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89 lines
3.1 KiB
Plaintext
89 lines
3.1 KiB
Plaintext
NAME
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rcmul - REDC multiplication
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SYNOPSIS
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rcmul(x, y, m)
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TYPES
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x integer
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y integer
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m odd positive integer
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return integer v, 0 <= v < m.
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DESCRIPTION
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Let B be the base calc uses for representing integers internally
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(B = 2^16 for 32-bit machines, 2^32 for 64-bit machines)
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and N the number of words (base-B digits) in the representation
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of m. Then rcmul(x,y,m) returns the value of B^-N * x * y % m,
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where the inverse implicit in B^-N is modulo m
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and the modulus operator % gives the least non-negative residue.
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The normal use of rcmul() may be said to be that of multiplying modulo m
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values encoded by rcin() and REDC functions, as in:
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rcin(x * y, m) = rcmul(rcin(x,m), rcin(y,m), m),
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or with only one factor encoded:
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x * y % m = rcmul(rcin(x,m), y, m).
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RUNTIME
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If the value of m in rcmul(x,y,m) is being used for the first time
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in a REDC function, the information required for the REDC
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algorithms is calculated and stored for future use, possibly
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replacing an already stored valued, in a table covering up to 5
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(i.e. MAXREDC) values of m. The runtime required for this is about
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two times that required for multiplying two N-word integers.
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Two algorithms are available for evaluating rcmul(x,y,m), the one
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which is usually faster for small N is used when N <
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config("redc2"); the other is usually faster for larger N. If
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config("redc2") is set at about 90 and x and y have both been
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reduced modulo m, the runtime required for rcmul(x,y,m) is at most
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about f times the runtime required for an N-word by N-word
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multiplication, where f increases from about 1.3 for N = 1 to near
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3 for N > 90. More runtime may be required if x and y have to be
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reduced modulo m.
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EXAMPLE
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Using a 64-bit machine with B = 2^32:
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> print rcin(4 * 5, 9), rcmul(rcin(4,9), rcin(5,9), 9), rcout(8, 9);
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8 8 2
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LIMITS
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none
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LINK LIBRARY
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void zredcmul(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res)
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SEE ALSO
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rcin, rcout, rcsq, rcpow
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## Copyright (C) 1999 Landon Curt Noll
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##
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## Calc is open software; you can redistribute it and/or modify it under
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## the terms of the version 2.1 of the GNU Lesser General Public License
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## as published by the Free Software Foundation.
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##
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## Calc is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## Public License for more details.
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##
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## A copy of version 2.1 of the GNU Lesser General Public License is
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## distributed with calc under the filename COPYING-LGPL. You should have
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## received a copy with calc; if not, write to Free Software Foundation, Inc.
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## 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
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##
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## @(#) $Revision: 29.1 $
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## @(#) $Id: rcmul,v 29.1 1999/12/14 09:16:04 chongo Exp $
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## @(#) $Source: /usr/local/src/cmd/calc/help/RCS/rcmul,v $
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##
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## Under source code control: 1996/02/25 02:22:21
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## File existed as early as: 1996
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##
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## chongo <was here> /\oo/\ http://reality.sgi.com/chongo/
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## Share and enjoy! :-) http://reality.sgi.com/chongo/tech/comp/calc/
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