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 2019-10-25, 19:48 #1 mgb   "Michael" Aug 2006 Usually at home 22·3·7 Posts Calculating inverses quickly. Do you think this would be faster than the extended Euclidean Algorithm for finding inverses? For a given a find a-1 (mod m) 1. Let m = r mod a 2. Find r -1 mod a 3. Let x = a - r -1 (ie, x = -r -1 mod a) 4. a-1 mod m = (xm + 1)/a If this algorithm is applied recursively, at step 2., it should be possible to reduce the question finding the inverse of a small number and then working back up the chain to find the original inverse. The reasoning is as follows:- aa-1 = km + 1, for some k < a, so km = -1 (mod a) That is, k = -m -1 (mod a) Adding 1, km + 1 = 0 (mod a) Whence, (km + 1)/a = a-1 (Moderator, I meant to post this in the computing forum but posted here by mistake.) Last fiddled with by mgb on 2019-10-25 at 20:45
2019-10-26, 13:48   #2
Dr Sardonicus

Feb 2017
Nowhere

633910 Posts

Quote:
 Originally Posted by mgb Do you think this would be faster than the extended Euclidean Algorithm for finding inverses? For a given a find a-1 (mod m) 1. Let m = r mod a 2. Find r -1 mod a 3. Let x = a - r -1 (ie, x = -r -1 mod a) 4. a-1 mod m = (xm + 1)/a If this algorithm is applied recursively, at step 2., it should be possible to reduce the question finding the inverse of a small number and then working back up the chain to find the original inverse.
If I understand you correctly, you're doing the same sequence of divisions with quotient and remainder as in the Euclidean algorithm.

Last fiddled with by Dr Sardonicus on 2019-10-26 at 13:51 Reason: Rephrasing

2019-10-26, 19:21   #3
mgb

"Michael"
Aug 2006
Usually at home

22×3×7 Posts

Quote:
 Originally Posted by Dr Sardonicus If I understand you correctly, you're doing the same sequence of divisions with quotient and remainder as in the Euclidean algorithm.
Yes but at step 2 the task is reduced to a smaller pair of numbers, to which the extended Euclidean algorithm can be applied. This is an immediate saving in processor time. Step 2 = Find r -1 mod a

Example:

Find 17 -1 mod 223
-223 -1 = 8 mod 17
(8*223 + 1)/17 = 105 = 17 -1 mod 223

Last fiddled with by mgb on 2019-10-26 at 19:41

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