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	Add exercises: [10] Fermat primality test - libzahl - big integer library
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	---
	commit 4d41ca9124a0bbb4a26a856c76943317653ebc77
	parent ae8fb895ff82b75f7708b89c60565a2347e4593f
	Author: Mattias Andrée <[email protected]>
	Date: Sun, 24 Jul 2016 21:45:58 +0200

	Add exercises: [10] Fermat primality test

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/exercises.tex \| 60 +++++++++++++++++++++++++++++…

	1 file changed, 60 insertions(+), 0 deletions(-)
	---
	diff --git a/doc/exercises.tex b/doc/exercises.tex
	@@ -130,6 +130,19 @@ a fast primality tester.



	+\item {[\textit{10}]} \textbf{Fermat primality test}
	+
	+$a^{p - 1} \equiv 1 ~(\text{Mod}~p) ~\forall~ 1 < a < p$
	+for all primes $p$ and for a few composites $p$,
	+which are know as pseudoprimes\footnote{If $p$ is composite
	+but passes the test for all $a$, $p$ is a Carmichael
	+number.}. Use this to implement a heuristic primality
	+tester. Try to mimic \texttt{zptest} as much as possible.
	+GNU~MP uses $a = 210$, but you don't have to. ($a$ is
	+called a base.)
	+
	+
	+
	\end{enumerate}


	@@ -288,4 +301,51 @@ enum zprimality ptest_fast(z_t p)



	+\item \textbf{Fermat primality test}
	+
	+Below is a simple implementation. It can be improved by
	+just testing against a fix base, such as $a = 210$, it
	+$t = 0$. It could also do an exhaustive test (all $a$
	+such that $1 < a < p$) if $t < 0$.
	+
	+\vspace{-1em}
	+\begin{alltt}
	+enum zprimality ptest_fermat(z_t witness, z_t p, int t)
	+\{
	+ enum zprimality rc = PROBABLY_PRIME;
	+ z_t a, p1, p3, temp;
	+ int c;
	+
	+ if ((c = zcmpu(p, 2)) <= 0) \{
	+ if (!c)
	+ return PRIME;
	+ if (witness && witness != p)
	+ zset(witness, p);
	+ return NONPRIME;
	+ \}
	+
	+ zinit(a), zinit(p1), zinit(p3), zinit(temp);
	+ zsetu(temp, 3), zsub(p3, p, temp);
	+ zsetu(temp, 1), zsub(p1, p, temp);
	+
	+ zsetu(temp, 2);
	+ while (t--) \{
	+ zrand(a, DEFAULT_RANDOM, UNIFORM, p3);
	+ zadd(a, a, temp);
	+ zmodpow(a, a, p1, p);
	+ if (zcmpu(a, 1)) \{
	+ if (witness)
	+ zswap(witness, a);
	+ rc = NONPRIME;
	+ break;
	+ \}
	+ \}
	+
	+ zfree(temp), zfree(p3), zfree(p1), zfree(a);
	+ return rc;
	+\}
	+\end{alltt}
	+
	+
	+
	\end{enumerate}