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Merge pull request #1117 from Kelimion/bigint

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big: Add Frobenius-Underwood, Miller-Rabin and primality testing.
This commit is contained in:
Jeroen van Rijn
2021-09-03 01:39:58 +02:00
committed by GitHub
parent e2f035d6ee eecc786bd2
commit 07f7d14d2c
8 changed files with 448 additions and 15 deletions
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4
core/math/big/build.bat
View File

@@ -1,10 +1,10 @@
@echo off
:odin run . -vet
odin run . -vet -define:MATH_BIG_USE_FROBENIUS_TEST=true
set TEST_ARGS=-fast-tests
:set TEST_ARGS=
:odin build . -build-mode:shared -show-timings -o:minimal -no-bounds-check -define:MATH_BIG_EXE=false && python test.py %TEST_ARGS%
odin build . -build-mode:shared -show-timings -o:size -no-bounds-check -define:MATH_BIG_EXE=false && python test.py %TEST_ARGS%
:odin build . -build-mode:shared -show-timings -o:size -no-bounds-check -define:MATH_BIG_EXE=false && python test.py %TEST_ARGS%
:odin build . -build-mode:shared -show-timings -o:size -define:MATH_BIG_EXE=false && python test.py %TEST_ARGS%
:odin build . -build-mode:shared -show-timings -o:speed -no-bounds-check -define:MATH_BIG_EXE=false && python test.py %TEST_ARGS%
:odin build . -build-mode:shared -show-timings -o:speed -define:MATH_BIG_EXE=false && python test.py -fast-tests %TEST_ARGS%

11
core/math/big/common.odin
View File

@@ -75,6 +75,17 @@ FACTORIAL_MAX_N := 1_000_000;
FACTORIAL_BINARY_SPLIT_CUTOFF := 6100;
FACTORIAL_BINARY_SPLIT_MAX_RECURSIONS := 100;
/*
`internal_int_is_prime` switchables.
Use Frobenius-Underwood for primality testing, or use Lucas-Selfridge (default).
*/
MATH_BIG_USE_FROBENIUS_TEST :: #config(MATH_BIG_USE_FROBENIUS_TEST, false);
/*
Runtime tunable to use Miller-Rabin primality testing only and skip the above.
*/
USE_MILLER_RABIN_ONLY := false;
/*
We don't allow these to be switched at runtime for two reasons:

20
core/math/big/example.odin
View File

@@ -26,6 +26,7 @@ Configuration:
_WARRAY %v
_TAB_SIZE %v
_MAX_WIN_SIZE %v
MATH_BIG_USE_FROBENIUS_TEST %v
Runtime tunable:
MUL_KARATSUBA_CUTOFF %v
SQR_KARATSUBA_CUTOFF %v
@@ -35,6 +36,7 @@ Runtime tunable:
FACTORIAL_MAX_N %v
FACTORIAL_BINARY_SPLIT_CUTOFF %v
FACTORIAL_BINARY_SPLIT_MAX_RECURSIONS %v
USE_MILLER_RABIN_ONLY %v
`, _DIGIT_BITS,
_LOW_MEMORY,
@@ -45,6 +47,8 @@ _MAX_COMBA,
_WARRAY,
_TAB_SIZE,
_MAX_WIN_SIZE,
MATH_BIG_USE_FROBENIUS_TEST,
MUL_KARATSUBA_CUTOFF,
SQR_KARATSUBA_CUTOFF,
MUL_TOOM_CUTOFF,
@@ -53,6 +57,7 @@ MAX_ITERATIONS_ROOT_N,
FACTORIAL_MAX_N,
FACTORIAL_BINARY_SPLIT_CUTOFF,
FACTORIAL_BINARY_SPLIT_MAX_RECURSIONS,
USE_MILLER_RABIN_ONLY,
);
}
@@ -79,11 +84,24 @@ print :: proc(name: string, a: ^Int, base := i8(10), print_name := true, newline
}
}
// printf :: fmt.printf;
printf :: fmt.printf;
demo :: proc() {
a, b, c, d, e, f, res := &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{}, &Int{};
defer destroy(a, b, c, d, e, f, res);
err: Error;
prime: bool;
set(a, "3317044064679887385961981"); // Composite: 1287836182261 × 2575672364521
trials := number_of_rabin_miller_trials(internal_count_bits(a));
{
SCOPED_TIMING(.is_prime);
prime, err = internal_int_is_prime(a, trials);
}
print("Candidate prime: ", a);
fmt.printf("%v Miller-Rabin trials needed.\n", trials);
fmt.printf("Is prime: %v, Error: %v\n", prime, err);
}
main :: proc() {

14
core/math/big/internal.odin
View File

@@ -1871,7 +1871,7 @@ internal_int_set_from_integer :: proc(dest: ^Int, src: $T, minimize := false, al
return nil;
}
internal_set :: proc { internal_int_set_from_integer, internal_int_copy };
internal_set :: proc { internal_int_set_from_integer, internal_int_copy, int_atoi };
internal_copy_digits :: #force_inline proc(dest, src: ^Int, digits: int, offset := int(0)) -> (err: Error) {
#force_inline internal_error_if_immutable(dest) or_return;
@@ -2019,8 +2019,18 @@ internal_invmod :: proc{ internal_int_inverse_modulo, };
/*
Helpers to extract values from the `Int`.
*/
internal_int_bitfield_extract_bool :: proc(a: ^Int, offset: int) -> (val: bool, err: Error) {
limb := offset / _DIGIT_BITS;
if limb < 0 || limb >= a.used { return false, .Invalid_Argument; }
i := _WORD(1 << _WORD((offset % _DIGIT_BITS)));
return bool(_WORD(a.digit[limb]) & i), nil;
}
internal_int_bitfield_extract_single :: proc(a: ^Int, offset: int) -> (bit: _WORD, err: Error) {
return #force_inline int_bitfield_extract(a, offset, 1);
limb := offset / _DIGIT_BITS;
if limb < 0 || limb >= a.used { return 0, .Invalid_Argument; }
i := _WORD(1 << _WORD((offset % _DIGIT_BITS)));
return 1 if ((_WORD(a.digit[limb]) & i) != 0) else 0, nil;
}
internal_int_bitfield_extract :: proc(a: ^Int, offset, count: int) -> (res: _WORD, err: Error) #no_bounds_check {

395
core/math/big/prime.odin
View File

@@ -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.
@@ -204,13 +206,404 @@ internal_int_kronecker :: proc(a, p: ^Int, allocator := context.allocator) -> (k
return;
}
/*
Miller-Rabin test of "a" to the base of "b" as described in HAC pp. 139 Algorithm 4.24.
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.
*/
internal_int_prime_miller_rabin :: proc(a, b: ^Int, allocator := context.allocator) -> (probably_prime: bool, err: Error) {
context.allocator = allocator;
n1, y, r := &Int{}, &Int{}, &Int{};
defer internal_destroy(n1, y, r);
/*
Ensure `b` > 1.
*/
if internal_lte(b, 1) { return false, nil; }
/*
Get `n1` = `a` - 1.
*/
internal_copy(n1, a) or_return;
internal_sub(n1, n1, 1) or_return;
/*
Set `2`**`s` * `r` = `n1`
*/
internal_copy(r, n1) or_return;
/*
Count the number of least significant bits which are zero.
*/
s := internal_count_lsb(r) or_return;
/*
Now divide `n` - 1 by `2`**`s`.
*/
internal_shr(r, r, s) or_return;
/*
Compute `y` = `b`**`r` mod `a`.
*/
internal_int_exponent_mod(y, b, r, a) or_return;
/*
If `y` != 1 and `y` != `n1` do.
*/
if !internal_eq(y, 1) && !internal_eq(y, n1) {
j := 1;
/*
While `j` <= `s` - 1 and `y` != `n1`.
*/
for j <= (s - 1) && !internal_eq(y, n1) {
internal_sqrmod(y, y, a) or_return;
/*
If `y` == 1 then composite.
*/
if internal_eq(y, 1) {
return false, nil;
}
j += 1;
}
/*
If `y` != `n1` then composite.
*/
if !internal_eq(y, n1) {
return false, nil;
}
}
/*
Probably prime now.
*/
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 {
if !internal_int_prime_frobenius_underwood(a) or_return { return; }
} 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;
}
/*
* floor of positive solution of (2^16) - 1 = (a + 4) * (2 * a + 5)
* TODO: Both values are smaller than N^(1/4), would have to use a bigint
* for `a` instead, but any `a` bigger than about 120 are already so rare that
* it is possible to ignore them and still get enough pseudoprimes.
* But it is still a restriction of the set of available pseudoprimes
* which makes this implementation less secure if used stand-alone.
*/
_FROBENIUS_UNDERWOOD_A :: 32764;
internal_int_prime_frobenius_underwood :: proc(N: ^Int, allocator := context.allocator) -> (result: bool, err: Error) {
context.allocator = allocator;
T1z, T2z, Np1z, sz, tz := &Int{}, &Int{}, &Int{}, &Int{}, &Int{};
defer internal_destroy(T1z, T2z, Np1z, sz, tz);
internal_init_multi(T1z, T2z, Np1z, sz, tz) or_return;
a, ap2: int;
frob: for a = 0; a < _FROBENIUS_UNDERWOOD_A; a += 1 {
switch a {
case 2, 4, 7, 8, 10, 14, 18, 23, 26, 28:
continue frob;
}
internal_set(T1z, i32((a * a) - 4));
j := internal_int_kronecker(T1z, N) or_return;
switch j {
case -1: break frob;
case 0: return false, nil;
}
}
// Tell it a composite and set return value accordingly.
if a >= _FROBENIUS_UNDERWOOD_A { return false, .Max_Iterations_Reached; }
// Composite if N and (a+4)*(2*a+5) are not coprime.
internal_set(T1z, u32((a + 4) * ((2 * a) + 5)));
internal_int_gcd_lcm(T1z, nil, T1z, N) or_return;
if !(T1z.used == 1 && T1z.digit[0] == 1) {
// Composite.
return false, nil;
}
ap2 = a + 2;
internal_add(Np1z, N, 1) or_return;
internal_set(sz, 1) or_return;
internal_set(tz, 2) or_return;
for i := internal_count_bits(Np1z) - 2; i >= 0; i -= 1 {
// temp = (sz * (a * sz + 2 * tz)) % N;
// tz = ((tz - sz) * (tz + sz)) % N;
// sz = temp;
internal_int_shl1(T2z, tz) or_return;
// a = 0 at about 50% of the cases (non-square and odd input)
if a != 0 {
internal_mul(T1z, sz, DIGIT(a)) or_return;
internal_add(T2z, T2z, T1z) or_return;
}
internal_mul(T1z, T2z, sz) or_return;
internal_sub(T2z, tz, sz) or_return;
internal_add(sz, sz, tz) or_return;
internal_mul(tz, sz, T2z) or_return;
internal_mod(tz, tz, N) or_return;
internal_mod(sz, T1z, N) or_return;
if bit, _ := internal_int_bitfield_extract_bool(Np1z, i); bit {
// temp = (a+2) * sz + tz
// tz = 2 * tz - sz
// sz = temp
if a == 0 {
internal_int_shl1(T1z, sz) or_return;
} else {
internal_mul(T1z, sz, DIGIT(ap2)) or_return;
}
internal_add(T1z, T1z, tz) or_return;
internal_int_shl1(T2z, tz) or_return;
internal_sub(tz, T2z, sz);
internal_swap(sz, T1z);
}
}
internal_set(T1z, u32((2 * a) + 5)) or_return;
internal_mod(T1z, T1z, N) or_return;
result = internal_is_zero(sz) && internal_eq(tz, T1z);
return;
}
/*
Returns the number of Rabin-Miller trials needed for a given bit size.
*/
number_of_rabin_miller_trials :: proc(bit_size: int) -> (number_of_trials: int) {
switch {
case bit_size <= 80:
return - 1; /* Use deterministic algorithm for size <= 80 bits */
return -1; /* Use deterministic algorithm for size <= 80 bits */
case bit_size >= 81 && bit_size < 96:
return 37; /* max. error = 2^(-96) */
case bit_size >= 96 && bit_size < 128:

2
core/math/big/private.odin
View File

@@ -1373,7 +1373,7 @@ _private_int_div_recursive :: proc(quotient, remainder, a, b: ^Int, allocator :=
_private_int_div_small :: proc(quotient, remainder, numerator, denominator: ^Int) -> (err: Error) {
ta, tb, tq, q := &Int{}, &Int{}, &Int{}, &Int{};
c: int;
defer internal_destroy(ta, tb, tq, q);
for {

16
core/math/big/radix.odin
View File

@@ -413,14 +413,14 @@ _log_bases :: [65]u32{
*/
RADIX_TABLE := "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";
RADIX_TABLE_REVERSE := [RADIX_TABLE_REVERSE_SIZE]u8{
0x3e, 0xff, 0xff, 0xff, 0x3f, 0x00, 0x01, 0x02, 0x03, 0x04, /* +,-./01234 */
0x05, 0x06, 0x07, 0x08, 0x09, 0xff, 0xff, 0xff, 0xff, 0xff, /* 56789:;<=> */
0xff, 0xff, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f, 0x10, 0x11, /* ?@ABCDEFGH */
0x12, 0x13, 0x14, 0x15, 0x16, 0x17, 0x18, 0x19, 0x1a, 0x1b, /* IJKLMNOPQR */
0x1c, 0x1d, 0x1e, 0x1f, 0x20, 0x21, 0x22, 0x23, 0xff, 0xff, /* STUVWXYZ[\ */
0xff, 0xff, 0xff, 0xff, 0x24, 0x25, 0x26, 0x27, 0x28, 0x29, /* ]^_`abcdef */
0x2a, 0x2b, 0x2c, 0x2d, 0x2e, 0x2f, 0x30, 0x31, 0x32, 0x33, /* ghijklmnop */
0x34, 0x35, 0x36, 0x37, 0x38, 0x39, 0x3a, 0x3b, 0x3c, 0x3d, /* qrstuvwxyz */
0x3e, 0xff, 0xff, 0xff, 0x3f, 0x00, 0x01, 0x02, 0x03, 0x04, /* +,-./01234 */
0x05, 0x06, 0x07, 0x08, 0x09, 0xff, 0xff, 0xff, 0xff, 0xff, /* 56789:;<=> */
0xff, 0xff, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f, 0x10, 0x11, /* ?@ABCDEFGH */
0x12, 0x13, 0x14, 0x15, 0x16, 0x17, 0x18, 0x19, 0x1a, 0x1b, /* IJKLMNOPQR */
0x1c, 0x1d, 0x1e, 0x1f, 0x20, 0x21, 0x22, 0x23, 0xff, 0xff, /* STUVWXYZ[\ */
0xff, 0xff, 0xff, 0xff, 0x24, 0x25, 0x26, 0x27, 0x28, 0x29, /* ]^_`abcdef */
0x2a, 0x2b, 0x2c, 0x2d, 0x2e, 0x2f, 0x30, 0x31, 0x32, 0x33, /* ghijklmnop */
0x34, 0x35, 0x36, 0x37, 0x38, 0x39, 0x3a, 0x3b, 0x3c, 0x3d, /* qrstuvwxyz */
};
RADIX_TABLE_REVERSE_SIZE :: 80;

1
core/math/big/tune.odin
View File

@@ -23,6 +23,7 @@ Category :: enum {
sqr,
bitfield_extract,
rm_trials,
is_prime,
};
Event :: struct {