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Brian Kessler 50649a967c math/big: implement Lehmer's extended GCD algorithm 
Updates #15833

The extended GCD algorithm can be implemented using
Lehmer's algorithm with additional updates for the
cosequences following Algorithm 10.45 from Cohen et al.
"Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
This brings the speed of the extended GCD calculation within
~2x of the base GCD calculation.  There is a slight degradation in
the non-extended GCD speed for small inputs (1-2 words) due to the
additional code to handle the extended updates.

name                          old time/op    new time/op    delta
GCD10x10/WithoutXY-4             262ns ± 1%     266ns ± 2%     ~     (p=0.333 n=5+5)
GCD10x10/WithXY-4               1.42µs ± 2%    0.74µs ± 3%  -47.90%  (p=0.008 n=5+5)
GCD10x100/WithoutXY-4            520ns ± 2%     539ns ± 1%   +3.81%  (p=0.008 n=5+5)
GCD10x100/WithXY-4              2.32µs ± 1%    1.67µs ± 0%  -27.80%  (p=0.008 n=5+5)
GCD10x1000/WithoutXY-4          1.40µs ± 1%    1.45µs ± 2%   +3.26%  (p=0.016 n=4+5)
GCD10x1000/WithXY-4             4.78µs ± 1%    3.43µs ± 1%  -28.37%  (p=0.008 n=5+5)
GCD10x10000/WithoutXY-4         10.0µs ± 0%    10.2µs ± 3%   +1.80%  (p=0.008 n=5+5)
GCD10x10000/WithXY-4            20.9µs ± 3%    17.9µs ± 1%  -14.20%  (p=0.008 n=5+5)
GCD10x100000/WithoutXY-4        96.8µs ± 0%    96.3µs ± 1%     ~     (p=0.310 n=5+5)
GCD10x100000/WithXY-4            196µs ± 3%     159µs ± 2%  -18.61%  (p=0.008 n=5+5)
GCD100x100/WithoutXY-4          2.53µs ±15%    2.34µs ± 0%   -7.35%  (p=0.008 n=5+5)
GCD100x100/WithXY-4             19.3µs ± 0%     3.9µs ± 1%  -79.58%  (p=0.008 n=5+5)
GCD100x1000/WithoutXY-4         4.23µs ± 0%    4.17µs ± 3%     ~     (p=0.127 n=5+5)
GCD100x1000/WithXY-4            22.8µs ± 1%     7.5µs ±10%  -67.00%  (p=0.008 n=5+5)
GCD100x10000/WithoutXY-4        19.1µs ± 0%    19.0µs ± 0%     ~     (p=0.095 n=5+5)
GCD100x10000/WithXY-4           75.1µs ± 2%    30.5µs ± 2%  -59.38%  (p=0.008 n=5+5)
GCD100x100000/WithoutXY-4        170µs ± 5%     167µs ± 1%     ~     (p=1.000 n=5+5)
GCD100x100000/WithXY-4           542µs ± 2%     267µs ± 2%  -50.79%  (p=0.008 n=5+5)
GCD1000x1000/WithoutXY-4        28.0µs ± 0%    27.1µs ± 0%   -3.29%  (p=0.008 n=5+5)
GCD1000x1000/WithXY-4            329µs ± 0%      42µs ± 1%  -87.12%  (p=0.008 n=5+5)
GCD1000x10000/WithoutXY-4       47.2µs ± 0%    46.4µs ± 0%   -1.65%  (p=0.016 n=5+4)
GCD1000x10000/WithXY-4           607µs ± 9%     123µs ± 1%  -79.70%  (p=0.008 n=5+5)
GCD1000x100000/WithoutXY-4       260µs ±17%     245µs ± 0%     ~     (p=0.056 n=5+5)
GCD1000x100000/WithXY-4         3.64ms ± 1%    0.93ms ± 1%  -74.41%  (p=0.016 n=4+5)
GCD10000x10000/WithoutXY-4       513µs ± 0%     507µs ± 0%   -1.22%  (p=0.008 n=5+5)
GCD10000x10000/WithXY-4         7.44ms ± 1%    1.00ms ± 0%  -86.58%  (p=0.008 n=5+5)
GCD10000x100000/WithoutXY-4     1.23ms ± 0%    1.23ms ± 1%     ~     (p=0.056 n=5+5)
GCD10000x100000/WithXY-4        37.3ms ± 0%     7.3ms ± 1%  -80.45%  (p=0.008 n=5+5)
GCD100000x100000/WithoutXY-4    24.2ms ± 0%    24.2ms ± 0%     ~     (p=0.841 n=5+5)
GCD100000x100000/WithXY-4        505ms ± 1%      56ms ± 1%  -88.92%  (p=0.008 n=5+5)

Change-Id: I25f42ab8c55033acb83cc32bb03c12c1963925e8
Reviewed-on: https://go-review.googlesource.com/78755
Reviewed-by: Robert Griesemer <gri@golang.org>
Run-TryBot: Robert Griesemer <gri@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
 		2018-05-08 17:24:36 +00:00
.github github: update Pull Request template 2018-02-21 02:07:46 +00:00
api syscall: introduce Pointer type and use it instead of uintptr 2018-04-18 08:50:42 +00:00
doc doc: document Go 1.10.2 2018-04-30 18:01:00 +00:00
lib/time all: use HTTPS for iana.org links 2018-02-13 18:36:48 +00:00
misc misc/ios: retry iOS launch even if no device app path was found 2018-05-08 15:00:57 +00:00
src math/big: implement Lehmer's extended GCD algorithm 2018-05-08 17:24:36 +00:00
test cmd/compile: simplify shift lowering on s390x 2018-05-08 16:19:56 +00:00
.gitattributes .gitattributes: prevent all magic line ending changes 2014-12-12 23:14:54 +00:00
.gitignore .gitignore: ignore src/cmd/dist/dist 2017-10-28 21:55:49 +00:00
AUTHORS A+C: update email address 2018-04-14 19:28:37 +00:00
CONTRIBUTING.md all: restore changes from faulty merge/revert 2018-02-12 20:13:59 +00:00
CONTRIBUTORS C: add Iskander Sharipov's individual CLA mail 2018-04-16 04:49:46 +00:00
LICENSE doc: revert copyright date to 2009 2016-06-01 22:40:04 +00:00
PATENTS
…
README.md README: update number of contributors 2018-04-03 22:10:32 +00:00
favicon.ico website: recreate 16px and 32px favicon 2016-08-25 15:43:32 +00:00
robots.txt
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