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security/nss/lib/freebl/mpi/doc/mul.txt
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| 1 Multiplication | |
| 2 | |
| 3 This describes the multiplication algorithm used by the MPI library. | |
| 4 | |
| 5 This is basically a standard "schoolbook" algorithm. It is slow -- | |
| 6 O(mn) for m = #a, n = #b -- but easy to implement and verify. | |
| 7 Basically, we run two nested loops, as illustrated here (R is the | |
| 8 radix): | |
| 9 | |
| 10 k = 0 | |
| 11 for j <- 0 to (#b - 1) | |
| 12 for i <- 0 to (#a - 1) | |
| 13 w = (a[j] * b[i]) + k + c[i+j] | |
| 14 c[i+j] = w mod R | |
| 15 k = w div R | |
| 16 endfor | |
| 17 c[i+j] = k; | |
| 18 k = 0; | |
| 19 endfor | |
| 20 | |
| 21 It is necessary that 'w' have room for at least two radix R digits. | |
| 22 The product of any two digits in radix R is at most: | |
| 23 | |
| 24 (R - 1)(R - 1) = R^2 - 2R + 1 | |
| 25 | |
| 26 Since a two-digit radix-R number can hold R^2 - 1 distinct values, | |
| 27 this insures that the product will fit into the two-digit register. | |
| 28 | |
| 29 To insure that two digits is enough for w, we must also show that | |
| 30 there is room for the carry-in from the previous multiplication, and | |
| 31 the current value of the product digit that is being recomputed. | |
| 32 Assuming each of these may be as big as R - 1 (and no larger, | |
| 33 certainly), two digits will be enough if and only if: | |
| 34 | |
| 35 (R^2 - 2R + 1) + 2(R - 1) <= R^2 - 1 | |
| 36 | |
| 37 Solving this equation shows that, indeed, this is the case: | |
| 38 | |
| 39 R^2 - 2R + 1 + 2R - 2 <= R^2 - 1 | |
| 40 | |
| 41 R^2 - 1 <= R^2 - 1 | |
| 42 | |
| 43 This suggests that a good radix would be one more than the largest | |
| 44 value that can be held in half a machine word -- so, for example, as | |
| 45 in this implementation, where we used a radix of 65536 on a machine | |
| 46 with 4-byte words. Another advantage of a radix of this sort is that | |
| 47 binary-level operations are easy on numbers in this representation. | |
| 48 | |
| 49 Here's an example multiplication worked out longhand in radix-10, | |
| 50 using the above algorithm: | |
| 51 | |
| 52 a = 999 | |
| 53 b = x 999 | |
| 54 ------------- | |
| 55 p = 98001 | |
| 56 | |
| 57 w = (a[jx] * b[ix]) + kin + c[ix + jx] | |
| 58 c[ix+jx] = w % RADIX | |
| 59 k = w / RADIX | |
| 60 product | |
| 61 ix jx a[jx] b[ix] kin w c[i+j] kout 000000 | |
| 62 0 0 9 9 0 81+0+0 1 8 000001 | |
| 63 0 1 9 9 8 81+8+0 9 8 000091 | |
| 64 0 2 9 9 8 81+8+0 9 8 000991 | |
| 65 8 0 008991 | |
| 66 1 0 9 9 0 81+0+9 0 9 008901 | |
| 67 1 1 9 9 9 81+9+9 9 9 008901 | |
| 68 1 2 9 9 9 81+9+8 8 9 008901 | |
| 69 9 0 098901 | |
| 70 2 0 9 9 0 81+0+9 0 9 098001 | |
| 71 2 1 9 9 9 81+9+8 8 9 098001 | |
| 72 2 2 9 9 9 81+9+9 9 9 098001 | |
| 73 | |
| 74 ------------------------------------------------------------------ | |
| 75 This Source Code Form is subject to the terms of the Mozilla Public | |
| 76 # License, v. 2.0. If a copy of the MPL was not distributed with this | |
| 77 # file, You can obtain one at http://mozilla.org/MPL/2.0/. |
