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	Add exercises: - libzahl - big integer library
	git clone git://git.suckless.org/libzahl
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	---
	commit 32f59d5fbed9dbfa913221ef37d8df9364b9963a
	parent 2864613b162ed4c7a28a79c8c4dcd24893cc2128
	Author: Mattias Andrée <[email protected]>
	Date: Thu, 28 Jul 2016 20:12:19 +0200

	Add exercises:

	[▶05] zlshu and zrshu
	[▶M15] Modular left-shift
	[▶08] Modular left-shift, extended
	[13] The totient

	Diffstat:
	M doc/exercises.tex \| 134 +++++++++++++++++++++++++++++…

	1 file changed, 134 insertions(+), 0 deletions(-)
	---
	diff --git a/doc/exercises.tex b/doc/exercises.tex
	@@ -207,6 +207,67 @@ $\varphi$ is the golden ratio.



	+\item {[\textit{$\RHD$05}]} \textbf{zlshu and zrshu}
	+
	+Why does libzahl have
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void zlsh(z_t, z_t, size_t);
	+ void zrsh(z_t, z_t, size_t);
	+\end{alltt}
	+\vspace{-1em}
	+
	+\noindent
	+rather than
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void zlsh(z_t, z_t, z_t);
	+ void zrsh(z_t, z_t, z_t);
	+ void zlshu(z_t, z_t, size_t);
	+ void zrshu(z_t, z_t, size_t);
	+\end{alltt}
	+\vspace{-1em}
	+
	+
	+
	+\item {[$\RHD$M15]} \textbf{Modular left-shift}
	+
	+Implement a function that calculates
	+$2^a \text{ mod } b$, using \texttt{zmod} and
	+only cheap functions. You can assume $a \ge 0$,
	+ $b \ge 1$. You can also assume that all
	+parameters are unique pointers.
	+
	+
	+
	+\item {[$\RHD$08]} \textbf{Modular left-shift, extended}
	+
	+{\small\textit{You should already have solved
	+``Modular left-shift'' before you solve this
	+problem.}}
	+
	+Extend the function you wrote in ``Modular left-shift''
	+to accept negative $b$ and non-unique pointers.
	+
	+
	+
	+\item {[13]} \textbf{The totient}
	+
	+The totient of $n$ is the number of integer $a$,
	+$0 < a < n$ that are relatively prime to $n$.
	+Implement Euler's totient function $\varphi(n)$
	+which calculates the totient of $n$. Its
	+formula is
	+
	+\( \displaystyle{
	+ \varphi(n) = n \prod_{p \in \textbf{P} : p \| n}
	+ \left ( 1 - \frac{1}{p} \right ).
	+}\)
	+
	+
	+
	\end{enumerate}


	@@ -228,6 +289,7 @@ void monus(z_t r, z_t a, z_t b)
	zsetu(r, 0);
	\}
	\end{alltt}
	+\vspace{-1em}


	\item \textbf{Modular powers of 2}
	@@ -372,6 +434,7 @@ ptest_fast(z_t p)
	return c ? NONPRIME : PROBABLY_PRIME;
	\}
	\end{alltt}
	+\vspace{-1em}



	@@ -420,6 +483,7 @@ ptest_fermat(z_t witness, z_t p, int t)
	return rc;
	\}
	\end{alltt}
	+\vspace{-1em}



	@@ -539,6 +603,76 @@ golden_pow(z_t r, z_t n)
	lucas(r, n);
	\}
	\end{alltt}
	+\vspace{-1em}
	+
	+
	+
	+\item \textbf{zlshu and zrshu}
	+
	+You are in big trouble, memorywise, of you
	+need to evaluate $2^{2^{64}}$.
	+
	+
	+
	+\item \textbf{Modular left-shift}
	+
	+\vspace{-1em}
	+\begin{alltt}
	+void modlsh(z_t r, z_t a, z_t b)
	+\{
	+ z_t t, at;
	+ size_t s = zbits(b);
	+
	+ zinit(t), zinit(at);
	+ zset(at, a);
	+ zsetu(r, 1);
	+ zsetu(t, s);
	+
	+ while (zcmp(at, t) > 0) \{
	+ zsub(at, at, t);
	+ zlsh(r, r, t);
	+ zmod(r, r, b);
	+ if (zzero(r))
	+ break;
	+ \}
	+ if (!zzero(a) && !zzero(b)) \{
	+ zlsh(r, r, a);
	+ zmod(r, r, b);
	+ \}
	+
	+ zfree(at), zfree(t);
	+\}
	+\end{alltt}
	+\vspace{-1em}
	+
	+It is worth noting that this function is
	+not necessarily faster than \texttt{zmodpow}.
	+
	+
	+
	+\item \textbf{Modular left-shift, extended}
	+
	+The sign of \texttt{b} shall not effect the
	+result. Use \texttt{zabs} to create a copy
	+of \texttt{b} with its absolute value.
	+
	+
	+
	+\item \textbf{The totient}
	+
	+\( \displaystyle{
	+ \varphi(n)
	+ = n \prod_{p \in \textbf{P} : p \| n} \left ( 1 - \frac{1}{p} \right )
	+ = n \prod_{p \in \textbf{P} : p \| n} \left ( \frac{p - 1}{p} \right )
	+}\)
	+
	+\noindent
	+So, if we set $a = n$ and $b = 1$, then we iterate
	+of all integers $p$, $2 \le p < n$. For which $p$
	+that is prime, we set $a \gets a \cdot (p - 1)$ and
	+$b \gets b \cdot p$. After the iteration, $b \| a$,
	+and $\varphi(n) = \frac{a}{b}$.
	+