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Russ Cox 3ba3b4893f math/big: use Montgomery for z.Exp(x, y, m) even for even m 
Montgomery multiplication can be used for Exp mod even m
by splitting it into two steps - Exp mod an odd number and
Exp mod a power of two - and then combining the two results
using the Chinese Remainder Theorem.

For more details, see Ç. K. Koç, “Montgomery Reduction with Even Modulus”,
IEE Proceedings: Computers and Digital Techniques, 141(5) 314-316, September 1994.
http://www.people.vcu.edu/~jwang3/CMSC691/j34monex.pdf

Thanks to Guido Vranken for suggesting that we use a faster algorithm.

name                   old time/op    new time/op    delta
ExpMont/Odd-16            240µs ± 2%     239µs ± 2%      ~     (p=0.853 n=10+10)
ExpMont/Even1-16          757µs ± 3%     249µs ± 6%   -67.17%  (p=0.000 n=10+10)
ExpMont/Even2-16          755µs ± 1%     244µs ± 4%   -67.64%  (p=0.000 n=8+10)
ExpMont/Even3-16          771µs ± 3%     240µs ± 2%   -68.90%  (p=0.000 n=10+10)
ExpMont/Even4-16          775µs ± 3%     241µs ± 2%   -68.91%  (p=0.000 n=10+10)
ExpMont/Even8-16          779µs ± 2%     241µs ± 3%   -69.06%  (p=0.000 n=9+10)
ExpMont/Even32-16         778µs ± 3%     240µs ± 4%   -69.11%  (p=0.000 n=9+9)
ExpMont/Even64-16         774µs ± 6%     186µs ± 2%   -76.00%  (p=0.000 n=10+10)
ExpMont/Even96-16         776µs ± 4%     186µs ± 6%   -76.09%  (p=0.000 n=9+10)
ExpMont/Even128-16        764µs ± 2%     143µs ± 3%   -81.24%  (p=0.000 n=10+10)
ExpMont/Even255-16        761µs ± 3%     109µs ± 2%   -85.73%  (p=0.000 n=10+10)
ExpMont/SmallEven1-16    45.6µs ± 1%    36.3µs ± 3%   -20.49%  (p=0.000 n=9+10)
ExpMont/SmallEven2-16    44.3µs ± 2%    37.5µs ± 2%   -15.26%  (p=0.000 n=9+10)
ExpMont/SmallEven3-16    44.1µs ± 5%    37.3µs ± 3%   -15.32%  (p=0.000 n=9+10)
ExpMont/SmallEven4-16    47.1µs ± 6%    38.0µs ± 5%   -19.40%  (p=0.000 n=10+9)

name                   old alloc/op   new alloc/op   delta
ExpMont/Odd-16           2.53kB ± 0%    2.53kB ± 0%      ~     (p=0.137 n=8+10)
ExpMont/Even1-16         2.57kB ± 0%    3.31kB ± 0%   +28.90%  (p=0.000 n=8+10)
ExpMont/Even2-16         2.57kB ± 0%    3.37kB ± 0%   +31.09%  (p=0.000 n=9+10)
ExpMont/Even3-16         2.57kB ± 0%    3.37kB ± 0%   +31.08%  (p=0.000 n=8+8)
ExpMont/Even4-16         2.57kB ± 0%    3.37kB ± 0%   +31.09%  (p=0.000 n=9+10)
ExpMont/Even8-16         2.57kB ± 0%    3.37kB ± 0%   +31.09%  (p=0.000 n=9+10)
ExpMont/Even32-16        2.57kB ± 0%    3.37kB ± 0%   +31.14%  (p=0.000 n=10+10)
ExpMont/Even64-16        2.57kB ± 0%    3.16kB ± 0%   +22.99%  (p=0.000 n=9+9)
ExpMont/Even96-16        2.57kB ± 0%    3.44kB ± 0%   +33.90%  (p=0.000 n=10+8)
ExpMont/Even128-16       2.57kB ± 0%    2.88kB ± 0%   +12.10%  (p=0.000 n=10+10)
ExpMont/Even255-16       2.57kB ± 0%    2.54kB ± 0%    -1.30%  (p=0.000 n=9+10)
ExpMont/SmallEven1-16      872B ± 0%     1232B ± 0%   +41.28%  (p=0.000 n=10+10)
ExpMont/SmallEven2-16      872B ± 0%     1233B ± 0%   +41.40%  (p=0.000 n=10+9)
ExpMont/SmallEven3-16      872B ± 0%     1289B ± 0%   +47.82%  (p=0.000 n=10+10)
ExpMont/SmallEven4-16      872B ± 0%     1241B ± 0%   +42.32%  (p=0.000 n=10+10)

name                   old allocs/op  new allocs/op  delta
ExpMont/Odd-16             21.0 ± 0%      21.0 ± 0%      ~     (all equal)
ExpMont/Even1-16           24.0 ± 0%      38.0 ± 0%   +58.33%  (p=0.000 n=10+10)
ExpMont/Even2-16           24.0 ± 0%      40.0 ± 0%   +66.67%  (p=0.000 n=10+10)
ExpMont/Even3-16           24.0 ± 0%      40.0 ± 0%   +66.67%  (p=0.000 n=10+10)
ExpMont/Even4-16           24.0 ± 0%      40.0 ± 0%   +66.67%  (p=0.000 n=10+10)
ExpMont/Even8-16           24.0 ± 0%      40.0 ± 0%   +66.67%  (p=0.000 n=10+10)
ExpMont/Even32-16          24.0 ± 0%      40.0 ± 0%   +66.67%  (p=0.000 n=10+10)
ExpMont/Even64-16          24.0 ± 0%      39.0 ± 0%   +62.50%  (p=0.000 n=10+10)
ExpMont/Even96-16          24.0 ± 0%      42.0 ± 0%   +75.00%  (p=0.000 n=10+10)
ExpMont/Even128-16         24.0 ± 0%      40.0 ± 0%   +66.67%  (p=0.000 n=10+10)
ExpMont/Even255-16         24.0 ± 0%      38.0 ± 0%   +58.33%  (p=0.000 n=10+10)
ExpMont/SmallEven1-16      16.0 ± 0%      35.0 ± 0%  +118.75%  (p=0.000 n=10+10)
ExpMont/SmallEven2-16      16.0 ± 0%      35.0 ± 0%  +118.75%  (p=0.000 n=10+10)
ExpMont/SmallEven3-16      16.0 ± 0%      37.0 ± 0%  +131.25%  (p=0.000 n=10+10)
ExpMont/SmallEven4-16      16.0 ± 0%      36.0 ± 0%  +125.00%  (p=0.000 n=10+10)

Change-Id: Ib7f70290f8f087b78805ec3120baf17dd7737ac3
Reviewed-on: https://go-review.googlesource.com/c/go/+/420897
Run-TryBot: Russ Cox <rsc@golang.org>
Reviewed-by: Roland Shoemaker <roland@golang.org>
Auto-Submit: Russ Cox <rsc@golang.org>
TryBot-Result: Gopher Robot <gobot@golang.org>
2022-11-02 14:43:52 +00:00
.github .github: update issue label for pkgsite-removal 2022-09-07 16:00:20 +00:00
api os/exec: add the Cancel and WaitDelay fields 2022-10-25 03:34:36 +00:00
doc doc/go1.20: go version supports non-executable Go binaries 2022-10-24 15:56:26 +00:00
lib/time lib/time, time/tzdata: update to 2022b 2022-08-11 20:03:19 +00:00
misc cmd/go: don't install most GOROOT .a files in pkg 2022-10-28 23:35:08 +00:00
src math/big: use Montgomery for z.Exp(x, y, m) even for even m 2022-11-02 14:43:52 +00:00
test test/bench/go1: eliminate start-up time 2022-11-01 17:07:14 +00:00
.gitattributes
.gitignore internal/buildcfg: move build configuration out of cmd/internal/objabi 2021-04-16 19:20:53 +00:00
codereview.cfg codereview.cfg: add codereview.cfg for master branch 2021-02-19 18:44:53 +00:00
CONTRIBUTING.md
LICENSE
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README.md README.md: update wiki link 2022-04-26 16:21:18 +00:00
SECURITY.md SECURITY.md: replace golang.org with go.dev 2022-04-26 19:59:47 +00:00

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