Use a different algorithm to minimize rejection. This is essentially
the same algorithm implemented in the Linux kernel for
__get_random_u32_below(), but written in a more readable way, and
avoiding microopimizations that make it less readable.
Which (the Linux kernel implementation) is itself based on Daniel
Lemire's algorithm from "Fast Random Integer Generation in an Interval",
linked below. However, I couldn't really understand that paper very
much, so I had to reconstruct the proofs from scratch, just from what I
could understand from the Linux kernel implementation source code.
I constructed some graphical explanation of how it works, and why it
is optimal, because I needed to visualize it to understand it. It is
published in the GitHub pull request linked below.
Here goes a wordy explanation of why this algorithm based on
multiplication is better optimized than my original implementation based
on masking.
masking:
It discards the extra bits of entropy that are not necessary for
this operation. This works as if dividing the entire space of
possible csrand() values into smaller spaces of a size that is
a smaller power of 2. Each of those smaller spaces has a
rejection band, so we get as many rejection bands as spaces
there are. For smaller values of 'n', the size of each
rejection band is smaller, but having more rejection bands
compensates for this, and results in the same inefficiency as
for large values of 'n'.
multiplication:
It divides the entire space of possible random numbers in
chunks of size exactly 'n', so that there is only one rejection
band that is the remainder of `2^64 % n`. The worst case is
still similar to the masking algorithm, a rejection band that is
almost half the entire space (n = 2^63 + 1), but for lower
values of 'n', by only having one small rejection band, it is
much faster than the masking algorithm.
This algorithm, however, has one caveat: the implementation
is harder to read, since it relies on several bitwise tricky
operations to perform operations like `2^64 % n`, `mult % 2^64`,
and `mult / 2^64`. And those operations are different depending
on the number of bits of the maximum possible random number
generated by the function. This means that while this algorithm
could also be applied to get uniform random numbers in the range
[0, n-1] quickly from a function like rand(3), which only
produces 31 bits of (non-CS) random numbers, it would need to be
implemented differently. However, that's not a concern for us,
it's just a note so that nobody picks this code and expects it
to just work with rand(3) (which BTW I tried for testing it, and
got a bit confused until I realized this).
Finally, here's some light testing of this implementation, just to know
that I didn't goof it. I pasted this function into a standalone
program, and run it many times to find if it has any bias (I tested also
to see how many iterations it performs, and it's also almost always 1,
but that test is big enough to not paste it here).
int main(int argc, char *argv[])
{
printf("%lu\n", csrand_uniform(atoi(argv[1])));
}
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 1 | wc -l
341
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 1 | wc -l
339
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 1 | wc -l
338
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 2 | wc -l
336
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 2 | wc -l
328
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 2 | wc -l
335
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 0 | wc -l
332
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 0 | wc -l
331
$ seq 1 1000 | while read _; do ./a.out 3; done | grep 0 | wc -l
327
This isn't a complete test for a cryptographically-secure random number
generator, of course, but I leave that for interested parties.
Link: <https://git.kernel.org/pub/scm/linux/kernel/git/torvalds/linux.git/commit/?id=e9a688bcb19348862afe30d7c85bc37c4c293471>
Link: <https://github.com/shadow-maint/shadow/pull/624#discussion_r1059574358>
Link: <https://arxiv.org/abs/1805.10941>
Cc: "Jason A. Donenfeld" <Jason@zx2c4.com>
Cc: Cristian Rodríguez <crrodriguez@opensuse.org>
Cc: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Cc: Björn Esser <besser82@fedoraproject.org>
Cc: Yann Droneaud <ydroneaud@opteya.com>
Cc: Joseph Myers <joseph@codesourcery.com>
Cc: Sam James <sam@gentoo.org>
Cc: Serge Hallyn <serge@hallyn.com>
Cc: Iker Pedrosa <ipedrosa@redhat.com>
[Daniel Lemire: Added link to research paper in source code]
Cc: Daniel Lemire <daniel@lemire.me>
Signed-off-by: Alejandro Colomar <alx@kernel.org>
1a0e13f94e