# (wrong string) –Lehmer primality test || Mattias Andrée

From: <git_AT_suckless.org>
Date: Mon, 25 Jul 2016 01:13:40 +0200 (CEST)

commit 611db6bdb6a5a08b628f571451a94a1147a0e16f
Author: Mattias Andrée <maandree_AT_kth.se>
AuthorDate: Mon Jul 25 00:50:52 2016 +0200
Commit: Mattias Andrée <maandree_AT_kth.se>
CommitDate: Mon Jul 25 00:50:52 2016 +0200

Add exercise:  Lucas–Lehmer primality test

Signed-off-by: Mattias Andrée <maandree_AT_kth.se>

diff --git a/doc/exercises.tex b/doc/exercises.tex
index f26e877..46fa6dd 100644
--- a/doc/exercises.tex
+++ b/doc/exercises.tex
_AT_@ -143,6 +143,43 @@ called a base.)

+\item {[\textit{11}]} \textbf{Lucas–Lehmer primality test}
+
+The Lucas–Lehmer primality test can be used to determine
+whether a Mersenne numbers $M_n = 2^n - 1$ is a prime (a
+Mersenne prime). $M_n$ is a prime if and only if
+$s_{n - 1} \equiv 0 ~(\text{Mod}~M_n)$, where
+
+$$\displaystyle{ + s_i = \left \{ \begin{array}{ll} + 4 & \text{if} ~ i = 0 \\ + s_{i - 1}^2 - 2 & \text{otherwise}. + \end{array} \right . +}$$
+
+\noindent
+The Lucas–Lehmer primality test requires that $n \ge 3$,
+however $M_2 = 2^2 - 1 = 3$ is a prime. Implement a version
+of the Lucas–Lehmer primality test that takes $n$ as the
+input. For some more fun, when you are done, you can
+implement a version that takes $M_n$ as the input.
+
+For improved performance, instead of using \texttt{zmod},
+you can use the recursive function
+%
+$$\displaystyle{ + k ~\mbox{Mod}~ 2^n - 1 = + \left ( + (k ~\mbox{Mod}~ 2^n) + \lfloor k \div 2^n \rfloor + \right ) ~\mbox{Mod}~ 2^n - 1, +}$$
+%
+where $k ~\mbox{Mod}~ 2^n$ is efficiently calculated
+using \texttt{zand($k$, $2^n - 1$)}. (This optimisation
+is not part of the difficulty rating of this problem.)
+
+
+
\end{enumerate}

_AT_@ -348,4 +385,52 @@ enum zprimality ptest_fermat(z_t witness, z_t p, int t)

+\item \textbf{Lucas–Lehmer primality test}
+
+\vspace{-1em}
+\begin{alltt}
+enum zprimality ptest_llt(z_t n)
+\{
+ z_t s, M;
+ int c;
+ size_t p;
+
+ if ((c = zcmpu(n, 2)) <= 0)
+ return c ? NONPRIME : PRIME;
+
+ if (n->used > 1) \{
+ \textcolor{c}{/* \textrm{An optimised implementation would not need this} */}
+ errno = ENOMEM;
+ return (enum zprimality)(-1);
+ \}
+
+ zinit(s), zinit(M), zinit(2);
+
+ p = (size_t)(n->chars);
+ zsetu(s, 1), zsetu(M, 0);
+ zbset(M, M, p, 1), zsub(M, M, s);
+ zsetu(s, 4);
+ zsetu(two, 2);
+
+ p -= 2;
+ while (p--) \{
+ zsqr(s, s);
+ zsub(s, s, two);
+ zmod(s, s, M);
+ \}
+ c = zzero(s);
+
+ zfree(two), zfree(M), zfree(s);
+ return c ? PRIME : NONPRIME;
+\}
+\end{alltt}
+
+$M_n$ is composite if $n$ is composite, therefore,
+if you do not expect prime-only values on $n$, the
+performance can be improve by using some other
+primality test (or this same test if $n$ is a
+Mersenne number) to first check that $n$ is prime.
+
+
+
\end{enumerate}
Received on Mon Jul 25 2016 - 01:13:40 CEST

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