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110 lines
3.8 KiB
Plaintext
110 lines
3.8 KiB
Plaintext
Squaring Algorithm
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When you are squaring a value, you can take advantage of the fact that
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half the multiplications performed by the more general multiplication
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algorithm (see 'mul.txt' for a description) are redundant when the
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multiplicand equals the multiplier.
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In particular, the modified algorithm is:
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k = 0
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for j <- 0 to (#a - 1)
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w = c[2*j] + (a[j] ^ 2);
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k = w div R
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for i <- j+1 to (#a - 1)
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w = (2 * a[j] * a[i]) + k + c[i+j]
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c[i+j] = w mod R
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k = w div R
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endfor
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c[i+j] = k;
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k = 0;
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endfor
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On the surface, this looks identical to the multiplication algorithm;
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however, note the following differences:
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- precomputation of the leading term in the outer loop
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- i runs from j+1 instead of from zero
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- doubling of a[i] * a[j] in the inner product
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Unfortunately, the construction of the inner product is such that we
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need more than two digits to represent the inner product, in some
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cases. In a C implementation, this means that some gymnastics must be
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performed in order to handle overflow, for which C has no direct
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abstraction. We do this by observing the following:
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If we have multiplied a[i] and a[j], and the product is more than half
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the maximum value expressible in two digits, then doubling this result
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will overflow into a third digit. If this occurs, we take note of the
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overflow, and double it anyway -- C integer arithmetic ignores
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overflow, so the two digits we get back should still be valid, modulo
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the overflow.
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Having doubled this value, we now have to add in the remainders and
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the digits already computed by earlier steps. If we did not overflow
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in the previous step, we might still cause an overflow here. That
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will happen whenever the maximum value expressible in two digits, less
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the amount we have to add, is greater than the result of the previous
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step. Thus, the overflow computation is:
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u = 0
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w = a[i] * a[j]
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if(w > (R - 1)/ 2)
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u = 1;
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w = w * 2
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v = c[i + j] + k
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if(u == 0 && (R - 1 - v) < w)
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u = 1
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If there is an overflow, u will be 1, otherwise u will be 0. The rest
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of the parameters are the same as they are in the above description.
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------------------------------------------------------------------
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***** BEGIN LICENSE BLOCK *****
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Version: MPL 1.1/GPL 2.0/LGPL 2.1
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The contents of this file are subject to the Mozilla Public License Version
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1.1 (the "License"); you may not use this file except in compliance with
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the License. You may obtain a copy of the License at
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http://www.mozilla.org/MPL/
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Software distributed under the License is distributed on an "AS IS" basis,
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WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
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for the specific language governing rights and limitations under the
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License.
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The Original Code is the MPI Arbitrary Precision Integer Arithmetic
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library.
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The Initial Developer of the Original Code is
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Michael J. Fromberger <sting@linguist.dartmouth.edu>
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Portions created by the Initial Developer are Copyright (C) 1998, 2000
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the Initial Developer. All Rights Reserved.
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Contributor(s):
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Alternatively, the contents of this file may be used under the terms of
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either the GNU General Public License Version 2 or later (the "GPL"), or
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the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
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in which case the provisions of the GPL or the LGPL are applicable instead
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of those above. If you wish to allow use of your version of this file only
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under the terms of either the GPL or the LGPL, and not to allow others to
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use your version of this file under the terms of the MPL, indicate your
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decision by deleting the provisions above and replace them with the notice
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and other provisions required by the GPL or the LGPL. If you do not delete
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the provisions above, a recipient may use your version of this file under
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the terms of any one of the MPL, the GPL or the LGPL.
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***** END LICENSE BLOCK *****
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$Id: square.txt,v 1.2 2005/02/02 22:28:22 gerv%gerv.net Exp $
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