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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
3.3 KiB
Plaintext
96 lines
3.3 KiB
Plaintext
NAME
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rcpow - REDC powers
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SYNOPSIS
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rcpow(x, k, m)
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TYPES
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x integer
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k nonnegative 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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rcpow(x,k,m) returns the value of B^-N * (B^N * x)^k % m, w here
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the inverse implicit in B^-N is modulo m and the modulus operator %
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gives the least nonnegative residue. Note that rcpow(x,0,m) =
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rcin(1,m), rcpow(x,1,m) = x % m; rcpow(x,2,m) = rcsq(x,m).
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The normal use of rcpow() may be said to be that of finding the
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encoded value of the k-th power of an integer modulo m:
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rcin(x^k, m) = rcpow(rcin(x,m), k, m),
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from which one gets:
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x^k % m = rcout(rcpow(rcin(x,m), k, m), m).
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If x^k % m is to be evaluated for the same k and m and several
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values of x, it may be worth while to first evaluate:
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a = minv(rcpow(1, k, m), m);
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and use:
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x^k % m = a * rcpow(x, k, m) % m.
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RUNTIME
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If the value of m in rcpow(x,k,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 rcpow(x,k,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 0 <= x < m, the runtime
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required for rcpow(x,k,m) is at most about f times the runtime
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required for ilog2(k) N-word by N-word multiplications, where f
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increases from about 1.3 for N = 1 to near 4 for N > 90. More
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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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; m = 1234567;
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; x = 15;
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; print rcout(rcpow((rcin(x,m), m - 1, m), m), pmod(x, m-1, m)
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783084 783084
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LIMITS
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none
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LINK LIBRARY
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void zredcpower(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res)
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SEE ALSO
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rcin, rcout, rcmul, rcsq
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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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