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131 lines
6.4 KiB
HTML
131 lines
6.4 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>Exact Remainder (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="Exact Remainder (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="Division-Algorithms.html#Division-Algorithms" rel="up" title="Division Algorithms">
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<link href="Small-Quotient-Division.html#Small-Quotient-Division" rel="next" title="Small Quotient Division">
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<link href="Exact-Division.html#Exact-Division" rel="prev" title="Exact Division">
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<style type="text/css">
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a.summary-letter {text-decoration: none}
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blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
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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="Exact-Remainder"></a>
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<div class="header">
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<p>
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Next: <a href="Small-Quotient-Division.html#Small-Quotient-Division" accesskey="n" rel="next">Small Quotient Division</a>, Previous: <a href="Exact-Division.html#Exact-Division" accesskey="p" rel="prev">Exact Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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="Exact-Remainder-1"></a>
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<h4 class="subsection">15.2.6 Exact Remainder</h4>
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<a name="index-Exact-remainder"></a>
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<p>If the exact division algorithm is done with a full subtraction at each stage
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and the dividend isn’t a multiple of the divisor, then low zero limbs are
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produced but with a remainder in the high limbs. For dividend <em>a</em>,
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divisor <em>d</em>, quotient <em>q</em>, and <em>b = 2^mp_bits_per_limb</em>, this remainder
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<em>r</em> is of the form
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</p>
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<div class="example">
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<pre class="example">a = q*d + r*b^n
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</pre></div>
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<p><em>n</em> represents the number of zero limbs produced by the subtractions,
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that being the number of limbs produced for <em>q</em>. <em>r</em> will be in the
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range <em>0<=r<d</em> and can be viewed as a remainder, but one shifted up by
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a factor of <em>b^n</em>.
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</p>
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<p>Carrying out full subtractions at each stage means the same number of cross
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products must be done as a normal division, but there’s still some single limb
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divisions saved. When <em>d</em> is a single limb some simplifications arise,
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providing good speedups on a number of processors.
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</p>
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<p>The functions <code>mpn_divexact_by3</code>, <code>mpn_modexact_1_odd</code> and the
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internal <code>mpn_redc_X</code> functions differ subtly in how they return <em>r</em>,
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leading to some negations in the above formula, but all are essentially the
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same.
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</p>
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<a name="index-Divisibility-algorithm"></a>
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<a name="index-Congruence-algorithm"></a>
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<p>Clearly <em>r</em> is zero when <em>a</em> is a multiple of <em>d</em>, and this
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leads to divisibility or congruence tests which are potentially more efficient
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than a normal division.
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</p>
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<p>The factor of <em>b^n</em> on <em>r</em> can be ignored in a GCD when <em>d</em> is
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odd, hence the use of <code>mpn_modexact_1_odd</code> by <code>mpn_gcd_1</code> and
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<code>mpz_kronecker_ui</code> etc (see <a href="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms">Greatest Common Divisor Algorithms</a>).
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</p>
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<p>Montgomery’s REDC method for modular multiplications uses operands of the form
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of <em>x*b^-n</em> and <em>y*b^-n</em> and on calculating <em>(x*b^-n)*(y*b^-n)</em> uses the factor of <em>b^n</em> in the exact
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remainder to reach a product in the same form <em>(x*y)*b^-n</em>
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(see <a href="Modular-Powering-Algorithm.html#Modular-Powering-Algorithm">Modular Powering Algorithm</a>).
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</p>
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<p>Notice that <em>r</em> generally gives no useful information about the ordinary
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remainder <em>a mod d</em> since <em>b^n mod d</em> could be anything. If
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however <em>b^n ≡ 1 mod d</em>, then <em>r</em> is the negative of the
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ordinary remainder. This occurs whenever <em>d</em> is a factor of
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<em>b^n-1</em>, as for example with 3 in <code>mpn_divexact_by3</code>. For a 32 or
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64 bit limb other such factors include 5, 17 and 257, but no particular use
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has been found for this.
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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="Small-Quotient-Division.html#Small-Quotient-Division" accesskey="n" rel="next">Small Quotient Division</a>, Previous: <a href="Exact-Division.html#Exact-Division" accesskey="p" rel="prev">Exact Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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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