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Some folks might think: “you still use RCS”?!? And we will say, hey, at least we switched from SCCS to RCS back in … I think it was around 1994 ... at least we are keeping up! :-) :-) :-) Logs say that SCCS version 18 became RCS version 19 on 1994 March 18. RCS served us well. But now it is time to move on. And so we are switching to git. Calc releases produce a lot of file changes. In the 125 releases of calc since 1996, when I started managing calc releases, there have been 15473 file mods!
96 lines
3.5 KiB
Plaintext
96 lines
3.5 KiB
Plaintext
NAME
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rcin - encode for REDC algorithms
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SYNOPSIS
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rcin(x, m)
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TYPES
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x 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) and N the
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number of words (base-B digits) in the representation of m. Then
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rcin(x,m) returns the value of B^N * x % m, where the modulus
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operator % here gives the least nonnegative residue.
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If y = rcin(x,m), x % m may be evaluated by x % m = rcout(y, m).
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The "encoding" method of using rcmul(), rcsq(), and rcpow() for
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evaluating products, squares and powers modulo m correspond to the
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formulae:
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rcin(x * y, m) = rcmul(rcin(x,m), rcin(y,m), m);
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rcin(x^2, m) = rcsq(rcin(x,m), m);
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rcin(x^k, m) = rcpow(rcin(x,m), k, m).
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Here k is any nonnegative integer. Using these formulae may be
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faster than direct evaluation of x * y % m, x^2 % m, x^k % m.
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Some encoding and decoding may be bypassed by formulae like:
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x * y % m = rcin(rcmul(x, y, m), m).
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If m is a divisor of B^N - h for some integer h, rcin(x,m) may be
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computed by using rcin(x,m) = h * x % m. In particular, if
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m is a divisor of B^N - 1 and 0 <= x < m, then rcin(x,m) = x.
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For example if B = 2^16 or 2^32, this is so for m = (B^N - 1)/d
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for the divisors d = 3, 5, 15, 17, ...
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RUNTIME
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The first time a particular value for m is used in rcin(x, m),
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the information required for the REDC algorithms is
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calculated and stored for future use in a table covering up to
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5 (i.e. MAXREDC) values of m. The runtime required for this is about
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two that required for multiplying two N-word integers.
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Two algorithms are available for evaluating rcin(x, m), the one
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which is usually faster for small N is used when N <
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config("pow2"); the other is usually faster for larger N. If
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config("pow2") is set at about 200 and x has both been reduced
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modulo m, the runtime required for rcin(x, m) is at most about f
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times the runtime required for an N-word by N-word multiplication,
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where f increases from about 1.3 for N = 1 to near 2 for N > 200.
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More runtime may be required if x has to be 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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; for (i = 0; i < 9; i++) print rcin(x, 9),:; print;
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0 4 8 3 7 2 6 1 5
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LIMITS
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none
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LINK LIBRARY
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void zredcencode(REDC *rp, ZVALUE z1, ZVALUE *res)
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SEE ALSO
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rcout, rcmul, 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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## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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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://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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