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big: Add internal_int_is_prime.

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Jeroen van Rijn
2021-09-02 18:31:08 +02:00
parent e639c61499
commit 31918d3b8f
8 changed files with 299 additions and 18 deletions
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core/math/big/prime.odin
+227 -6
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@@ -10,6 +10,8 @@
*/
package math_big
import rnd "core:math/rand";
/*
Determines if an Integer is divisible by one of the _PRIME_TABLE primes.
Returns true if it is, false if not.
@@ -207,7 +209,7 @@ internal_int_kronecker :: proc(a, p: ^Int, allocator := context.allocator) -> (k
/*
Miller-Rabin test of "a" to the base of "b" as described in HAC pp. 139 Algorithm 4.24.
Sets result to 0 if definitely composite or 1 if probably prime.
Sets result to `false` if definitely composite or `true` if probably prime.
Randomly the chance of error is no more than 1/4 and often very much lower.
Assumes `a` and `b` not to be `nil` and to have been initialized.
@@ -224,13 +226,13 @@ internal_int_prime_miller_rabin :: proc(a, b: ^Int, allocator := context.allocat
if internal_gt(b, 1) { return false, nil; }
/*
Get n1 = a - 1.
Get `n1` = `a` - 1.
*/
internal_copy(n1, a) or_return;
internal_sub(n1, n1, 1) or_return;
/*
Set 2**s * r = n1
Set `2`**`s` * `r` = `n1`
*/
internal_copy(r, n1) or_return;
@@ -240,17 +242,17 @@ internal_int_prime_miller_rabin :: proc(a, b: ^Int, allocator := context.allocat
s := internal_count_lsb(r) or_return;
/*
Now divide n - 1 by 2**s.
Now divide `n` - 1 by `2`**`s`.
*/
internal_shr(r, r, s) or_return;
/*
Compute y = b**r mod a.
Compute `y` = `b`**`r` mod `a`.
*/
internal_int_exponent_mod(y, b, r, a) or_return;
/*
If y != 1 and y != n1 do.
If `y` != 1 and `y` != `n1` do.
*/
if !internal_eq(y, 1) && !internal_eq(y, n1) {
j := 1;
@@ -285,6 +287,225 @@ internal_int_prime_miller_rabin :: proc(a, b: ^Int, allocator := context.allocat
return true, nil;
}
/*
`a` is the big Int to test for primality.
`miller_rabin_trials` can be one of the following:
< 0: For `a` up to 3_317_044_064_679_887_385_961_981, set `miller_rabin_trials` to negative to run a predetermined
number of trials for a deterministic answer.
= 0: Run Miller-Rabin with bases 2, 3 and one random base < `a`. Non-deterministic.
> 0: Run Miller-Rabin with bases 2, 3 and `miller_rabin_trials` number of random bases. Non-deterministic.
`miller_rabin_only`:
`false` Also use either Frobenius-Underwood or Lucas-Selfridge, depending on the compile-time `MATH_BIG_USE_FROBENIUS_TEST` choice.
`true` Run Rabin-Miller trials but skip Frobenius-Underwood / Lucas-Selfridge.
`r` takes a pointer to an instance of `core:math/rand`'s `Rand` and may be `nil` to use the global one.
Returns `is_prime` (bool), where:
`false` Definitively composite.
`true` Probably prime if `miller_rabin_trials` >= 0, with increasing certainty with more trials.
Deterministically prime if `miller_rabin_trials` = 0 for `a` up to 3_317_044_064_679_887_385_961_981.
Assumes `a` not to be `nil` and to have been initialized.
*/
internal_int_is_prime :: proc(a: ^Int, miller_rabin_trials := int(-1), miller_rabin_only := USE_MILLER_RABIN_ONLY, r: ^rnd.Rand = nil, allocator := context.allocator) -> (is_prime: bool, err: Error) {
context.allocator = allocator;
miller_rabin_trials := miller_rabin_trials;
// Default to `no`.
is_prime = false;
b, res := &Int{}, &Int{};
defer internal_destroy(b, res);
// Some shortcuts
// `N` > 3
if a.used == 1 {
if a.digit[0] == 0 || a.digit[0] == 1 {
return;
}
if a.digit[0] == 2 {
return true, nil;
}
}
// `N` must be odd.
if internal_is_even(a) {
return;
}
// `N` is not a perfect square: floor(sqrt(`N`))^2 != `N`
if internal_int_is_square(a) or_return { return; }
// Is the input equal to one of the primes in the table?
for p in _private_prime_table {
if internal_eq(a, p) {
return true, nil;
}
}
// First perform trial division
if internal_int_prime_is_divisible(a) or_return { return; }
// Run the Miller-Rabin test with base 2 for the BPSW test.
internal_set(b, 2) or_return;
if !internal_int_prime_miller_rabin(a, b) or_return { return; }
// Rumours have it that Mathematica does a second M-R test with base 3.
// Other rumours have it that their strong L-S test is slightly different.
// It does not hurt, though, beside a bit of extra runtime.
b.digit[0] += 1;
if !internal_int_prime_miller_rabin(a, b) or_return { return; }
// Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite
// slow so if speed is an issue, set `USE_MILLER_RABIN_ONLY` to use M-R tests with
// bases 2, 3 and t random bases.
if !miller_rabin_only {
if miller_rabin_trials >= 0 {
when MATH_BIG_USE_FROBENIUS_TEST {
// err = mp_prime_frobenius_underwood(a, &res);
// if ((err != MP_OKAY) && (err != MP_ITER)) {
// goto LBL_B;
// }
// if (!res) {
// goto LBL_B;
// }
} else {
// if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) {
// goto LBL_B;
// }
// if (!res) {
// goto LBL_B;
// }
}
}
}
// Run at least one Miller-Rabin test with a random base.
// Don't replace this with `min`, because we try known deterministic bases
// for certain sized inputs when `miller_rabin_trials` is negative.
if miller_rabin_trials == 0 {
miller_rabin_trials = 1;
}
// Only recommended if the input range is known to be < 3_317_044_064_679_887_385_961_981
// It uses the bases necessary for a deterministic M-R test if the input is smaller than 3_317_044_064_679_887_385_961_981
// The caller has to check the size.
// TODO: can be made a bit finer grained but comparing is not free.
if miller_rabin_trials < 0 {
p_max := 0;
// Sorenson, Jonathan; Webster, Jonathan (2015), "Strong Pseudoprimes to Twelve Prime Bases".
// 0x437ae92817f9fc85b7e5 = 318_665_857_834_031_151_167_461
atoi(b, "437ae92817f9fc85b7e5", 16) or_return;
if internal_lt(a, b) {
p_max = 12;
} else {
/* 0x2be6951adc5b22410a5fd = 3_317_044_064_679_887_385_961_981 */
atoi(b, "2be6951adc5b22410a5fd", 16) or_return;
if internal_lt(a, b) {
p_max = 13;
} else {
return false, .Invalid_Argument;
}
}
// We did bases 2 and 3 already, skip them
for ix := 2; ix < p_max; ix += 1 {
internal_set(b, _private_prime_table[ix]);
if !internal_int_prime_miller_rabin(a, b) or_return { return; }
}
} else if miller_rabin_trials > 0 {
// Perform `miller_rabin_trials` M-R tests with random bases between 3 and "a".
// See Fips 186.4 p. 126ff
// The DIGITs have a defined bit-size but the size of a.digit is a simple 'int',
// the size of which can depend on the platform.
size_a := internal_count_bits(a);
mask := (1 << uint(ilog2(size_a))) - 1;
/*
Assuming the General Rieman hypothesis (never thought to write that in a
comment) the upper bound can be lowered to 2*(log a)^2.
E. Bach, "Explicit bounds for primality testing and related problems,"
Math. Comp. 55 (1990), 355-380.
size_a = (size_a/10) * 7;
len = 2 * (size_a * size_a);
E.g.: a number of size 2^2048 would be reduced to the upper limit
floor(2048/10)*7 = 1428
2 * 1428^2 = 4078368
(would have been ~4030331.9962 with floats and natural log instead)
That number is smaller than 2^28, the default bit-size of DIGIT on 32-bit platforms.
*/
/*
How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame
does exactly 1. In words: one. Look at the end of _GMP_is_prime() in
Math-Prime-Util-GMP-0.50/primality.c if you do not believe it.
The function rand() goes to some length to use a cryptographically
good PRNG. That also means that the chance to always get the same base
in the loop is non-zero, although very low.
-- NOTE(Jeroen): This is not yet true in Odin, but I have some ideas.
If the BPSW test and/or the addtional Frobenious test have been
performed instead of just the Miller-Rabin test with the bases 2 and 3,
a single extra test should suffice, so such a very unlikely event will not do much harm.
To preemptivly answer the dangling question: no, a witness does not need to be prime.
*/
for ix := 0; ix < miller_rabin_trials; ix += 1 {
// rand() guarantees the first digit to be non-zero
internal_rand(b, _DIGIT_TYPE_BITS, r) or_return;
// Reduce digit before casting because DIGIT might be bigger than
// an unsigned int and "mask" on the other side is most probably not.
l: int;
fips_rand := (uint)(b.digit[0] & DIGIT(mask));
if fips_rand > (uint)(max(int) - _DIGIT_BITS) {
l = max(int) / _DIGIT_BITS;
} else {
l = (int(fips_rand) + _DIGIT_BITS) / _DIGIT_BITS;
}
// Unlikely.
if (l < 0) {
ix -= 1;
continue;
}
internal_rand(b, l) or_return;
// That number might got too big and the witness has to be smaller than "a"
l = internal_count_bits(b);
if l >= size_a {
l = (l - size_a) + 1;
internal_shr(b, b, l) or_return;
}
// Although the chance for b <= 3 is miniscule, try again.
if internal_lte(b, 3) {
ix -= 1;
continue;
}
if !internal_int_prime_miller_rabin(a, b) or_return { return; }
}
}
// Passed the test.
return true, nil;
}
/*
Returns the number of Rabin-Miller trials needed for a given bit size.