - 3) \bmod 5$
\item $(1000!) \bmod 7$
\end{subproblems}
\end{problem}
\begin{problem}
There are quite a few tricks one can use to make things simpler when
doing modular arithmetic. Here's one.
\begin{subproblems}
\item Compute $(3^i \bmod 10)$ for $i = 0, 1, 2, 3, 4, 5$.
\item Do you notice a pattern? Do you think the pattern will continue?
\item Compute $3^{1247} \bmod 10$.
\end{subproblems}
\end{problem}
\section{C\ae sar ciphers}
The \emph{C\ae sar cipher} is one of the earliest and most simple
forms of cipher. It is named after Julius C\ae sar\footnote{I'm just
using \ae\ because it looks really cool. \ae\ \ae\ \ae.}, who used
it to encrypt messages to his generals. To encrypt a message using a
C\ae sar cipher, the first step is to convert the letters in the
message to numbers from $0$ to $25$: A is $0$, B is $1$, and so on.
So the message \textsf{PHISH PHRIEZ} would become \[ 15 \; 7 \; 8 \;
18 \; 7 \quad 15 \; 7 \; 17 \; 8 \; 4 \; 25. \] Next, add some
specified amount to each number. For example, let's add $3$.
However, the key point is that we do this addition \emph{modulo $26$}:
any numbers larger than $25$ wrap back around starting with $0$.
Performing this operation, we get \[ 18 \; 10 \; 11 \; 21 \; 10 \quad
18 \; 10 \; 20 \; 11 \; 7 \; 2. \] Note how the $25$ wrapped around:
$25 + 3 \equiv 2 \pmod{26}$. Finally, we convert back to letters:
\textsf{SKLVK SKULHC} would be our encrypted message. The recipient
of the message, of course, simply has to reverse the process, assuming
that they know the secret number ($3$, in this example): they just
convert the letters to numbers, subtract $3$ modulo $26$ (which is the
same as adding $23$) and convert back to letters to read the secret
message.
In practice, converting to numbers is unnecessary; it is easy to
``count letters'' in your head. For example, to add $3$ to P, you can
just think ``P\dots Q, R, S''.
\begin{problem}
You have intercepted the following encrypted message to your
mailman, who you suspect is actually an evil robot from the planet
Zorkotron. What should you do?
\begin{quote}\textsf{
USVV KVV REWKXC IYE KBO YEB YXVI BOWKSXSXQ RYZO BOWOWLOB SP IYE
ROKB DRO GYBN ZKBKUOOD SD GSVV MKECO IYE DY COVP-NOCDBEMD CY NY
XYD CKI SD}
\end{quote}
\end{problem}
\section{Vigen\`ere ciphers}
As you discovered for yourself in the previous section, a C\ae sar
cipher is not very secure: there are only $26$ possible secret numbers
(actually, $25$, since $0$ is not a very interesting secret number!),
and it is entirely feasible to just try them all.
The \term{Vigen\`ere cipher}, originally invented by Giovan Battista
Bellaso (but later misattributed to Blaise de Vigen\`ere), is similar
to the C\ae sar cipher, but uses a secret \emph{word} (or phrase)
instead of a secret \emph{number}. The secret word is often called
the ``keyword''. Intuitively, a Vigen\`ere cipher is much more
secure, since it is much harder to guess a secret word than it is to
guess a secret number: there are only $25$ possible secret numbers,
but there are infinitely many possible secret words!
Here's how it works. In a C\ae sar cipher, you add the same amount to
every letter of the message; in a Vigen\`ere, you add different
amounts to different letters, as determined by the secret word. Let's
suppose the secret word is \textsf{PHISH}, and we want to encrypt the
message \textsf{DO NOT EAT THE MONKEY}. We first line up the secret word
underneath the message, repeating it as many times as necessary:
\begin{center}\sffamily
\begin{tabular}{@{\extracolsep{-1 ex}}*{21}c}
D&O& &N&O&T& &E&A&T& &T&H&E& &M&O&N&K&E&Y \\
P&H& &I&S&H& &P&H&I& &S&H&P& &H&I&S&H&P&H
\end{tabular}
\end{center}
Now, ``add'' each letter of the message to the corresponding keyword
letter with addition modulo $26$, remembering that A corresponds to
$0$ and Z to $25$. For example, $\mathrm{D} + \mathrm{P} =
\mathrm{S}$, since $3 + 15 = 18$; as another example, $\mathrm{O} +
\mathrm{S} = \mathrm{G}$, since $14 + 18 \equiv 6 \pmod{26}$.
\begin{center}\sffamily
\begin{tabular}{@{\extracolsep{-1 ex}}*{22}c}
&D&O& &N&O&T& &E&A&T& &T&H&E& &M&O&N&K&E&Y \\
$+$&P&H& &I&S&H& &P&H&I& &S&H&P& &H&I&S&H&P&H \\
\hline
&S&V& &V&G&A& &T&H&B& &L&O&T& &T&W&F&R&T&F
\end{tabular}
\end{center}
So the secret message is \textsf{SVVGATHBLOTTWFRTF} (we often omit the
spaces from encrypted messages this way; leaving them in just gives
more information to anyone trying to break the encryption, and if you
know the secret word it is not hard to figure out where the spaces go
once you have decrypted the message).
\begin{problem}
Your evil mail-robot (who you destroyed) was carrying a
suspicious-looking piece of mail with the following encrypted message:
\begin{quote}
\textsf{MTKMQFBMERKVGVHTUCNSWFDINKJ \\ XPBDIAHXNIYBJFXEIYWWGTNSJQZ}
\end{quote}
The word ``apricot'' is written next to it. What should you do?
\end{problem}
\begin{problem}
Although Vigen\`ere ciphers are certainly more secure than C\ae sar
ciphers, they are not unbreakable. Your answer to this problem
should be a message encrypted using a Vigen\`ere cipher. I will
attempt to decrypt the message \emph{without} knowing the secret
word. If I cannot decrypt it, you will get an automatic score of
$5$ on this assignment, no matter what you turn in for the other
problems. In fact, if you are feeling particularly ambitious, you
could just send me a code and not do any of the other problems, and
hope I can't solve it---but I don't recommend it. \smiley
There are a few requirements:
\begin{enumerate}
\item The encrypted message must contain at least 100 letters.
\item The original message must be written in English, using complete
sentences and correct grammar and spelling.
\item The secret word or phrase must be \emph{no longer than} ten letters.
\item Don't bother trying to cheat and send me gibberish. If I
can't decrypt your message, before giving you your score of $5$ I
will require proof that the message really was a Vigen\`ere
cipher---that is, you must tell me what the secret word was and I
will then check that I can in fact decrypt the message using the
secret word.
\end{enumerate}
Whether I succeed in decrypting your message or not, next week I can
(if you like) explain how to go about trying to decrypt Vigen\`ere
ciphers without knowing the keyword.
\end{problem}
There is more that could be said about modular arithmetic and
cryptography. In particular, we haven't yet talked about public-key
cryptography and the RSA system, which is the basis of much modern
cryptography. For example, your computer uses some variant of RSA
every time you connect to a secure web site, like when you make a
purchase from Amazon, so that no one observing the data being
transmitted between your computer and Amazon can steal your credit
card number.
If you find this cryptography stuff interesting, let me know and we
could spend another week on it if you want.
\end{document}