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Change-Id: I4dc846227af95a73ee9a3074d0c379ff0fa955df
Reviewed-on: https://go-review.googlesource.com/115798
Reviewed-by: Ian Lance Taylor <iant@golang.org>
Run-TryBot: Ian Lance Taylor <iant@golang.org>
After x.ProbablyPrime(n) passes the n Miller-Rabin rounds,
add a Baillie-PSW test before declaring x probably prime.
Although the provable error bounds are unchanged, the empirical
error bounds drop dramatically: there are no known inputs
for which Baillie-PSW gives the wrong answer. For example,
before this CL, big.NewInt(443*1327).ProbablyPrime(1) == true.
Now it is (correctly) false.
The new Baillie-PSW test is two pieces: an added Miller-Rabin
round with base 2, and a so-called extra strong Lucas test.
(See the references listed in prime.go for more details.)
The Lucas test takes about 3.5x as long as the Miller-Rabin round,
which is close to theoretical expectations.
name time/op
ProbablyPrime/Lucas 2.91ms ± 2%
ProbablyPrime/MillerRabinBase2 850µs ± 1%
ProbablyPrime/n=0 3.75ms ± 3%
The speed of prime testing for a prime input does get slower:
name old time/op new time/op delta
ProbablyPrime/n=1 849µs ± 1% 4521µs ± 1% +432.31% (p=0.000 n=10+9)
ProbablyPrime/n=5 4.31ms ± 3% 7.87ms ± 1% +82.70% (p=0.000 n=10+10)
ProbablyPrime/n=10 8.52ms ± 3% 12.28ms ± 1% +44.11% (p=0.000 n=10+10)
ProbablyPrime/n=20 16.9ms ± 2% 21.4ms ± 2% +26.35% (p=0.000 n=9+10)
However, because the Baillie-PSW test is only added when the old
ProbablyPrime(n) would return true, testing composites runs at
the same speed as before, except in the case where the result
would have been incorrect and is now correct.
In particular, the most important use of this code is for
generating random primes in crypto/rand. That use spends
essentially all its time testing composites, so it is not
slowed down by the new Baillie-PSW check:
name old time/op new time/op delta
Prime 104ms ±22% 111ms ±16% ~ (p=0.165 n=10+10)
Thanks to Serhat Şevki Dinçer for CL 20170, which this CL builds on.
Fixes#13229.
Change-Id: Id26dde9b012c7637c85f2e96355d029b6382812a
Reviewed-on: https://go-review.googlesource.com/30770
Run-TryBot: Russ Cox <rsc@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Robert Griesemer <gri@golang.org>
A later CL will be adding more code here.
It will help to keep it separate from the other code.
Change-Id: I971ba53de819cd10991b51fdec665984939a5f9b
Reviewed-on: https://go-review.googlesource.com/30709
Reviewed-by: Brad Fitzpatrick <bradfitz@golang.org>
