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120 lines
5.9 KiB
HTML
120 lines
5.9 KiB
HTML
4 years ago
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 6.1.0.
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Copyright 1991, 1993-2015 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.3 or any later
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version published by the Free Software Foundation; with no Invariant Sections,
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with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover
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Texts being "You have freedom to copy and modify this GNU Manual, like GNU
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software". A copy of the license is included in
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GNU Free Documentation License. -->
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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Perfect Square Algorithm (GNU MP 6.1.0)</title>
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<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
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<meta name="keywords" content="Perfect Square Algorithm (GNU MP 6.1.0)">
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<meta name="resource-type" content="document">
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<meta name="distribution" content="global">
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<meta name="Generator" content="makeinfo">
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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<link href="index.html#Top" rel="start" title="Top">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" rel="up" title="Root Extraction Algorithms">
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<link href="Perfect-Power-Algorithm.html#Perfect-Power-Algorithm" rel="next" title="Perfect Power Algorithm">
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<link href="Nth-Root-Algorithm.html#Nth-Root-Algorithm" rel="prev" title="Nth Root Algorithm">
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<style type="text/css">
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<!--
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a.summary-letter {text-decoration: none}
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blockquote.indentedblock {margin-right: 0em}
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blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
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blockquote.smallquotation {font-size: smaller}
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div.display {margin-left: 3.2em}
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div.example {margin-left: 3.2em}
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div.lisp {margin-left: 3.2em}
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kbd {font-style: oblique}
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pre.display {font-family: inherit}
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pre.smalldisplay {font-family: inherit; font-size: smaller}
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pre.smalllisp {font-size: smaller}
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span.nolinebreak {white-space: nowrap}
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span.roman {font-family: initial; font-weight: normal}
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span.sansserif {font-family: sans-serif; font-weight: normal}
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ul.no-bullet {list-style: none}
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-->
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</style>
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</head>
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<body lang="en">
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<a name="Perfect-Square-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Perfect-Power-Algorithm.html#Perfect-Power-Algorithm" accesskey="n" rel="next">Perfect Power Algorithm</a>, Previous: <a href="Nth-Root-Algorithm.html#Nth-Root-Algorithm" accesskey="p" rel="prev">Nth Root Algorithm</a>, Up: <a href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" accesskey="u" rel="up">Root Extraction Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<a name="Perfect-Square"></a>
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<h4 class="subsection">15.5.3 Perfect Square</h4>
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<a name="index-Perfect-square-algorithm"></a>
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<p>A significant fraction of non-squares can be quickly identified by checking
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whether the input is a quadratic residue modulo small integers.
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</p>
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<p><code>mpz_perfect_square_p</code> first tests the input mod 256, which means just
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examining the low byte. Only 44 different values occur for squares mod 256,
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so 82.8% of inputs can be immediately identified as non-squares.
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</p>
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<p>On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total
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99.25% of inputs identified as non-squares. On a 64-bit system 97 is tested
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too, for a total 99.62%.
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</p>
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<p>These moduli are chosen because they’re factors of <em>2^24<!-- /@w -->-1</em> (or
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<em>2^48<!-- /@w -->-1</em> for 64-bits), and such a remainder can be quickly taken just
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using additions (see <code>mpn_mod_34lsub1</code>).
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</p>
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<p>When nails are in use moduli are instead selected by the <samp>gen-psqr.c</samp>
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program and applied with an <code>mpn_mod_1</code>. The same <em>2^24<!-- /@w -->-1</em> or
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<em>2^48<!-- /@w -->-1</em> could be done with nails using some extra bit shifts, but
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this is not currently implemented.
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</p>
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<p>In any case each modulus is applied to the <code>mpn_mod_34lsub1</code> or
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<code>mpn_mod_1</code> remainder and a table lookup identifies non-squares. By
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using a “modexact” style calculation, and suitably permuted tables, just one
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multiply each is required, see the code for details. Moduli are also combined
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to save operations, so long as the lookup tables don’t become too big.
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<samp>gen-psqr.c</samp> does all the pre-calculations.
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</p>
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<p>A square root must still be taken for any value that passes these tests, to
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verify it’s really a square and not one of the small fraction of non-squares
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that get through (i.e. a pseudo-square to all the tested bases).
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</p>
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<p>Clearly more residue tests could be done, <code>mpz_perfect_square_p</code> only
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uses a compact and efficient set. Big inputs would probably benefit from more
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residue testing, small inputs might be better off with less. The assumed
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distribution of squares versus non-squares in the input would affect such
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considerations.
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</p>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Perfect-Power-Algorithm.html#Perfect-Power-Algorithm" accesskey="n" rel="next">Perfect Power Algorithm</a>, Previous: <a href="Nth-Root-Algorithm.html#Nth-Root-Algorithm" accesskey="p" rel="prev">Nth Root Algorithm</a>, Up: <a href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" accesskey="u" rel="up">Root Extraction Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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