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diff --git a/security/nss/lib/freebl/mpi/doc/mul.txt b/security/nss/lib/freebl/mpi/doc/mul.txt deleted file mode 100644 index 436d01bf7..000000000 --- a/security/nss/lib/freebl/mpi/doc/mul.txt +++ /dev/null @@ -1,109 +0,0 @@ -Multiplication - -This describes the multiplication algorithm used by the MPI library. - -This is basically a standard "schoolbook" algorithm. It is slow -- -O(mn) for m = #a, n = #b -- but easy to implement and verify. -Basically, we run two nested loops, as illustrated here (R is the -radix): - -k = 0 -for j <- 0 to (#b - 1) - for i <- 0 to (#a - 1) - w = (a[j] * b[i]) + k + c[i+j] - c[i+j] = w mod R - k = w div R - endfor - c[i+j] = k; - k = 0; -endfor - -It is necessary that 'w' have room for at least two radix R digits. -The product of any two digits in radix R is at most: - - (R - 1)(R - 1) = R^2 - 2R + 1 - -Since a two-digit radix-R number can hold R^2 - 1 distinct values, -this insures that the product will fit into the two-digit register. - -To insure that two digits is enough for w, we must also show that -there is room for the carry-in from the previous multiplication, and -the current value of the product digit that is being recomputed. -Assuming each of these may be as big as R - 1 (and no larger, -certainly), two digits will be enough if and only if: - - (R^2 - 2R + 1) + 2(R - 1) <= R^2 - 1 - -Solving this equation shows that, indeed, this is the case: - - R^2 - 2R + 1 + 2R - 2 <= R^2 - 1 - - R^2 - 1 <= R^2 - 1 - -This suggests that a good radix would be one more than the largest -value that can be held in half a machine word -- so, for example, as -in this implementation, where we used a radix of 65536 on a machine -with 4-byte words. Another advantage of a radix of this sort is that -binary-level operations are easy on numbers in this representation. - -Here's an example multiplication worked out longhand in radix-10, -using the above algorithm: - - a = 999 - b = x 999 - ------------- - p = 98001 - -w = (a[jx] * b[ix]) + kin + c[ix + jx] -c[ix+jx] = w % RADIX -k = w / RADIX - product -ix jx a[jx] b[ix] kin w c[i+j] kout 000000 -0 0 9 9 0 81+0+0 1 8 000001 -0 1 9 9 8 81+8+0 9 8 000091 -0 2 9 9 8 81+8+0 9 8 000991 - 8 0 008991 -1 0 9 9 0 81+0+9 0 9 008901 -1 1 9 9 9 81+9+9 9 9 008901 -1 2 9 9 9 81+9+8 8 9 008901 - 9 0 098901 -2 0 9 9 0 81+0+9 0 9 098001 -2 1 9 9 9 81+9+8 8 9 098001 -2 2 9 9 9 81+9+9 9 9 098001 - ------------------------------------------------------------------- -The contents of this file are subject to the Mozilla Public -License Version 1.1 (the "License"); you may not use this file -except in compliance with the License. You may obtain a copy of -the License at http://www.mozilla.org/MPL/ - -Software distributed under the License is distributed on an "AS -IS" basis, WITHOUT WARRANTY OF ANY KIND, either express or -implied. See the License for the specific language governing -rights and limitations under the License. - -The Original Code is the MPI Arbitrary Precision Integer Arithmetic -library. - -The Initial Developer of the Original Code is -Michael J. Fromberger <sting@linguist.dartmouth.edu> - -Portions created by Michael J. Fromberger are -Copyright (C) 1998, 2000 Michael J. Fromberger. All Rights Reserved. - -Contributor(s): - -Alternatively, the contents of this file may be used under the -terms of the GNU General Public License Version 2 or later (the -"GPL"), in which case the provisions of the GPL are applicable -instead of those above. If you wish to allow use of your -version of this file only under the terms of the GPL and not to -allow others to use your version of this file under the MPL, -indicate your decision by deleting the provisions above and -replace them with the notice and other provisions required by -the GPL. If you do not delete the provisions above, a recipient -may use your version of this file under either the MPL or the GPL. - -$Id$ - - |