\chapter{Modular Arithmetic: The Number System Cryptography Lives In}
\label{ch:modular}

\section{Clock arithmetic, taken seriously}

You already compute modulo twelve every day: four hours after ten o'clock is
two o'clock. Wrap-around arithmetic --- add, overflow the dial, keep the
remainder --- is the entire idea of \emph{modular arithmetic}. Cryptography's
only twist is the size of the clock: Ed25519's dial has
\[
p = 2^{255} - 19
\]
hours on it --- a 77-digit prime. Everything else --- the wrap-around, the
remainder-taking, the algebra --- is the twelve-hour clock you already know.

Formally, we write $a \equiv b \pmod{n}$ when $n$ divides $a - b$, and we
collect all integers with the same remainder into one object: the world
$\Zmod{n}$ has exactly $n$ elements, $\{0, 1, \dots, n-1\}$, with addition
and multiplication that wrap. In Lean/Mathlib this world is a first-class
type, and computation in it is exactly what you would hope:

\begin{lstlisting}[language=Lean]
#eval (7 + 8 : ZMod 12)    -- 3     (fifteen o'clock is three o'clock)
#eval (5 * 9 : ZMod 12)    -- 9
#eval (3 - 7 : ZMod 12)    -- 8     (subtraction wraps -- no truncation!)

def p : Nat := 2^255 - 19
#eval (2^255 : ZMod p)     -- 19    (of course: 2^255 = p + 19)
\end{lstlisting}

Note the third line with relief: unlike \lean{Nat}, subtraction in
\lean{ZMod n} is a total, well-behaved inverse of addition. Every element has
a negative. In algebra terms, $\Zmod{n}$ is a \emph{commutative ring} --- and
Lean knows it, so the \lean{ring} tactic from Chapter~\ref{ch:automation}
works there natively.

\section{Division: where primes enter}

Rings give us $+$, $-$, $\times$. Cryptography also needs $\div$ --- and
division is where the modulus stops being a size parameter and starts being a
design decision. Dividing by $a$ means multiplying by some $a^{-1}$ with
$a \cdot a^{-1} = 1$. Does such an inverse exist? On the twelve-hour clock,
try to invert $4$: the multiples of $4$ are $4, 8, 0, 4, 8, 0, \dots$ ---
the value $1$ never appears. $4$ has no inverse mod $12$, because $4$ and
$12$ share the factor~$4$.

\begin{bigidea}
In $\Zmod{n}$, the element $a$ is invertible exactly when
$\gcd(a, n) = 1$. So if $n = p$ is \textbf{prime}, \emph{every} nonzero
element is invertible: you can divide freely, and $\Zmod{p}$ earns the title
of \textbf{field}, written $\Fp$. Fields are the arithmetic paradise where
linear algebra, polynomial factoring, and elliptic-curve geometry all work.
This --- and only this --- is why cryptographic moduli are prime: primality
is the entry ticket to division.
\end{bigidea}

\begin{center}
\begin{tikzpicture}[scale=0.62,every node/.style={font=\small}]
  % Z/12 clock: multiples of 4 stuck in a cycle
  \begin{scope}
    \draw[ink2,thick] (0,0) circle (2.2);
    \foreach \i in {0,...,11} \node[color=ink] at ({90-\i*30}:1.8) {\i};
    \foreach \s/\t in {0/4, 4/8, 8/0}
      \draw[-{Stealth},accent,thick] ({90-\s*30}:2.55) arc ({90-\s*30}:{90-\t*30+8}:2.55);
    \node[color=accent,align=center] at (0,-3.4) {$\Zmod{12}$: stepping by $4$\\ visits only $\{0,4,8\}$ --- never $1$};
  \end{scope}
  % Z/11 clock: multiples of 4 reach 1
  \begin{scope}[xshift=9.5cm]
    \draw[ink2,thick] (0,0) circle (2.2);
    \foreach \i in {0,...,10} \node[color=ink] at ({90-\i*32.72}:1.8) {\i};
    \foreach \s/\t in {0/4, 4/8, 8/1}
      \draw[-{Stealth},proven,thick] ({90-\s*32.72}:2.55) arc ({90-\s*32.72}:{90-\t*32.72+8}:2.55);
    \node[color=proven,align=center] at (0,-3.4) {$\Zmod{11}$: stepping by $4$\\ reaches $1$ in three steps: $4^{-1}=3$};
  \end{scope}
\end{tikzpicture}
\end{center}

How do you \emph{find} $a^{-1}$ mod $p$? Two classical answers, both of which
appear in real Ed25519 code: the extended Euclidean algorithm, and Fermat's
little theorem, which says $a^{p-1} \equiv 1 \pmod p$ for $a \not\equiv 0$
--- hence $a^{p-2}$ \emph{is} the inverse. The dalek library computes
inverses as $a^{p-2}$ with a hand-crafted chain of $254$ squarings and $11$
multiplications; one of the proofs you will meet in Chapter~\ref{ch:field}
verifies precisely that this chain computes what Fermat promises.

\section{\texorpdfstring{Why $2^{255}-19$?}{Why 2**255-19?} A prime chosen for machines}

Any large prime makes a field. Why this one? Because arithmetic mod $p$ is
computed by \emph{machines with 64-bit words}, and $p = 2^{255}-19$ is shaped
for them:

\begin{itemize}[leftmargin=1.4em]
\item \textbf{Reduction is a multiply-by-19.} Numbers at or beyond $2^{255}$
  overflow the dial; but since $2^{255} \equiv 19 \pmod p$, any overflow bit
  re-enters the sum carrying a factor of just $19$. Reducing mod $p$ costs
  one small multiplication --- no long division, ever.
\item \textbf{255 bits split evenly into five limbs of 51.} A 64-bit word
  holding a 51-bit limb leaves 13 bits of headroom for carries to accumulate
  lazily --- the delayed-carry style from Chapter~\ref{ch:why}, now by design
  rather than accident. The bound $a_i < 2^{51+\varepsilon}$ will follow us
  through Chapters~\ref{ch:denotation} and~\ref{ch:field}.
\item \textbf{It sits just below a power of two}, so 255-bit values fit in 32
  bytes with one bit to spare --- and Ed25519 spends that spare bit storing a
  point's sign. Engineering all the way down.
\end{itemize}

The Pasta curves verified in the companion projects (Pallas/Vesta, used in
zero-knowledge proof systems) choose their $\approx 2^{254}$ primes by a
different criterion --- friendliness to fast Fourier transforms --- and
represent elements in \emph{Montgomery form}, a representation trick we will
touch in Chapter~\ref{ch:denotation}. Different shapes, same theory: pick a
prime whose field your machine can love.

\section{Modular arithmetic in Lean, hands on}

Statements about $\Fp$ are ordinary Lean theorems, and the automation you
already own applies:

\begin{lstlisting}[language=Lean]
-- ring identities hold verbatim in ZMod n:
example (x y : ZMod 12) : (x + y)^2 = x^2 + 2*x*y + y^2 := by ring

-- small finite worlds: check every case
example : ∀ x : ZMod 12, 4 * x ≠ 1 := by decide

-- inverses exist in prime fields (Mathlib knows ZMod 11 is a field):
example (a : ZMod 11) (h : a ≠ 0) : a * a⁻¹ = 1 :=
  ZMod.mul_inv_cancel_of_ne_zero h
\end{lstlisting}

What about the crown fact, $2^{255} \equiv 19$? For \emph{concrete numeric}
statements at 77-digit scale, tactic choice suddenly matters: \lean{decide}
would ask the kernel to grind case analysis it cannot afford, while
\lean{norm_num} and reflexivity-by-computation on efficient numerals handle
it comfortably. Verifying real cryptography is partly a \emph{performance
engineering} discipline --- Chapter~\ref{ch:prime} turns this observation
into a principle when the question becomes primality itself.

\begin{pitfall}
In \lean{ZMod n}, the numeral \lean{57896044618658...} and the numeral
\lean{19} can be \emph{the same element}. Equality of elements is not
equality of the numerals you typed. When a surprising \lean{rfl} succeeds ---
or two ``different'' constants turn out equal --- remember you are on the
clock face, not the number line. This is a feature: the entire verification
story of Chapter~\ref{ch:denotation} consists of choosing, deliberately, when
to view a pile of bytes as an integer and when as a clock position.
\end{pitfall}

\begin{tryit}
Open \code{exercises/Ch06.lean}. You will: compute your first inverses with
\lean{\#eval}; expand $(x+y)^3$ in \lean{ZMod 7} with one tactic; show $4$
has no inverse in \lean{ZMod 12} by \lean{decide}; and prove the reduction
identity $2^{255} = 19$ in \lean{ZMod p}.
\end{tryit}

\section*{Exercises}

\exercise{Compute by hand (yes, hand): $3^{-1}$ in $\Zmod{7}$, and check that
stepping by $3$ on a 7-hour clock visits every position. Then confirm both
with \lean{\#eval}.}

\exercise{Prove in Lean: \lean{∀ x : ZMod 12, 4 * x ≠ 1}, then change $12$
to $13$ and watch the same tactic refuse --- find the witness it is
implicitly pointing at.}

\exercise{Fermat's little theorem is \lean{ZMod.pow_card_sub_one_eq_one} in
Mathlib. Use it to prove that in \lean{ZMod 11}, \lean{a\textasciicircum{}9 * a = 1} whenever
\lean{a ≠ 0}.}

\exercise{(Paper) The Ed25519 code reduces a 256-bit value $x$ by writing
$x = x_{\mathrm{low}} + 2^{255}\, x_{\mathrm{high}}$ and returning
$x_{\mathrm{low}} + 19\, x_{\mathrm{high}}$. Prove informally that this
preserves the value mod $p$, and bound how much smaller the result is. You
have just re-derived the reduction step you will verify formally in
Chapter~\ref{ch:field}.}

\begin{checkpoint}
You should now be able to: compute in $\Zmod{n}$ and explain the notation;
state exactly when division works and why primality guarantees it; give two
independent reasons the constant $19$ appears throughout curve25519
codebases; and prove small modular facts in Lean with \lean{decide},
\lean{ring}, and a Mathlib lemma found by name.
\end{checkpoint}
