Miller–Rabin primality test: Difference between revisions
(Add Nim version (translated from Go)) |
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:- end_module test_is_prime. |
:- end_module test_is_prime. |
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</lang> |
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=={{header|Nim}}== |
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<lang nim> |
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## Nim currently doesn't have a BigInt standard library |
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## so we translate the version from Go which uses a |
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## deterministic approach, which is correct for all |
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## possible values in uint32. |
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proc isPrime*(n: uint32): bool = |
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# bases of 2, 7, 61 are sufficient to cover 2^32 |
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case n |
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of 0, 1: return false |
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of 2, 7, 61: return true |
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else: discard |
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var |
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nm1 = n-1 |
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d = nm1.int |
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s = 0 |
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n = n.uint64 |
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while d mod 2 == 0: |
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d = d shr 1 |
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s += 1 |
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for a in [2, 7, 61]: |
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var |
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x = 1.uint64 |
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p = a.uint64 |
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dr = d |
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while dr > 0: |
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if dr mod 2 == 1: |
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x = x * p mod n |
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p = p * p mod n |
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dr = dr shr 1 |
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if x == 1 or x.uint32 == nm1: |
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continue |
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var r = 1 |
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while true: |
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if r >= s: |
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return false |
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x = x * x mod n |
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if x == 1: |
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return false |
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if x.uint32 == nm1: |
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break |
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r += 1 |
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return true |
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proc isPrime*(n: int32): bool = |
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## Overload for int32 |
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n >= 0 and n.uint32.isPrime |
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when isMainModule: |
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const primeNumber1000 = 7919 # source: https://en.wikipedia.org/wiki/List_of_prime_numbers |
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var |
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i = 0u32 |
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numberPrimes = 0 |
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while true: |
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if isPrime(i): |
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if numberPrimes == 999: |
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break |
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numberPrimes += 1 |
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i += 1 |
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assert i == primeNumber1000 |
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assert isPrime(2u32) |
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assert isPrime(31u32) |
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assert isPrime(37u32) |
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assert isPrime(1123u32) |
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assert isPrime(492366587u32) |
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assert isPrime(1645333507u32) |
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</lang> |
</lang> |
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Revision as of 14:20, 18 November 2018
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at Miller–Rabin primality test. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime or not.
The algorithm, as modified by Michael O. Rabin to avoid the generalized Riemann hypothesis, is a probabilistic algorithm.
The pseudocode, from Wikipedia is:
Input: n > 2, an odd integer to be tested for primality;
k, a parameter that determines the accuracy of the test
Output: composite if n is composite, otherwise probably prime
write n − 1 as 2s·d with d odd by factoring powers of 2 from n − 1
LOOP: repeat k times:
pick a randomly in the range [2, n − 1]
x ← ad mod n
if x = 1 or x = n − 1 then do next LOOP
for r = 1 .. s − 1
x ← x2 mod n
if x = 1 then return composite
if x = n − 1 then do next LOOP
return composite
return probably prime
- The nature of the test involves big numbers, so the use of "big numbers" libraries (or similar features of the language of your choice) are suggested, but not mandatory.
- Deterministic variants of the test exist and can be implemented as extra (not mandatory to complete the task)
Ada
ordinary integers
It's easy to get overflows doing exponential calculations. Therefore I implemented my own function for that.
For Number types >= 2**64 you may have to use an external library -- see below.
First, a package Miller_Rabin is specified. The same package is used elsewhere in Rosetta Code, such as for the Carmichael 3 strong pseudoprimes the Extensible prime generator, and the Emirp primes.
<lang Ada>generic
type Number is range <>;
package Miller_Rabin is
type Result_Type is (Composite, Probably_Prime);
function Is_Prime (N : Number; K : Positive := 10) return Result_Type;
end Miller_Rabin;</lang>
The implementation of that package is as follows:
<lang Ada>with Ada.Numerics.Discrete_Random;
package body Miller_Rabin is
function Is_Prime (N : Number; K : Positive := 10)
return Result_Type
is
subtype Number_Range is Number range 2 .. N - 1;
package Random is new Ada.Numerics.Discrete_Random (Number_Range);
function Mod_Exp (Base, Exponent, Modulus : Number) return Number is
Result : Number := 1;
begin
for E in 1 .. Exponent loop
Result := Result * Base mod Modulus;
end loop;
return Result;
end Mod_Exp;
Generator : Random.Generator;
D : Number := N - 1;
S : Natural := 0;
X : Number;
begin
-- exclude 2 and even numbers
if N = 2 then
return Probably_Prime;
elsif N mod 2 = 0 then
return Composite;
end if;
-- write N-1 as 2**S * D, with D mod 2 /= 0
while D mod 2 = 0 loop
D := D / 2;
S := S + 1;
end loop;
-- initialize RNG
Random.Reset (Generator);
for Loops in 1 .. K loop
X := Mod_Exp(Random.Random (Generator), D, N);
if X /= 1 and X /= N - 1 then
Inner : for R in 1 .. S - 1 loop
X := Mod_Exp (X, 2, N);
if X = 1 then return Composite; end if;
exit Inner when X = N - 1;
end loop Inner;
if X /= N - 1 then return Composite; end if;
end if;
end loop;
return Probably_Prime; end Is_Prime;
end Miller_Rabin;</lang>
Finally, the program itself:
<lang Ada>with Ada.Text_IO, Miller_Rabin;
procedure Mr_Tst is
type Number is range 0 .. (2**48)-1;
package Num_IO is new Ada.Text_IO.Integer_IO (Number); package Pos_IO is new Ada.Text_IO.Integer_IO (Positive); package MR is new Miller_Rabin(Number); use MR;
N : Number; K : Positive;
begin
for I in Number(2) .. 1000 loop
if Is_Prime (I) = Probably_Prime then
Ada.Text_IO.Put (Number'Image (I));
end if;
end loop;
Ada.Text_IO.Put_Line (".");
Ada.Text_IO.Put ("Enter a Number: "); Num_IO.Get (N);
Ada.Text_IO.Put ("Enter the count of loops: "); Pos_IO.Get (K);
Ada.Text_IO.Put_Line ("What is it? " & Result_Type'Image (Is_Prime(N, K)));
end MR_Tst;</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997. Enter a Number: 1234567 Enter the count of loops: 20 What is it? COMPOSITE
using an external library to handle big integers
Using the big integer implementation from a cryptographic library [1].
<lang Ada>with Ada.Text_IO, Crypto.Types.Big_Numbers, Ada.Numerics.Discrete_Random;
procedure Miller_Rabin is
Bound: constant Positive := 256; -- can be any multiple of 32
package LN is new Crypto.Types.Big_Numbers (Bound); use type LN.Big_Unsigned; -- all computations "mod 2**Bound"
function "+"(S: String) return LN.Big_Unsigned
renames LN.Utils.To_Big_Unsigned;
function Is_Prime (N : LN.Big_Unsigned; K : Positive := 10) return Boolean is
subtype Mod_32 is Crypto.Types.Mod_Type;
use type Mod_32;
package R_32 is new Ada.Numerics.Discrete_Random (Mod_32);
Generator : R_32.Generator;
function Random return LN.Big_Unsigned is
X: LN.Big_Unsigned := LN.Big_Unsigned_Zero;
begin
for I in 1 .. Bound/32 loop
X := (X * 2**16) * 2**16;
X := X + R_32.Random(Generator);
end loop;
return X;
end Random;
D: LN.Big_Unsigned := N - LN.Big_Unsigned_One;
S: Natural := 0;
A, X: LN.Big_Unsigned;
begin
-- exclude 2 and even numbers
if N = 2 then
return True;
elsif N mod 2 = LN.Big_Unsigned_Zero then
return False;
else
-- write N-1 as 2**S * D, with odd D
while D mod 2 = LN.Big_Unsigned_Zero loop
D := D / 2;
S := S + 1;
end loop;
-- initialize RNG
R_32.Reset (Generator);
-- run the real test
for Loops in 1 .. K loop
loop
A := Random;
exit when (A > 1) and (A < (N - 1));
end loop;
X := LN.Mod_Utils.Pow(A, D, N); -- X := (Random**D) mod N
if X /= 1 and X /= N - 1 then
Inner:
for R in 1 .. S - 1 loop
X := LN.Mod_Utils.Pow(X, LN.Big_Unsigned_Two, N);
if X = 1 then
return False;
end if;
exit Inner when X = N - 1;
end loop Inner;
if X /= N - 1 then
return False;
end if;
end if;
end loop;
end if;
return True; end Is_Prime;
S: constant String :=
"4547337172376300111955330758342147474062293202868155909489";
T: constant String :=
"4547337172376300111955330758342147474062293202868155909393";
K: constant Positive := 10;
begin
Ada.Text_IO.Put_Line("Prime(" & S & ")=" & Boolean'Image(Is_Prime(+S, K)));
Ada.Text_IO.Put_Line("Prime(" & T & ")=" & Boolean'Image(Is_Prime(+T, K)));
end Miller_Rabin;</lang>
- Output:
Prime(4547337172376300111955330758342147474062293202868155909489)=TRUE Prime(4547337172376300111955330758342147474062293202868155909393)=FALSE
Using the built-in Miller-Rabin test from the same library:
<lang Ada>with Ada.Text_IO, Crypto.Types.Big_Numbers, Ada.Numerics.Discrete_Random;
procedure Miller_Rabin is
Bound: constant Positive := 256; -- can be any multiple of 32
package LN is new Crypto.Types.Big_Numbers (Bound); use type LN.Big_Unsigned; -- all computations "mod 2**Bound"
function "+"(S: String) return LN.Big_Unsigned
renames LN.Utils.To_Big_Unsigned;
S: constant String :=
"4547337172376300111955330758342147474062293202868155909489";
T: constant String :=
"4547337172376300111955330758342147474062293202868155909393";
K: constant Positive := 10;
begin
Ada.Text_IO.Put_Line("Prime(" & S & ")="
& Boolean'Image (LN.Mod_Utils.Passed_Miller_Rabin_Test(+S, K)));
Ada.Text_IO.Put_Line("Prime(" & T & ")="
& Boolean'Image (LN.Mod_Utils.Passed_Miller_Rabin_Test(+T, K)));
end Miller_Rabin;</lang>
The output is the same.
ALGOL 68
<lang algol68>MODE LINT=LONG INT; MODE LOOPINT = INT;
MODE POWMODSTRUCT = LINT; PR READ "prelude/pow_mod.a68" PR;
PROC miller rabin = (LINT n, LOOPINT k)BOOL: (
IF n<=3 THEN TRUE
ELIF NOT ODD n THEN FALSE
ELSE
LINT d := n - 1;
INT s := 0;
WHILE NOT ODD d DO
d := d OVER 2;
s +:= 1
OD;
TO k DO
LINT a := 2 + ENTIER (random*(n-3));
LINT x := pow mod(a, d, n);
IF x /= 1 THEN
TO s DO
IF x = n-1 THEN done FI;
x := x*x %* n
OD;
else: IF x /= n-1 THEN return false FI;
done: EMPTY
FI
OD;
TRUE EXIT
return false: FALSE
FI
);
FOR i FROM 937 TO 1000 DO
IF miller rabin(i, 10) THEN
print((" ",whole(i,0)))
FI
OD</lang>
- Output:
937 941 947 953 967 971 977 983 991 997
AutoHotkey
ahk forum: discussion <lang AutoHotkey>MsgBox % MillerRabin(999983,10) ; 1 MsgBox % MillerRabin(999809,10) ; 1 MsgBox % MillerRabin(999727,10) ; 1 MsgBox % MillerRabin(52633,10) ; 0 MsgBox % MillerRabin(60787,10) ; 0 MsgBox % MillerRabin(999999,10) ; 0 MsgBox % MillerRabin(999995,10) ; 0 MsgBox % MillerRabin(999991,10) ; 0
MillerRabin(n,k) { ; 0: composite, 1: probable prime (n < 2**31)
d := n-1, s := 0
While !(d&1)
d>>=1, s++
Loop %k% {
Random a, 2, n-2 ; if n < 4,759,123,141, it is enough to test a = 2, 7, and 61.
x := PowMod(a,d,n)
If (x=1 || x=n-1)
Continue
Cont := 0
Loop % s-1 {
x := PowMod(x,2,n)
If (x = 1)
Return 0
If (x = n-1) {
Cont = 1
Break
}
}
IfEqual Cont,1, Continue
Return 0
}
Return 1
}
PowMod(x,n,m) { ; x**n mod m
y := 1, i := n, z := x
While i>0
y := i&1 ? mod(y*z,m) : y, z := mod(z*z,m), i >>= 1
Return y
}</lang>
bc
Requires a bc with long names.
(A previous version worked with GNU bc.) <lang bc>seed = 1 /* seed of the random number generator */ scale = 0
/* Random number from 0 to 32767. */ define rand() {
/* Cheap formula (from POSIX) for random numbers of low quality. */ seed = (seed * 1103515245 + 12345) % 4294967296 return ((seed / 65536) % 32768)
}
/* Random number in range [from, to]. */ define rangerand(from, to) {
auto b, h, i, m, n, r
m = to - from + 1
h = length(m) / 2 + 1 /* want h iterations of rand() % 100 */
b = 100 ^ h % m /* want n >= b */
while (1) {
n = 0 /* pick n in range [b, 100 ^ h) */
for (i = h; i > 0; i--) {
r = rand()
while (r < 68) { r = rand(); } /* loop if the modulo bias */
n = (n * 100) + (r % 100) /* append 2 digits to n */
}
if (n >= b) { break; } /* break unless the modulo bias */
}
return (from + (n % m))
}
/* n is probably prime? */ define miller_rabin_test(n, k) {
auto d, r, a, x, s
if (n <= 3) { return (1); }
if ((n % 2) == 0) { return (0); }
/* find s and d so that d * 2^s = n - 1 */
d = n - 1
s = 0
while((d % 2) == 0) {
d /= 2
s += 1
}
while (k-- > 0) {
a = rangerand(2, n - 2)
x = (a ^ d) % n
if (x != 1) {
for (r = 0; r < s; r++) {
if (x == (n - 1)) { break; }
x = (x * x) % n
}
if (x != (n - 1)) {
return (0)
}
}
}
return (1)
}
for (i = 1; i < 1000; i++) {
if (miller_rabin_test(i, 10) == 1) {
i
}
} quit</lang>
Bracmat
<lang bracmat>( 1:?seed & ( rand
=
. mod$(!seed*1103515245+12345.4294967296):?seed
& mod$(div$(!seed.65536).32768)
)
& ( rangerand
= from to b h i m n r length
. !arg:(?from,?to)
& !to+-1*!from+1:?m
& @(!m:? [?length)
& div$(!length+1.2)+1:?h
& 100^mod$(!h.!m):?b
& whl
' ( 0:?n
& !h+1:?i
& whl
' ( !i+-1:>0:?i
& rand$:?r
& whl'(!r:<68&rand$:?r)
& !n*100+mod$(!r.100):?n
)
& !n:>!b
)
& !from+mod$(!n.!m)
)
& ( miller-rabin-test
= n k d r a x s return
. !arg:(?n,?k)
& ( !n:~>3&1
| mod$(!n.2):0
| !n+-1:?d
& 0:?s
& whl
' ( mod$(!d.2):0
& !d*1/2:?d
& 1+!s:?s
)
& 1:?return
& whl
' ( !k+-1:?k:~<0
& rangerand$(2,!n+-2):?a
& mod$(!a^!d.!n):?x
& ( !x:1
| 0:?r
& whl
' ( !r+1:~>!s:?r
& !n+-1:~!x
& mod$(!x*!x.!n):?x
)
& ( !n+-1:!x
| 0:?return&~
)
)
)
& !return
)
)
& 0:?i & :?primes & whl
' ( 1+!i:<1000:?i
& ( miller-rabin-test$(!i,10):1
& !primes !i:?primes
|
)
)
& !primes:? [-11 ?last & out$!last );</lang>
- Output:
937 941 947 953 967 971 977 983 991 997
C
miller-rabin.h <lang c>#ifndef _MILLER_RABIN_H_
- define _MILLER_RABIN_H
- include <gmp.h>
bool miller_rabin_test(mpz_t n, int j);
- endif</lang>
miller-rabin.c
For decompose (and header primedecompose.h),
see Prime decomposition.
<lang c>#include <stdbool.h>
- include <gmp.h>
- include "primedecompose.h"
- define MAX_DECOMPOSE 100
bool miller_rabin_test(mpz_t n, int j) {
bool res; mpz_t f[MAX_DECOMPOSE]; mpz_t s, d, a, x, r; mpz_t n_1, n_3; gmp_randstate_t rs; int l=0, k;
res = false; gmp_randinit_default(rs);
mpz_init(s); mpz_init(d); mpz_init(a); mpz_init(x); mpz_init(r); mpz_init(n_1); mpz_init(n_3);
if ( mpz_cmp_si(n, 3) <= 0 ) { // let us consider 1, 2, 3 as prime
gmp_randclear(rs);
return true;
}
if ( mpz_odd_p(n) != 0 ) {
mpz_sub_ui(n_1, n, 1); // n-1
mpz_sub_ui(n_3, n, 3); // n-3
l = decompose(n_1, f);
mpz_set_ui(s, 0);
mpz_set_ui(d, 1);
for(k=0; k < l; k++) {
if ( mpz_cmp_ui(f[k], 2) == 0 )
mpz_add_ui(s, s, 1);
else
mpz_mul(d, d, f[k]);
} // 2^s * d = n-1
while(j-- > 0) {
mpz_urandomm(a, rs, n_3); // random from 0 to n-4
mpz_add_ui(a, a, 2); // random from 2 to n-2
mpz_powm(x, a, d, n);
if ( mpz_cmp_ui(x, 1) == 0 ) continue;
mpz_set_ui(r, 0);
while( mpz_cmp(r, s) < 0 ) {
if ( mpz_cmp(x, n_1) == 0 ) break; mpz_powm_ui(x, x, 2, n); mpz_add_ui(r, r, 1);
}
if ( mpz_cmp(x, n_1) == 0 ) continue;
goto flush; // woops
}
res = true;
}
flush:
for(k=0; k < l; k++) mpz_clear(f[k]); mpz_clear(s); mpz_clear(d); mpz_clear(a); mpz_clear(x); mpz_clear(r); mpz_clear(n_1); mpz_clear(n_3); gmp_randclear(rs); return res;
}</lang> Testing <lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdbool.h>
- include <gmp.h>
- include "miller-rabin.h"
- define PREC 10
- define TOP 4000
int main() {
mpz_t num;
mpz_init(num);
mpz_set_ui(num, 1);
while ( mpz_cmp_ui(num, TOP) < 0 ) {
if ( miller_rabin_test(num, PREC) ) {
gmp_printf("%Zd maybe prime\n", num);
} /*else {
gmp_printf("%Zd not prime\n", num);
}*/ // remove the comment iff you're interested in
// sure non-prime.
mpz_add_ui(num, num, 1);
}
mpz_clear(num); return EXIT_SUCCESS;
}</lang>
Deterministic up to 341,550,071,728,321
<lang c>// calcul a^n%mod size_t power(size_t a, size_t n, size_t mod) {
size_t power = a; size_t result = 1;
while (n)
{
if (n & 1)
result = (result * power) % mod;
power = (power * power) % mod;
n >>= 1;
}
return result;
}
// n−1 = 2^s * d with d odd by factoring powers of 2 from n−1 bool witness(size_t n, size_t s, size_t d, size_t a) {
size_t x = power(a, d, n); size_t y;
while (s) {
y = (x * x) % n;
if (y == 1 && x != 1 && x != n-1)
return false;
x = y;
--s;
}
if (y != 1)
return false;
return true;
}
/*
* if n < 1,373,653, it is enough to test a = 2 and 3; * if n < 9,080,191, it is enough to test a = 31 and 73; * if n < 4,759,123,141, it is enough to test a = 2, 7, and 61; * if n < 1,122,004,669,633, it is enough to test a = 2, 13, 23, and 1662803; * if n < 2,152,302,898,747, it is enough to test a = 2, 3, 5, 7, and 11; * if n < 3,474,749,660,383, it is enough to test a = 2, 3, 5, 7, 11, and 13; * if n < 341,550,071,728,321, it is enough to test a = 2, 3, 5, 7, 11, 13, and 17. */
bool is_prime_mr(size_t n) {
if (((!(n & 1)) && n != 2 ) || (n < 2) || (n % 3 == 0 && n != 3))
return false;
if (n <= 3)
return true;
size_t d = n / 2;
size_t s = 1;
while (!(d & 1)) {
d /= 2;
++s;
}
if (n < 1373653)
return witness(n, s, d, 2) && witness(n, s, d, 3);
if (n < 9080191)
return witness(n, s, d, 31) && witness(n, s, d, 73);
if (n < 4759123141)
return witness(n, s, d, 2) && witness(n, s, d, 7) && witness(n, s, d, 61);
if (n < 1122004669633)
return witness(n, s, d, 2) && witness(n, s, d, 13) && witness(n, s, d, 23) && witness(n, s, d, 1662803);
if (n < 2152302898747)
return witness(n, s, d, 2) && witness(n, s, d, 3) && witness(n, s, d, 5) && witness(n, s, d, 7) && witness(n, s, d, 11);
if (n < 3474749660383)
return witness(n, s, d, 2) && witness(n, s, d, 3) && witness(n, s, d, 5) && witness(n, s, d, 7) && witness(n, s, d, 11) && witness(n, s, d, 13);
return witness(n, s, d, 2) && witness(n, s, d, 3) && witness(n, s, d, 5) && witness(n, s, d, 7) && witness(n, s, d, 11) && witness(n, s, d, 13) && witness(n, s, d, 17);
}</lang> Inspiration from http://stackoverflow.com/questions/4424374/determining-if-a-number-is-prime
C#
<lang csharp>public static class RabinMiller {
public static bool IsPrime(int n, int k)
{
if ((n < 2) || (n % 2 == 0)) return (n == 2);
int s = n - 1;
while (s % 2 == 0) s >>= 1;
Random r = new Random();
for (int i = 0; i < k; i++)
{
int a = r.Next(n - 1) + 1;
int temp = s;
long mod = 1;
for (int j = 0; j < temp; ++j) mod = (mod * a) % n;
while (temp != n - 1 && mod != 1 && mod != n - 1)
{
mod = (mod * mod) % n;
temp *= 2;
}
if (mod != n - 1 && temp % 2 == 0) return false;
}
return true;
}
}</lang>
[2] Corrections made 6/21/2017
<lang csharp>// Miller-Rabin primality test as an extension method on the BigInteger type.
// Based on the Ruby implementation on this page.
public static class BigIntegerExtensions
{
public static bool IsProbablePrime(this BigInteger source, int certainty)
{
if(source == 2 || source == 3)
return true;
if(source < 2 || source % 2 == 0)
return false;
BigInteger d = source - 1; int s = 0;
while(d % 2 == 0)
{
d /= 2;
s += 1;
}
// There is no built-in method for generating random BigInteger values. // Instead, random BigIntegers are constructed from randomly generated // byte arrays of the same length as the source. RandomNumberGenerator rng = RandomNumberGenerator.Create(); byte[] bytes = new byte[source.ToByteArray().LongLength]; BigInteger a;
for(int i = 0; i < certainty; i++)
{
do
{
// This may raise an exception in Mono 2.10.8 and earlier.
// http://bugzilla.xamarin.com/show_bug.cgi?id=2761
rng.GetBytes(bytes);
a = new BigInteger(bytes);
}
while(a < 2 || a >= source - 2);
BigInteger x = BigInteger.ModPow(a, d, source);
if(x == 1 || x == source - 1)
continue;
for(int r = 1; r < s; r++)
{
x = BigInteger.ModPow(x, 2, source);
if(x == 1)
return false;
if(x == source - 1)
break;
}
if(x != source - 1)
return false;
}
return true; }
}</lang>
Clojure
Random Approach
<lang lisp>(ns test-p.core
(:require [clojure.math.numeric-tower :as math]) (:require [clojure.set :as set]))
(def WITNESSLOOP "witness") (def COMPOSITE "composite")
(defn m* [p q m]
" Computes (p*q) mod m " (mod (*' p q) m))
(defn power
"modular exponentiation (i.e. b^e mod m"
[b e m]
(loop [b b, e e, x 1]
(if (zero? e)
x
(if (even? e) (recur (m* b b m) (quot e 2) x)
(recur (m* b b m) (quot e 2) (m* b x m))))))
- Sequence of random numbers to use in the test
(defn rand-num [n]
" random number between 2 and n-2 " (bigint (math/floor (+' 2 (*' (- n 4) (rand))))))
- Unique set of random numbers
(defn unique-random-numbers [n k]
" k unique random numbers between 2 and n-2 "
(loop [a-set #{}]
(cond
(>= (count a-set) k) a-set
:else (recur (conj a-set (rand-num n))))))
(defn find-d-s [n]
" write n − 1 as 2s·d with d odd "
(loop [d (dec n), s 0]
(if (odd? d)
[d s]
(recur (quot d 2) (inc s)))))
(defn random-test
([n] (random-test n (min 1000 (bigint (/ n 2)))))
([n k]
" Random version of primality test"
(let [[d s] (find-d-s n)
; Individual Primality Test
single-test (fn [a s]
(let [z (power a d n)]
(if (some #{z} [1 (dec n)])
WITNESSLOOP
(loop [x (power z 2 n), r s]
(cond
(= x 1) COMPOSITE
(= x (dec n)) WITNESSLOOP
(= r 0) COMPOSITE
:else (recur (power x 2 n) (dec r)))))))]
; Apply Test
;(not-any? #(= COMPOSITE (local-test % s))
; (unique-random-numbers n k))))
(not-any? #(= COMPOSITE (single-test % s)) (unique-random-numbers n k)))))
- Testing
(println "Primes beteen 900-1000:") (doseq [q (range 900 1000)
:when (random-test q)] (print " " q))
(println) (println "Is Prime?" 4547337172376300111955330758342147474062293202868155909489 (random-test 4547337172376300111955330758342147474062293202868155909489)) (println "Is Prime?" 4547337172376300111955330758342147474062293202868155909393 (random-test 4547337172376300111955330758342147474062293202868155909393)) (println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(random-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(random-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
</lang>
- Output:
Primes beteen 900-1000: 907 911 919 929 937 941 947 953 967 971 977 983 991 997 Is Prime? 4547337172376300111955330758342147474062293202868155909489N true Is Prime? 4547337172376300111955330758342147474062293202868155909393N false Is Prime? 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153N true Is Prime? 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153N false
Deterministic Approach
<lang lisp>(ns test-p.core
(:require [clojure.math.numeric-tower :as math]))
(def WITNESSLOOP "witness") (def COMPOSITE "composite")
(defn m* [p q m]
" Computes (p*q) mod m " (mod (*' p q) m))
(defn power
"modular exponentiation (i.e. b^e mod m"
[b e m]
(loop [b b, e e, x 1]
(if (zero? e)
x
(if (even? e) (recur (m* b b m) (quot e 2) x)
(recur (m* b b m) (quot e 2) (m* b x m))))))
(defn find-d-s [n] " write n − 1 as 2s·d with d odd " (loop [d (dec n), s 0]
(if (odd? d) [d s] (recur (quot d 2) (inc s)))))
- Deterministic Test
(defn individual-deterministic-test [a d n s]
" Deterministic Primality Test "
(let [z (power a d n)]
(if (= z 1)
WITNESSLOOP
(loop [x z, r s]
(cond
(= x (dec n)) WITNESSLOOP
(zero? r) COMPOSITE
:else (recur (power x 2 n) (dec r)))))))
(defn deterministic-test [n]
" Sequence of Primality Tests "
(cond
(some #{n} [0 1 4]) false
(some #{n} [2 3]) true
(even? n) false
:else (let [[d s] (find-d-s n)]
(cond
(< n 2047) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 ])
(< n 1373653) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3])
(< n 9090191) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [31 73])
(< n 25326001) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5])
(< n 3215031751) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7])
(< n 1122004669633) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 13 23 1662803])
(< n 2152302898747) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11])
(< n 2152302898747) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11])
(< n 3474749660383) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13])
(< n 341550071728321) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17])
(< n 3825123056546,413,051) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23])
(< n (math/expt 2 64) ) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23 29 31 37])
(< n 318665857834031151167461) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23 29 31 37])
(< n 3317044064679887385961981) (not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) [2 3 5 7 11 13 17 19 23 29 31 37 41])
:else (let [k (min (dec n) (int (math/expt (Math/log n) 2)))]
(not-any? #(= COMPOSITE (individual-deterministic-test % d n s)) (range 2 (inc k))))))))
- Testing
(println "Primes beteen 900-1000:") (doseq [q (range 900 1000)
:when (deterministic-test q)] (print " " q))
(println) (println "Is Prime?" 4547337172376300111955330758342147474062293202868155909489 (deterministic-test 4547337172376300111955330758342147474062293202868155909489)) (println "Is Prime?" 4547337172376300111955330758342147474062293202868155909393 (deterministic-test 4547337172376300111955330758342147474062293202868155909393)) println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
</lang>
- Output:
Primes beteen 900-1000:
907 911 919 929 937 941 947 953 967 971 977 983 991 997
Is Prime? 4547337172376300111955330758342147474062293202868155909489N true
Is Prime? 4547337172376300111955330758342147474062293202868155909393N false
(println "Is Prime?" 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
(println "Is Prime?" 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153
(deterministic-test 743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153))
Common Lisp
<lang lisp>(defun factor-out (number divisor)
"Return two values R and E such that NUMBER = DIVISOR^E * R,
and R is not divisible by DIVISOR."
(do ((e 0 (1+ e))
(r number (/ r divisor)))
((/= (mod r divisor) 0) (values r e))))
(defun mult-mod (x y modulus) (mod (* x y) modulus))
(defun expt-mod (base exponent modulus)
"Fast modular exponentiation by repeated squaring."
(labels ((expt-mod-iter (b e p)
(cond ((= e 0) p)
((evenp e)
(expt-mod-iter (mult-mod b b modulus)
(/ e 2)
p))
(t
(expt-mod-iter b
(1- e)
(mult-mod b p modulus))))))
(expt-mod-iter base exponent 1)))
(defun random-in-range (lower upper)
"Return a random integer from the range [lower..upper]." (+ lower (random (+ (- upper lower) 1))))
(defun miller-rabin-test (n k)
"Test N for primality by performing the Miller-Rabin test K times.
Return NIL if N is composite, and T if N is probably prime."
(cond ((= n 1) nil)
((< n 4) t)
((evenp n) nil)
(t
(multiple-value-bind (d s) (factor-out (- n 1) 2)
(labels ((strong-liar? (a)
(let ((x (expt-mod a d n)))
(or (= x 1)
(loop repeat s
for y = x then (mult-mod y y n)
thereis (= y (- n 1)))))))
(loop repeat k
always (strong-liar? (random-in-range 2 (- n 2)))))))))</lang>
CL-USER> (last (loop for i from 1 to 1000
when (miller-rabin-test i 10)
collect i)
10)
(937 941 947 953 967 971 977 983 991 997)
D
<lang d>import std.random;
bool isProbablePrime(in ulong n, in uint k=10) /*nothrow*/ @safe /*@nogc*/ {
static ulong modPow(ulong b, ulong e, in ulong m)
pure nothrow @safe @nogc {
ulong result = 1;
while (e > 0) {
if ((e & 1) == 1)
result = (result * b) % m;
b = (b ^^ 2) % m;
e >>= 1;
}
return result;
}
if (n < 2 || n % 2 == 0)
return n == 2;
ulong d = n - 1;
ulong s = 0;
while (d % 2 == 0) {
d /= 2;
s++;
}
assert(2 ^^ s * d == n - 1);
outer:
foreach (immutable _; 0 .. k) {
immutable ulong a = uniform(2, n);
ulong x = modPow(a, d, n);
if (x == 1 || x == n - 1)
continue;
foreach (immutable __; 1 .. s) {
x = modPow(x, 2, n);
if (x == 1)
return false;
if (x == n - 1)
continue outer;
}
return false;
}
return true;
}
void main() { // Demo code.
import std.stdio, std.range, std.algorithm;
iota(2, 30).filter!isProbablePrime.writeln;
}</lang>
- Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
E
<lang e>def millerRabinPrimalityTest(n :(int > 0), k :int, random) :boolean {
if (n <=> 2 || n <=> 3) { return true }
if (n <=> 1 || n %% 2 <=> 0) { return false }
var d := n - 1
var s := 0
while (d %% 2 <=> 0) {
d //= 2
s += 1
}
for _ in 1..k {
def nextTrial := __continue
def a := random.nextInt(n - 3) + 2 # [2, n - 2] = [0, n - 4] + 2 = [0, n - 3) + 2
var x := a**d %% n # Note: Will do optimized modular exponentiation
if (x <=> 1 || x <=> n - 1) { nextTrial() }
for _ in 1 .. (s - 1) {
x := x**2 %% n
if (x <=> 1) { return false }
if (x <=> n - 1) { nextTrial() }
}
return false
}
return true
}</lang> <lang e>for i ? (millerRabinPrimalityTest(i, 1, entropy)) in 4..1000 {
print(i, " ")
} println()</lang>
EchoLisp
EchoLisp natively implement the prime? function = Miller-Rabin tests for big integers. The definition is as follows : <lang scheme> (lib 'bigint)
- output
- #t if n probably prime
(define (miller-rabin n (k 7) (composite #f)(x)) (define d (1- n)) (define s 0) (define a 0) (while (even? d) (set! s (1+ s)) (set! d (quotient d 2)))
(for [(i k)] (set! a (+ 2 (random (- n 3)))) (set! x (powmod a d n)) #:continue (or (= x 1) (= x (1- n))) (set! composite (for [(r (in-range 1 s))] (set! x (powmod x 2 n)) #:break (= x 1) => #t #:break (= x (1- n)) => #f #t )) #:break composite => #f )
(not composite))
- output
(miller-rabin #101)
→ #t
(miller-rabin #111)
→ #f
(define big-prime (random-prime 1e+100)) 3461396142610375479080862553800503306376298093021233334170610435506057862777898396429 6627816219192601527 (miller-rabin big-prime)
→ #t
(miller-rabin (1+ (factorial 100)))
→ #f
(prime? (1+ (factorial 100))) ;; native
→ #f
</lang>
Erlang
This implementation of a Miller-Rabin method was revised to permit use of integers of arbitrary precision.
<lang erlang>-module(miller_rabin).
-export([is_prime/1, power/2]).
is_prime(1) -> false; is_prime(2) -> true; is_prime(3) -> true; is_prime(N) when N > 3, ((N rem 2) == 0) -> false; is_prime(N) when ((N rem 2) ==1), N < 341550071728321 ->
is_mr_prime(N, proving_bases(N));
is_prime(N) when ((N rem 2) == 1) -> is_mr_prime(N, random_bases(N, 100)).
proving_bases(N) when N < 1373653 ->
[2, 3];
proving_bases(N) when N < 9080191 ->
[31, 73];
proving_bases(N) when N < 25326001 -> [2, 3, 5]; proving_bases(N) when N < 3215031751 -> [2, 3, 5, 7]; proving_bases(N) when N < 4759123141 ->
[2, 7, 61];
proving_bases(N) when N < 1122004669633 -> [2, 13, 23, 1662803]; proving_bases(N) when N < 2152302898747 -> [2, 3, 5, 7, 11]; proving_bases(N) when N < 3474749660383 ->
[2, 3, 5, 7, 11, 13];
proving_bases(N) when N < 341550071728321 ->
[2, 3, 5, 7, 11, 13, 17].
is_mr_prime(N, As) when N>2, N rem 2 == 1 ->
{D, S} = find_ds(N),
%% this is a test for compositeness; the two case patterns disprove
%% compositeness.
not lists:any(fun(A) ->
case mr_series(N, A, D, S) of
[1|_] -> false; % first elem of list = 1
L -> not lists:member(N-1, L) % some elem of list = N-1
end
end,
As).
find_ds(N) ->
find_ds(N-1, 0).
find_ds(D, S) ->
case D rem 2 == 0 of
true ->
find_ds(D div 2, S+1);
false ->
{D, S}
end.
mr_series(N, A, D, S) when N rem 2 == 1 ->
Js = lists:seq(0, S), lists:map(fun(J) -> pow_mod(A, power(2, J)*D, N) end, Js).
pow_mod(B, E, M) ->
case E of
0 -> 1;
_ -> case ((E rem 2) == 0) of
true -> (power(pow_mod(B, (E div 2), M), 2)) rem M;
false -> (B*pow_mod(B, E-1, M)) rem M
end
end.
random_bases(N, K) ->
[basis(N) || _ <- lists:seq(1, K)].
basis(N) when N>2 ->
1 + random:uniform(N-3). % random:uniform returns a single random number in range 1 -> N-3, to which is added 1, shifting the range to 2 -> N-2
power(B, E) ->
power(B, E, 1).
power(_, 0, Acc) ->
Acc;
power(B, E, Acc) ->
power(B, E - 1, B * Acc).</lang>
Fortran
Direct translation
For the module PrimeDecompose, see Prime decomposition. <lang fortran>
module Miller_Rabin use PrimeDecompose implicit none
integer, parameter :: max_decompose = 100
private :: int_rrand, max_decompose
contains
function int_rrand(from, to) integer(huge) :: int_rrand integer(huge), intent(in) :: from, to
real :: o call random_number(o) int_rrand = floor(from + o * real(max(from,to) - min(from, to))) end function int_rrand
function miller_rabin_test(n, k) result(res) logical :: res integer(huge), intent(in) :: n integer, intent(in) :: k integer(huge), dimension(max_decompose) :: f integer(huge) :: s, d, i, a, x, r
res = .true. f = 0
if ( (n <= 2) .and. (n > 0) ) return
if ( mod(n, 2) == 0 ) then
res = .false.
return
end if
call find_factors(n-1, f)
s = count(f == 2)
d = (n-1) / (2 ** s)
loop: do i = 1, k
a = int_rrand(2_huge, n-2)
x = mod(a ** d, n)
if ( x == 1 ) cycle
do r = 0, s-1
if ( x == ( n - 1 ) ) cycle loop
x = mod(x*x, n)
end do
if ( x == (n-1) ) cycle
res = .false.
return
end do loop
res = .true.
end function miller_rabin_test
end module Miller_Rabin</lang> Testing <lang fortran>program TestMiller
use Miller_Rabin implicit none
integer, parameter :: prec = 30 integer(huge) :: i
! this is limited since we're not using a bignum lib call do_test( (/ (i, i=1, 29) /) )
contains
subroutine do_test(a) integer(huge), dimension(:), intent(in) :: a
integer :: i
do i = 1, size(a,1)
print *, a(i), miller_rabin_test(a(i), prec)
end do
end subroutine do_test
end program TestMiller</lang>
Possible improvements: create bindings to the GMP library, change integer(huge) into something like type(huge_integer), write a lots of interfaces to allow to use big nums naturally (so that the code will be unchanged, except for the changes said above)
With some avoidance of overflow
Integer overflow is a severe risk, and even 64-bit integers won't get you far when the formulae are translated as MOD(A**D,N) - what is needed is a method for raising to a power that incorporates the modulus along the way. There is no library routine for that, so... <lang Fortran> MODULE MRTEST !Try the Miller-Rabin primality test.
CONTAINS !Working only with in-built integers.
LOGICAL FUNCTION MRPRIME(N,TRIALS) !Could N be a prime number?
USE DFPORT !To get RAND.
INTEGER N !The number.
INTEGER TRIALS !The count of trials to make.
INTEGER D,S !Represents a number in a special form.
INTEGER TRIAL
INTEGER A,X,R
Catch some annoying cases.
IF (N .LE. 4) THEN !A single-digit number?
MRPRIME = N.GT.1 .AND. N.LE.3 !Yes. Some special values.
RETURN !Thus allow 2 to be reported as prime.
END IF !Yet, test for 2 as a possible factor for larger numbers.
MRPRIME = .FALSE. !Pessimism prevails.
IF (MOD(N,2).EQ.0 .OR. MOD(N,3).EQ.0) RETURN !Thus.
Construct D such that N - 1 = D*2**S. By here, N is odd, and greater than three.
D = N - 1 !Thus, D becomes an even number.
S = 1 !So, it has at least one power of two.
10 D = D/2 !Divide it out.
IF (MOD(D,2).EQ.0) THEN !If there is another,
S = S + 1 !Count it,
GO TO 10 !And divide it out also.
END IF !So, D is no longer even. N = 1 + D*2**S
WRITE (6,11) N,D,S
11 FORMAT("For ",I0,", D=",I0,",S=",I0)
Convince through repetition..
T:DO TRIAL = 1,TRIALS !Some trials yield a definite result.
A = RAND(0)*(N - 2) + 2 !For small N, the birthday problem.
X = MODEXP(N,A,D) !A**D mod N.
WRITE (6,22) TRIAL,A,X,INT8(A)**D,N,MOD(INT8(A)**D,N)
22 FORMAT(6X,"Trial ",I0,",A=",I4,",X=",I4,
1 "=MOD(",I0,",",I0,")=",I0)
IF (X.EQ.1 .OR. X.EQ.N - 1) CYCLE T !Pox. A prime yields these.
DO R = 1,S - 1 !Step through the powers of two in N - 1.
X = MODEXP(N,X,2) !X**2 mod N.
WRITE (6,23) R,X
23 FORMAT (14X,"R=",I4,",X=",I0)
IF (X.EQ.1) RETURN !Definitely composite. No prime does this.
IF (X.EQ.N - 1) CYCLE T !Pox. Try something else.
END DO !Another power of two?
RETURN !Definitely composite.
END DO T !Have another go.
MRPRIME = .TRUE. !Would further trials yield greater assurance?
END FUNCTION MRPRIME !Are some numbers resistant to this scheme?
INTEGER FUNCTION MODEXP(N,X,P) !Calculate X**P mod N without overflowing...
C Relies on a.b mod n = (a mod n)(b mod n) mod n
INTEGER N,X,P !All presumed positive, and X < N.
INTEGER I !A stepper.
INTEGER*8 V,W !Broad scratchpads, otherwise N > 46340 may incur overflow in 32-bit.
V = 1 !=X**0
IF (P.GT.0) THEN !Something to do?
I = P !Yes. Get a copy I can mess with.
W = X !=X**1, X**2, X**4, X**8, ... except, all are mod N.
1 IF (MOD(I,2).EQ.1) V = MOD(V*W,N) !Incorporate W if the low-end calls for it.
I = I/2 !Used. Shift the next one down.
IF (I.GT.0) THEN !Still something to do?
W = MOD(W**2,N) !Yes. Square W ready for the next bit up.
GO TO 1 !Consider it.
END IF !Don't square W if nothing remains. It might overflow.
END IF !Negative powers are ignored.
MODEXP = V !Done, in lb(P) iterations!
END FUNCTION MODEXP !"Bit" presence by arithmetic: works for non-binary arithmetic too.
PROGRAM POKEMR
USE MRTEST
INTEGER I
LOGICAL HIC
DO I = 3,36,2
HIC = MRPRIME(I,6)
WRITE (6,11) I,HIC
11 FORMAT (I6,1X,L)
END DO
END</lang>
Output:
3 T
For 5, D=1,S=2
Trial 1,A= 2,X= 2=MOD(2,5)=2
R= 1,X=4
Trial 2,A= 2,X= 2=MOD(2,5)=2
R= 1,X=4
Trial 3,A= 3,X= 3=MOD(3,5)=3
R= 1,X=4
Trial 4,A= 4,X= 4=MOD(4,5)=4
Trial 5,A= 4,X= 4=MOD(4,5)=4
Trial 6,A= 2,X= 2=MOD(2,5)=2
R= 1,X=4
5 T
For 7, D=3,S=1
Trial 1,A= 4,X= 1=MOD(64,7)=1
Trial 2,A= 3,X= 6=MOD(27,7)=6
Trial 3,A= 3,X= 6=MOD(27,7)=6
Trial 4,A= 5,X= 6=MOD(125,7)=6
Trial 5,A= 2,X= 1=MOD(8,7)=1
Trial 6,A= 4,X= 1=MOD(64,7)=1
7 T
9 F
For 11, D=5,S=1
Trial 1,A= 7,X= 10=MOD(16807,11)=10
Trial 2,A= 9,X= 1=MOD(59049,11)=1
Trial 3,A= 7,X= 10=MOD(16807,11)=10
Trial 4,A= 6,X= 10=MOD(7776,11)=10
Trial 5,A= 9,X= 1=MOD(59049,11)=1
Trial 6,A= 10,X= 10=MOD(100000,11)=10
11 T
For 13, D=3,S=2
Trial 1,A= 9,X= 1=MOD(729,13)=1
Trial 2,A= 12,X= 12=MOD(1728,13)=12
Trial 3,A= 5,X= 8=MOD(125,13)=8
R= 1,X=12
Trial 4,A= 6,X= 8=MOD(216,13)=8
R= 1,X=12
Trial 5,A= 11,X= 5=MOD(1331,13)=5
R= 1,X=12
Trial 6,A= 9,X= 1=MOD(729,13)=1
13 T
15 F
For 17, D=1,S=4
Trial 1,A= 15,X= 15=MOD(15,17)=15
R= 1,X=4
R= 2,X=16
Trial 2,A= 16,X= 16=MOD(16,17)=16
Trial 3,A= 4,X= 4=MOD(4,17)=4
R= 1,X=16
Trial 4,A= 14,X= 14=MOD(14,17)=14
R= 1,X=9
R= 2,X=13
R= 3,X=16
Trial 5,A= 15,X= 15=MOD(15,17)=15
R= 1,X=4
R= 2,X=16
Trial 6,A= 6,X= 6=MOD(6,17)=6
R= 1,X=2
R= 2,X=4
R= 3,X=16
17 T
For 19, D=9,S=1
Trial 1,A= 17,X= 1=MOD(118587876497,19)=1
Trial 2,A= 9,X= 1=MOD(387420489,19)=1
Trial 3,A= 7,X= 1=MOD(40353607,19)=1
Trial 4,A= 10,X= 18=MOD(1000000000,19)=18
Trial 5,A= 8,X= 18=MOD(134217728,19)=18
Trial 6,A= 15,X= 18=MOD(38443359375,19)=18
19 T
21 F
For 23, D=11,S=1
Trial 1,A= 16,X= 1=MOD(17592186044416,23)=1
Trial 2,A= 13,X= 1=MOD(1792160394037,23)=1
Trial 3,A= 14,X= 22=MOD(4049565169664,23)=22
Trial 4,A= 3,X= 1=MOD(177147,23)=1
Trial 5,A= 14,X= 22=MOD(4049565169664,23)=22
Trial 6,A= 11,X= 22=MOD(285311670611,23)=22
23 T
For 25, D=3,S=3
Trial 1,A= 15,X= 0=MOD(3375,25)=0
R= 1,X=0
R= 2,X=0
25 F
27 F
For 29, D=7,S=2
Trial 1,A= 24,X= 1=MOD(4586471424,29)=1
Trial 2,A= 15,X= 17=MOD(170859375,29)=17
R= 1,X=28
Trial 3,A= 22,X= 28=MOD(2494357888,29)=28
Trial 4,A= 3,X= 12=MOD(2187,29)=12
R= 1,X=28
Trial 5,A= 7,X= 1=MOD(823543,29)=1
Trial 6,A= 8,X= 17=MOD(2097152,29)=17
R= 1,X=28
29 T
For 31, D=15,S=1
Trial 1,A= 24,X= 30=MOD(6795192965888212992,31)=1
Trial 2,A= 4,X= 1=MOD(1073741824,31)=1
Trial 3,A= 7,X= 1=MOD(4747561509943,31)=1
Trial 4,A= 19,X= 1=MOD(-3265617043834753317,31)=-15
Trial 5,A= 18,X= 1=MOD(6746640616477458432,31)=1
Trial 6,A= 23,X= 30=MOD(8380818432457522983,31)=23
31 T
33 F
For 35, D=17,S=1
Trial 1,A= 12,X= 17=MOD(2218611106740436992,35)=17
35 F
In this run, 32-bit integers falter for 19 in calculating 179, and 64-bit integers falter for 31 with 1915 by showing a negative number. Other 64-bit overflows however do not show a negative (as with 2315) because there is about an even chance that the high-order bit will be on or off. The compiler option for checking integer overflow does not report such faults with 64-bit integers, at least with the Compaq V6.6 F90/95 compiler. In this context, one misses the IF OVERFLOW ... that was part of Fortran II but which has been omitted from later versions.
Thus, there is no avoiding a special MODEXP function, even for small test numbers.
FreeBASIC
Using the task pseudo code
Up to 2^63-1
<lang freebasic>' version 29-11-2016 ' compile with: fbc -s console
' TRUE/FALSE are built-in constants since FreeBASIC 1.04 ' But we have to define them for older versions.
- Ifndef TRUE
#Define FALSE 0 #Define TRUE Not FALSE
- EndIf
Function mul_mod(a As ULongInt, b As ULongInt, modulus As ULongInt) As ULongInt
' returns a * b mod modulus Dim As ULongInt x, y = a ' a mod modulus, but a is already smaller then modulus
While b > 0
If (b And 1) = 1 Then
x = (x + y) Mod modulus
End If
y = (y Shl 1) Mod modulus
b = b Shr 1
Wend
Return x
End Function
Function pow_mod(b As ULongInt, power As ULongInt, modulus As ULongInt) As ULongInt
' returns b ^ power mod modulus Dim As ULongInt x = 1
While power > 0
If (power And 1) = 1 Then
' x = (x * b) Mod modulus
x = mul_mod(x, b, modulus)
End If
' b = (b * b) Mod modulus
b = mul_mod(b, b, modulus)
power = power Shr 1
Wend
Return x
End Function
Function miller_rabin_test(n As ULongInt, k As Integer) As Byte
If n > 9223372036854775808ull Then ' limit 2^63, pow_mod/mul_mod can't handle bigger numbers
Print "number is to big, program will end"
Sleep
End
End If
' 2 is a prime, if n is smaller then 2 or n is even then n = composite If n = 2 Then Return TRUE If (n < 2) OrElse ((n And 1) = 0) Then Return FALSE
Dim As ULongInt a, x, n_one = n - 1, d = n_one Dim As UInteger s
While (d And 1) = 0
d = d Shr 1
s = s + 1
Wend
While k > 0
k = k - 1
a = Int(Rnd * (n -2)) +2 ' 2 <= a < n
x = pow_mod(a, d, n)
If (x = 1) Or (x = n_one) Then Continue While
For r As Integer = 1 To s -1
x = pow_mod(x, 2, n)
If x = 1 Then Return FALSE
If x = n_one Then Continue While
Next
If x <> n_one Then Return FALSE
Wend
Return TRUE
End Function ' ------=< MAIN >=------
Randomize Timer
Dim As Integer total Dim As ULongInt y, limit = 2^63-1
For y = limit - 1000 To limit
If miller_rabin_test(y, 5) = TRUE Then
total = total + 1
Print y,
End If
Next
Print : Print Print total; " primes between "; limit - 1000; " and "; y -1
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
9223372036854774893 9223372036854774917 9223372036854774937 9223372036854774959 9223372036854775057 9223372036854775073 9223372036854775097 9223372036854775139 9223372036854775159 9223372036854775181 9223372036854775259 9223372036854775279 9223372036854775291 9223372036854775337 9223372036854775351 9223372036854775399 9223372036854775417 9223372036854775421 9223372036854775433 9223372036854775507 9223372036854775549 9223372036854775643 9223372036854775783 23 primes between 9223372036854774808 and 9223372036854775808
Using Big Integer library
<lang freebasic>' version 05-04-2017 ' compile with: fbc -s console
' TRUE/FALSE are built-in constants since FreeBASIC 1.04 ' But we have to define them for older versions.
- Ifndef TRUE
#Define FALSE 0 #Define TRUE Not FALSE
- EndIf
- Include Once "gmp.bi"
- Macro big_int(a)
Dim As Mpz_ptr a = Allocate( Len( __mpz_struct)) Mpz_init(a)
- EndMacro
Dim Shared As __gmp_randstate_struct rnd_
Function miller_rabin(big_n As Mpz_ptr, num_of_tests As ULong) As Byte
If mpz_cmp_ui(big_n, 1) < 1 Then
Print "Numbers smaller then 1 not allowed"
Sleep 5000
End If
If mpz_cmp_ui(big_n, 2) = 0 OrElse mpz_cmp_ui(big_n, 3) = 0 Then
Return TRUE ' 2 = prime , 3 = prime
End If
If mpz_tstbit(big_n, 0) = 0 Then Return FALSE ' even number, no prime
Dim As ULong r, s Dim As Byte return_value = TRUE
big_int(n_1) : big_int(n_2) : big_int(a) : big_int(d) : big_int(x)
mpz_sub_ui(n_1, big_n, 1) : mpz_sub_ui(n_2, big_n, 2) : mpz_set(d, n_1)
While mpz_tstbit(d, 0) = 0
mpz_fdiv_q_2exp(d, d, 1)
s += 1
Wend
While num_of_tests > 0
num_of_tests -= 1
mpz_urandomm(a, @rnd_, n_2)
mpz_add_ui(a, a, 2)
mpz_powm(x, a, d, big_n)
If mpz_cmp_ui(x, 1) = 0 Or mpz_cmp(x, n_1) = 0 Then Continue While
For r = 1 To s -1
mpz_powm_ui(x, x, 2, big_n)
If mpz_cmp_ui(x, 1) = 0 Then
return_value = FALSE
Exit While
End If
If mpz_cmp(x, n_1) = 0 Then Continue While
Next
If mpz_cmp(x, n_1) <> 0 Then
Return_value = FALSE
Exit while
End If
Wend
mpz_clear(n_1) : mpz_clear(a) : mpz_clear(d) mpz_clear(n_2) : mpz_clear(x)
Return return_value
End Function
' ------=< MAIN >=------
Dim As Long x Dim As String tmp Dim As ZString Ptr gmp_str : gmp_str = Allocate(1000000) big_int(big_n)
Randomize Timer gmp_randinit_mt(@rnd_) For x = 0 To 200 'create seed for random generator
tmp += Str(Int(Rnd * 10))
Next Mpz_set_str(big_n, tmp, 10) gmp_randseed(@rnd_, big_n) ' seed the random number generator
For x = 2 To 100
mpz_set_ui(big_n, x)
If miller_rabin(big_n, 5) = TRUE Then
Print Using "####"; x;
End If
Next
Print : Print For x = 2 To 3300
mpz_set_ui(big_n, 1)
mpz_mul_2exp(big_n, big_n, x)
mpz_sub_ui(big_n, big_n, 1)
If miller_rabin(big_n, 5) = TRUE Then
gmp_str = Mpz_get_str(0, 10, big_n)
Print "2^";Str(x);"-1 = prime"
End If
Next
gmp_randclear(@rnd_) mpz_clear(big_n) DeAllocate(gmp_str)
' empty keyboard buffer Print : While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 2^2-1 = prime 2^3-1 = prime 2^5-1 = prime 2^7-1 = prime 2^13-1 = prime 2^17-1 = prime 2^19-1 = prime 2^31-1 = prime 2^61-1 = prime 2^89-1 = prime 2^107-1 = prime 2^127-1 = prime 2^521-1 = prime 2^607-1 = prime 2^1279-1 = prime 2^2203-1 = prime 2^2281-1 = prime 2^3217-1 = prime
FunL
Direct implementation of the task algorithm.
<lang funl>import util.rnd
def isProbablyPrimeMillerRabin( n, k ) =
d = n - 1 s = 0
while 2|d s++ d /= 2
repeat k a = rnd( 2, n ) x = a^d mod n
if x == 1 or x == n - 1 then continue
repeat s - 1
x = x^2 mod n
if x == 1 then return false
if x == n - 1 then break
else
return false
true
for i <- 3..100
if isProbablyPrimeMillerRabin( i, 5 ) println( i )</lang>
- Output:
3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
Go
- Library
Go has it in math/big in standard library as ProbablyPrime. The argument n to ProbablyPrime is the input k of the pseudocode in the task description.
- Deterministic
Below is a deterministic test for 32 bit unsigned integers. Intermediate results in the code below include a 64 bit result from multiplying two 32 bit numbers. Since 64 bits is the largest fixed integer type in Go, a 32 bit number is the largest that is convenient to test.
The main difference between this algorithm and the pseudocode in the task description is that k numbers are not chosen randomly, but instead are the three numbers 2, 7, and 61. These numbers provide a deterministic primality test up to 2^32. <lang go>package main
import "log"
func main() {
// max uint32 is not prime
c := uint32(1<<32 - 1)
// a few primes near the top of the range. source: prime pages.
for _, p := range []uint32{1<<32 - 5, 1<<32 - 17, 1<<32 - 65, 1<<32 - 99} {
for ; c > p; c-- {
if prime(c) {
log.Fatalf("prime(%d) returned true", c)
}
}
if !prime(p) {
log.Fatalf("prime(%d) returned false", p)
}
c--
}
}
func prime(n uint32) bool {
// bases of 2, 7, 61 are sufficient to cover 2^32
switch n {
case 0, 1:
return false
case 2, 7, 61:
return true
}
// compute s, d where 2^s * d = n-1
nm1 := n - 1
d := nm1
s := 0
for d&1 == 0 {
d >>= 1
s++
}
n64 := uint64(n)
for _, a := range []uint32{2, 7, 61} {
// compute x := a^d % n
x := uint64(1)
p := uint64(a)
for dr := d; dr > 0; dr >>= 1 {
if dr&1 != 0 {
x = x * p % n64
}
p = p * p % n64
}
if x == 1 || uint32(x) == nm1 {
continue
}
for r := 1; ; r++ {
if r >= s {
return false
}
x = x * x % n64
if x == 1 {
return false
}
if uint32(x) == nm1 {
break
}
}
}
return true
}</lang>
Haskell
- Ideas taken from Primality proving
- Functions witns and isMillerRabinPrime follow closely the code outlined in J/Essays
- A useful powerMod function is taken from Multiplicative order#Haskell
- Original Rosetta code has been simplified to be easier to follow
Another Miller Rabin test can be found in D. Amos's Haskell for Math module Primes.hs <lang Haskell>module Primes where
import System.Random import System.IO.Unsafe
-- Miller-Rabin wrapped up as an (almost deterministic) pure function isPrime :: Integer -> Bool isPrime n = unsafePerformIO (isMillerRabinPrime 100 n)
isMillerRabinPrime :: Int -> Integer -> IO Bool
isMillerRabinPrime k n
| even n = return (n==2)
| n < 100 = return (n `elem` primesTo100)
| otherwise = do ws <- witnesses k n
return $ and [test n (pred n) evens (head odds) a | a <- ws]
where
(evens,odds) = span even (iterate (`div` 2) (pred n))
test :: Integral nat => nat -> nat -> [nat] -> nat -> nat -> Bool test n n_1 evens d a = x `elem` [1,n_1] || n_1 `elem` powers
where x = powerMod n a d powers = map (powerMod n a) evens
witnesses :: (Num a, Ord a, Random a) => Int -> a -> IO [a] witnesses k n
| n < 9080191 = return [31,73]
| n < 4759123141 = return [2,7,61]
| n < 3474749660383 = return [2,3,5,7,11,13]
| n < 341550071728321 = return [2,3,5,7,11,13,17]
| otherwise = do g <- newStdGen
return $ take k (randomRs (2,n-1) g)
primesTo100 :: [Integer] primesTo100 = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
-- powerMod m x n = x^n `mod` m powerMod :: Integral nat => nat -> nat -> nat -> nat powerMod m x n = f (n - 1) x x `rem` m
where
f d a y = if d==0 then y else g d a y
g i b y | even i = g (i `quot` 2) (b*b `rem` m) y
| otherwise = f (i-1) b (b*y `rem` m)
</lang>
- Sample output:
Testing in GHCi: ~> isPrime 4547337172376300111955330758342147474062293202868155909489 True *~> isPrime 4547337172376300111955330758342147474062293202868155909393 False *~> dropWhile (<900) $ filter isPrime [2..1000] [907,911,919,929,937,941,947,953,967,971,977,983,991,997]
Perhaps a slightly clearer (less monadic) version. Transcription of pseudocode.
- The code above likely has better complexity.
<lang Haskell> import Control.Monad (liftM) import Data.Bits (Bits, testBit, shiftR) import System.Random (Random, getStdGen, randomRs) import System.IO.Unsafe (unsafePerformIO) import Prelude hiding (even, odd)
odd :: (Integral a, Bits a) => a -> Bool odd = (`testBit` 0)
even :: (Integral a, Bits a) => a -> Bool even = not . odd
-- modPow - Recursive modular exponentiation by taking successive powers of two modPow :: (Integral a, Bits a) => a -> a -> a -> a modPow _ 0 _ = 1 modPow base ex m = let term
| testBit ex 0 = base `mod` m
| otherwise = 1
in (term * modPow (base^2 `mod` m) (ex `shiftR` 1) m) `mod` m
isPrime :: (Integral a, Bits a, Random a) => a -> a -> Bool isPrime n k
| n < 4 = if n > 1 then True else False -- Deal with 0-3.
| even n = False
| otherwise = let randPool = unsafePerformIO $ randNums (n - 2)
in witness k randPool
where
randNums upper = do
g <- getStdGen
return (randomRs (2, upper) g)
(d, r) = let decompose d r
| odd d = (d, r)
| otherwise = decompose (d `shiftR` 1) (r + 1)
in decompose (n - 1) 0
witness 0 _ = True
witness k (a:rands)
| x == 1 || x == n - 1 = witness (k - 1) rands
| otherwise = check x (r - 1)
where
x = modPow a d n
check _ 0 = False
check x count
| x' == 1 = False
| x' == n - 1 = witness (k - 1) rands
| otherwise = check x' (count - 1)
where x' = modPow x 2 n
-- main function for testing main :: IO() main = do
[n,k] <- liftM (map (\x -> read x :: Integer) . words) getLine print $ isPrime n k
</lang>
- Sample Output:
4547337172376300111955330758342147474062293202868155909489 10 True 4547337172376300111955330758342147474062293202868155909393 10 False 226801 7 False 94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881 50 True
Icon and Unicon
The following runs in both languages: <lang unicon>procedure main(A)
every n := !A do write(n," is ",(mrp(n,5),"probably prime")|"composite")
end
procedure mrp(n, k)
if n = 2 then return ""
if n%2 = 0 then fail
nm1 := decompose(n-1)
s := nm1[1]
d := nm1[2]
every !k do {
a := ?(n-2)+1
x := (a^d)%n
if x = (1|(n-1)) then next
every !(s-1) do {
x := (x*x)%n
if x = 1 then fail
if x = (n-1) then break next
}
fail
}
return ""
end
procedure decompose(nm1)
s := 1
d := nm1
while d%2 = 0 do {
d /:= 2
s +:= 1
}
return [s,d]
end</lang>
Sample run:
->mrp 219 221 223 225 227 229 219 is composite 221 is composite 223 is probably prime 225 is composite 227 is probably prime 229 is probably prime ->
J
See Primality Tests essay on the J wiki.
Java
The Miller-Rabin primality test is part of the standard library (java.math.BigInteger) <lang java>import java.math.BigInteger;
public class MillerRabinPrimalityTest {
public static void main(String[] args) {
BigInteger n = new BigInteger(args[0]);
int certainty = Integer.parseInt(args[1]);
System.out.println(n.toString() + " is " + (n.isProbablePrime(certainty) ? "probably prime" : "composite"));
}
}</lang>
- Sample output:
java MillerRabinPrimalityTest 123456791234567891234567 1000000 123456791234567891234567 is probably prime
This is a translation of the Python solution for a deterministic test for n < 341,550,071,728,321: <lang java>import java.math.BigInteger;
public class Prime {
// this is the RabinMiller test, deterministically correct for n < 341,550,071,728,321 // http://rosettacode.org/wiki/Miller-Rabin_primality_test#Python:_Proved_correct_up_to_large_N public static boolean isPrime(BigInteger n, int precision) {
if (n.compareTo(new BigInteger("341550071728321")) >= 0) {
return n.isProbablePrime(precision);
}
int intN = n.intValue();
if (intN == 1 || intN == 4 || intN == 6 || intN == 8) return false;
if (intN == 2 || intN == 3 || intN == 5 || intN == 7) return true;
int[] primesToTest = getPrimesToTest(n);
if (n.equals(new BigInteger("3215031751"))) {
return false;
}
BigInteger d = n.subtract(BigInteger.ONE);
BigInteger s = BigInteger.ZERO;
while (d.mod(BigInteger.valueOf(2)).equals(BigInteger.ZERO)) {
d = d.shiftRight(1);
s = s.add(BigInteger.ONE);
}
for (int a : primesToTest) {
if (try_composite(a, d, n, s)) {
return false;
}
}
return true;
}
public static boolean isPrime(BigInteger n) {
return isPrime(n, 100);
}
public static boolean isPrime(int n) {
return isPrime(BigInteger.valueOf(n), 100);
}
public static boolean isPrime(long n) {
return isPrime(BigInteger.valueOf(n), 100);
}
private static int[] getPrimesToTest(BigInteger n) {
if (n.compareTo(new BigInteger("3474749660383")) >= 0) {
return new int[]{2, 3, 5, 7, 11, 13, 17};
}
if (n.compareTo(new BigInteger("2152302898747")) >= 0) {
return new int[]{2, 3, 5, 7, 11, 13};
}
if (n.compareTo(new BigInteger("118670087467")) >= 0) {
return new int[]{2, 3, 5, 7, 11};
}
if (n.compareTo(new BigInteger("25326001")) >= 0) {
return new int[]{2, 3, 5, 7};
}
if (n.compareTo(new BigInteger("1373653")) >= 0) {
return new int[]{2, 3, 5};
}
return new int[]{2, 3};
}
private static boolean try_composite(int a, BigInteger d, BigInteger n, BigInteger s) {
BigInteger aB = BigInteger.valueOf(a);
if (aB.modPow(d, n).equals(BigInteger.ONE)) {
return false;
}
for (int i = 0; BigInteger.valueOf(i).compareTo(s) < 0; i++) {
// if pow(a, 2**i * d, n) == n-1
if (aB.modPow(BigInteger.valueOf(2).pow(i).multiply(d), n).equals(n.subtract(BigInteger.ONE))) {
return false;
}
}
return true;
}
} </lang>
JavaScript
For the return values of this function, true means "probably prime" and false means "definitely composite."
<lang JavaScript>function probablyPrime(n, k) { if (n === 2 || n === 3) return true; if (n % 2 === 0 || n < 2) return false;
// Write (n - 1) as 2^s * d var s = 0, d = n - 1; while (d % 2 === 0) { d /= 2; ++s; }
WitnessLoop: do { // A base between 2 and n - 2 var x = Math.pow(2 + Math.floor(Math.random() * (n - 3)), d) % n;
if (x === 1 || x === n - 1) continue;
for (var i = s - 1; i--;) { x = x * x % n; if (x === 1) return false; if (x === n - 1) continue WitnessLoop; }
return false; } while (--k);
return true; }</lang>
Julia
The built-in isprime function uses the Miller-Rabin primality test. Here is the implementation of isprime from the Julia standard library (Julia version 0.2):
<lang julia>
witnesses(n::Union(Uint8,Int8,Uint16,Int16)) = (2,3)
witnesses(n::Union(Uint32,Int32)) = n < 1373653 ? (2,3) : (2,7,61)
witnesses(n::Union(Uint64,Int64)) =
n < 1373653 ? (2,3) :
n < 4759123141 ? (2,7,61) :
n < 2152302898747 ? (2,3,5,7,11) :
n < 3474749660383 ? (2,3,5,7,11,13) :
(2,325,9375,28178,450775,9780504,1795265022)
function isprime(n::Integer)
n == 2 && return true
(n < 2) | iseven(n) && return false
s = trailing_zeros(n-1)
d = (n-1) >>> s
for a in witnesses(n)
a < n || break
x = powermod(a,d,n)
x == 1 && continue
t = s
while x != n-1
(t-=1) <= 0 && return false
x = oftype(n, Base.widemul(x,x) % n)
x == 1 && return false
end
end
return true
end </lang>
Kotlin
Translating the pseudo-code directly rather than using the Java library method BigInteger.isProbablePrime(certainty): <lang scala>// version 1.1.2
import java.math.BigInteger import java.util.Random
val bigTwo = BigInteger.valueOf(2L)
fun isProbablyPrime(n: BigInteger, k: Int): Boolean {
require (n > bigTwo && n % bigTwo == BigInteger.ONE) { "Must be odd and greater than 2" }
var s = 0
val nn = n - BigInteger.ONE
var d: BigInteger
do {
s++
d = nn.shiftRight(s)
}
while (d % bigTwo == BigInteger.ZERO)
val rand = Random()
loop@ for (i in 1..k) {
var a: BigInteger
do {
a = BigInteger(n.bitLength(), rand)
}
while(a < bigTwo || a > nn) // make sure it's in the interval [2, n - 1]
var x = a.modPow(d, n)
if (x == BigInteger.ONE || x == nn) continue
for (r in 1 until s) {
x = (x * x) % n
if (x == BigInteger.ONE) return false
if (x == nn) break@loop
}
return false
}
return true
}
fun main(args: Array<String>) {
val k = 20 // say
// obtain all primes up to 100
println("The following numbers less than 100 are prime:")
for (i in 3..99 step 2)
if (isProbablyPrime(BigInteger.valueOf(i.toLong()), k)) print("$i ")
println("\n")
// check if some big numbers are probably prime
val bia = arrayOf(
BigInteger("4547337172376300111955330758342147474062293202868155909489"),
BigInteger("4547337172376300111955330758342147474062293202868155909393")
)
for (bi in bia)
println("$bi is ${if (isProbablyPrime(bi, k)) "probably prime" else "composite"}")
}</lang>
- Output:
The following numbers less than 100 are prime: 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 4547337172376300111955330758342147474062293202868155909489 is probably prime 4547337172376300111955330758342147474062293202868155909393 is composite
Liberty BASIC
<lang lb> DIM mersenne(11) mersenne(1)=7 mersenne(2)=31 mersenne(3)=127 mersenne(4)=8191 mersenne(5)=131071 mersenne(6)=524287 mersenne(7)=2147483647 mersenne(8)=2305843009213693951 mersenne(9)=618970019642690137449562111 mersenne(10)=162259276829213363391578010288127 mersenne(11)=170141183460469231731687303715884105727
dim SmallPrimes(1000)
data 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
data 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
data 73, 79, 83, 89, 97, 101, 103, 107, 109, 113
data 127, 131, 137, 139, 149, 151, 157, 163, 167, 173
data 179, 181, 191, 193, 197, 199, 211, 223, 227, 229
data 233, 239, 241, 251, 257, 263, 269, 271, 277, 281
data 283, 293, 307, 311, 313, 317, 331, 337, 347, 349
data 353, 359, 367, 373, 379, 383, 389, 397, 401, 409
data 419, 421, 431, 433, 439, 443, 449, 457, 461, 463
data 467, 479, 487, 491, 499, 503, 509, 521, 523, 541
data 547, 557, 563, 569, 571, 577, 587, 593, 599, 601
data 607, 613, 617, 619, 631, 641, 643, 647, 653, 659
data 661, 673, 677, 683, 691, 701, 709, 719, 727, 733
data 739, 743, 751, 757, 761, 769, 773, 787, 797, 809
data 811, 821, 823, 827, 829, 839, 853, 857, 859, 863
data 877, 881, 883, 887, 907, 911, 919, 929, 937, 941
data 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013
data 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069
data 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151
data 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223
data 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291
data 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373
data 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451
data 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511
data 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583
data 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657
data 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733
data 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811
data 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889
data 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987
data 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053
data 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129
data 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213
data 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287
data 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357
data 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423
data 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531
data 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617
data 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687
data 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741
data 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819
data 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903
data 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999
data 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079
data 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181
data 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257
data 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331
data 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413
data 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511
data 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571
data 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643
data 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727
data 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821
data 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907
data 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989
data 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057
data 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139
data 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231
data 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297
data 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409
data 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493
data 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583
data 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657
data 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751
data 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831
data 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937
data 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003
data 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087
data 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179
data 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279
data 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387
data 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443
data 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521
data 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639
data 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693
data 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791
data 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857
data 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939
data 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053
data 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133
data 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221
data 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301
data 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367
data 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473
data 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571
data 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673
data 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761
data 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833
data 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917
data 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997
data 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103
data 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207
data 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297
data 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411
data 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499
data 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561
data 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643
data 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723
data 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829
data 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919
print "Liberty Miller Rabin Demonstration"
print "Loading Small Primes"
for i=1 to 1000: read x : SmallPrimes(i)=x :next :NoOfSmallPrimes=1000
print NoOfSmallPrimes;" Primes Loaded"
'Prompt "Enter number to test:";resp$ 'x=val(resp$) 'goto [Jump]
For i=1 to 11
x=mersenne(i)
t1=time$("ms")
[TryAnother]
print
iterations=0 [Loop] iterations=iterations+1
if MillerRabin(x,7)=1 then
t2=time$("ms")
print "Composite, found in ";t2-t1;" milliseconds"
else
t2=time$("ms")
print x;" Probably Prime. Tested in ";t2-t1;" milliseconds"
playwave "tada.wav", async
end if
print
next
END
Function GCD( m,n )
' Find greatest common divisor with Extend Euclidian Algorithm
' Knuth Vol 1 P.13 Algorithm E
ap =1 :b =1 :a =0 :bp =0: c =m :d =n
[StepE2] q = int(c/d) :r = c-q*d
if r<>0 then
c=d :d=r :t=ap :ap=a :a=t-q*a :t=bp :bp=b :b=t-q*b 'print ap;" ";b;" ";a;" ";bp;" ";c;" ";d;" ";t;" ";q goto [StepE2]
end if
GCD=a*m+b*n
'print ap;" ";b;" ";a;" ";bp;" ";c;" ";d;" ";t;" ";q
End Function 'Extended Euclidian GCD
function IsEven( x )
if ( x MOD 2 )=0 then
IsEven=1
else
IsEven=0
end if
end function
function IsOdd( x )
if ( x MOD 2 )=0 then
IsOdd=0
else
IsOdd=1
end if
end function
Function FastExp(x, y, N)
if (y=1) then 'MOD(x,N)
FastExp=x-int(x/N)*N
goto [ExitFunction]
end if
if ( y and 1) = 0 then
dum1=y/2
dum2=y-int(y/2)*2 'MOD(y,2)
temp=FastExp(x,dum1,N)
z=temp*temp
FastExp=z-int(z/N)*N 'MOD(temp*temp,N)
goto [ExitFunction]
else
dum1=y-1
dum1=dum1/2
temp=FastExp(x,dum1,N)
dum2=temp*temp
temp=dum2-int(dum2/N)*N 'MOD(dum2,N)
z=temp*x
FastExp=z-int(z/N)*N 'MOD(temp*x,N)
goto [ExitFunction]
end if
[ExitFunction]
end function
Function MillerRabin(n,b)
'print "Miller Rabin" 't1=time$("ms")
if IsEven(n) then MillerRabin=1 goto [ExtFn] end if
i=0
[Loop]
i=i+1
if i>1000 then goto [Continue]
if ( n MOD SmallPrimes(i) )=0 then
MillerRabin=0
goto [ExtFn]
end if
goto [Loop]
[Continue]
if GCD(n,b)>1 then MillerRabin=1 goto [ExtFn] end if
q=n-1
t=0
while (int(q) AND 1 )=0 t=t+1 q=int(q/2) wend
r=FastExp(b, q, n)
if ( r <> 1 ) then
e=0
while ( e < (t-1) )
if ( r <> (n-1) ) then
r=FastExp(r, r, n)
else
Exit While
end if
e=e+1 wend [ExitLoop] end if
if ( (r=1) OR (r=(n-1)) ) then
MillerRabin=0
else
MillerRabin=1
end if
[ExtFn]
End Function </lang>
Mathematica
<lang Mathematica>MillerRabin[n_,k_]:=Module[{d=n-1,s=0,test=True},While[Mod[d,2]==0 ,d/=2 ;s++] Do[
a=RandomInteger[{2,n-1}]; x=PowerMod[a,d,n];
If[x!=1,
For[ r = 0, r < s, r++, If[x==n-1, Continue[]]; x = Mod[x*x, n]; ];
If[ x != n-1, test=False ];
];
,{k}]; Print[test] ]</lang>
- Example output (not using the PrimeQ builtin):
<lang mathematica>MillerRabin[17388,10] ->False</lang>
Maxima
<lang maxima>/* Miller-Rabin algorithm is builtin, see function primep. Here is another implementation */
/* find highest power of p, p^s, that divide n, and return s and n / p^s */
facpow(n, p) := block(
[s: 0], while mod(n, p) = 0 do (s: s + 1, n: quotient(n, p)), [s, n]
)$
/* check whether n is a strong pseudoprime to base a; s and d are given by facpow(n - 1, 2) */
sppp(n, a, s, d) := block(
[x: power_mod(a, d, n), q: false],
if x = 1 or x = n - 1 then true else (
from 2 thru s do (
x: mod(x * x, n),
if x = 1 then return(q: false) elseif x = n - 1 then return(q: true)
),
q
)
)$
/* Miller-Rabin primality test. For n < 341550071728321, the test is deterministic;
for larger n, the number of bases tested is given by the option variable primep_number_of_tests, which is used by Maxima in primep. The bound for deterministic test is also the same as in primep. */
miller_rabin(n) := block(
[v: [2, 3, 5, 7, 11, 13, 17], s, d, q: true, a],
if n < 19 then member(n, v) else (
[s, d]: facpow(n - 1, 2),
if n < 341550071728321 then ( /* see http://oeis.org/A014233 */
for a in v do (
if not sppp(n, a, s, d) then return(q: false)
),
q
) else (
thru primep_number_of_tests do (
a: 2 + random(n - 3),
if not sppp(n, a, s, d) then return(q: false)
),
q
)
)
)$</lang>
Mercury
This naive implementation of a Miller-Rabin method is adapted from the Prolog version on this page. The use of the form integer(N) to permit use of integers of arbitrary precision as done here is not efficient. It is better to define a tabled version of each known integer and to use the tabled versions. For example, suppose you want integer(2), then do
:- func n2 = integer.integer. :- pragma memo(n2/0). n2 = integer.integer(2).
and use n2 as the integer in your code. Performance will be greatly improved. Also Mercury has a package using Tom's Math for integers of arbitrary precision and another package to some of the functions of the GMP library for much faster operation with long integers. These can be found with instructions for use in Github.
<lang Mercury> %----------------------------------------------------------------------%
- - module primality.
- - interface.
- - import_module integer.
- - pred is_prime(integer::in, integer::out) is multi.
%----------------------------------------------------------------------%
- - implementation.
- - import_module bool, int, list, math, require, string.
%----------------------------------------------------------------------%
% is_prime/2 implements a Miller-Rabin primality test, is
% deterministic for N < 3.415e+14, and is probabilistic for
% larger N. Returns integer(0) if not prime, integer(1) if prime,
% and -integer(1) if fails.
% :- pred is_prime(integer::in, integer::out) is multi.
is_prime(N, P) :- N < integer(2), P = integer(0). is_prime(N, P) :- N = integer(2), P = integer(1). is_prime(N, P) :- N = integer(3), P = integer(1). is_prime(N, P) :- %% even numbers > 3: false N > integer(3), (N mod integer(2)) = integer(0), P = integer(0). %%-------------deterministic-------- is_prime(N, P) :- %% 3 < odd number < 3.415e+14 N > integer(3), (N mod integer(2)) = integer(1), N < integer(341550071728321), deterministic_witnesses(N, DList), is_mr_prime(N, DList, R), P = R. %%-------------probabilistic-------- is_prime(N, P) :- %% 3.415e+14 =< odd number N > integer(3), (N mod integer(2)) = integer(1), N >= integer(341550071728321), random_witnesses(N, 20, RList), is_mr_prime(N, RList, R), P = R. is_prime(_N, P) :- P = -integer(1).
%----------------------------------------------------------------------% % returns list of deterministic witnesses
- - pred deterministic_witnesses(integer::in,
list(integer)::out) is multi.
deterministic_witnesses(N, L) :- N < integer(1373653), L = [integer(2), integer(3)]. deterministic_witnesses(N, L) :- N < integer(9080191), L = [integer(31), integer(73)]. deterministic_witnesses(N, L) :- N < integer(25326001), L = [integer(2), integer(3), integer(5)]. deterministic_witnesses(N, L) :- N < integer(3215031751), L = [integer(2), integer(3), integer(5), integer(7)]. deterministic_witnesses(N, L) :- N < integer(4759123141), L = [integer(2), integer(7), integer(61)]. deterministic_witnesses(N, L) :- N < integer(1122004669633), L = [integer(2), integer(13), integer(23), integer(1662803)]. deterministic_witnesses(N, L) :- N < integer(2152302898747), L = [integer(2), integer(3), integer(5), integer(7), integer(11)]. deterministic_witnesses(N, L) :- N < integer(3474749660383), L = [integer(2), integer(3), integer(5), integer(7), integer(11),
integer(13)].
deterministic_witnesses(N, L) :- N < integer(341550071728321), L = [integer(2), integer(3), integer(5), integer(7), integer(11), integer(13), integer(17)]. deterministic_witnesses(_N, L) :- L = []. %% signals failure
%----------------------------------------------------------------------% %% random_witnesses/3 receives an integer, X, an int, K, and %% returns a list, P, of K pseudo-random integers in a range %% 1 to X-1.
- - pred random_witnesses(integer::in, int::in,
list(integer)::out) is det.
random_witnesses(X, K, P) :- A = integer(6364136223846793005), B = integer(1442695040888963407), C = X - integer(2), rw_loop(X, A, B, C, K, [], P).
- - pred rw_loop(integer::in, integer::in, integer::in, integer::in,
int::in, list(integer)::in, list(integer)::out) is det.
rw_loop(X, A, B, C, K, L, P) :- X1 = (((X * A) + B) mod C) + integer(1), ( if K = 0 then P = L else rw_loop(X1, A, B, C, K-1, [X1|L], P) ).
%----------------------------------------------------------------------% % is_mr_prime/2 receives integer N and list As and returns true if % N is probably prime, and false otherwise
- - pred is_mr_prime(integer::in, list(integer)::in, integer::out) is nondet.
is_mr_prime(N, As, R) :-
find_ds(N, L), L = [D|T], T = [S|_], outer_loop(N, As, D, S, R).
- - pred outer_loop(integer::in, list(integer)::in, integer::in,
integer::in, integer::out) is nondet.
outer_loop(N, As, D, S, R) :-
As = [A|At], Base = powm(A, D, N), %% = A^D mod N inner_loop(Base, N, integer(0), S, U),
( if U = integer(0) then R = integer(0) else ( if At = [] then R = integer(1)
else outer_loop(N, At, D, S, R)
)
).
- - pred inner_loop(integer::in, integer::in, integer::in,
integer::in, integer::out) is multi.
inner_loop(Base, N, Loop, S, U) :-
Next_Base = (Base * Base) mod N, Next_Loop = Loop + integer(1),
( if Loop = integer(0) then ( if Base = integer(1) then U = integer(1) % true
else if Base = N - integer(1) then U = integer(1) % true
else if Next_Loop = S then U = integer(0) % false
else inner_loop(Next_Base, N, Next_Loop, S, U)
)
else if Base = N - integer(1) then U = integer(1) % true else if Next_Loop = S then U = integer(0) % false else inner_loop(Next_Base, N, Next_Loop, S, U) ).
%----------------------------------------------------------------------% % find_ds/2 receives odd integer N % and returns [D, S] such that N-1 = 2^S * D
- - pred find_ds(integer::in, list(integer)::out) is multi.
find_ds(N, L) :-
A = N - integer(1),
find_ds1(A, integer(0), L).
- - pred find_ds1(integer::in, integer::in, list(integer)::out) is multi.
find_ds1(D, S, L) :- D mod integer(2) = integer(0), P = D div integer(2), Q = S + integer(1), find_ds1(P, Q, L). find_ds1(D, S, L) :- L = [D, S].
%----------------------------------------------------------------------%
- - func powm(integer, integer, integer) = integer.
% computes A^D mod N
powm(A, D, N) =
( if D = integer(0) then integer(1)
else ( if (D mod integer(2)) = integer(0) then
(integer.pow(powm(A, (D div integer(2)), N),
integer(2))) mod N
else (A * powm(A, D - integer(1), N)) mod N
)
).
%----------------------------------------------------------------------%
- - end_module primality.
% A means of testing the predicate is_prime/2 %----------------------------------------------------------------------%
- - module test_is_prime.
- - interface.
- - import_module io.
- - pred main(io::di, io::uo) is cc_multi.
%----------------------------------------------------------------------%
- - implementation.
- - import_module bool, char, int, integer, list, math, require, string.
- - import_module primality.
%----------------------------------------------------------------------%
% TEST THE IS_PRIME PREDICATE
% $ ./test_is_prime <integer>
%---------------------------------------------%
main(!IO) :-
command_line_arguments(Args, !IO), filter(is_all_digits, Args, CleanArgs), Arg1 = list.det_index0(CleanArgs, 0), M = integer.det_from_string(Arg1),
is_prime(M,P), io.format(" is_prime(%s) = ", [s(integer.to_string(M))], !IO), ( if P = integer(0) then io.write_string("false.\n", !IO) else if P = integer(1) then io.write_string("true.\n", !IO) else if P = -integer(1) then
io.write_string("N fails all tests.\n", !IO)
else io.write_string("Has reported neither true nor false
nor any error condition.\n", !IO)
).
%----------------------------------------------------------------------%
- - end_module test_is_prime.
</lang>
Nim
<lang nim>
- Nim currently doesn't have a BigInt standard library
- so we translate the version from Go which uses a
- deterministic approach, which is correct for all
- possible values in uint32.
proc isPrime*(n: uint32): bool =
# bases of 2, 7, 61 are sufficient to cover 2^32 case n of 0, 1: return false of 2, 7, 61: return true else: discard
var nm1 = n-1 d = nm1.int s = 0 n = n.uint64
while d mod 2 == 0: d = d shr 1 s += 1
for a in [2, 7, 61]:
var
x = 1.uint64
p = a.uint64
dr = d
while dr > 0:
if dr mod 2 == 1:
x = x * p mod n
p = p * p mod n
dr = dr shr 1
if x == 1 or x.uint32 == nm1:
continue
var r = 1
while true:
if r >= s:
return false
x = x * x mod n
if x == 1:
return false
if x.uint32 == nm1:
break
r += 1
return true
proc isPrime*(n: int32): bool =
## Overload for int32 n >= 0 and n.uint32.isPrime
when isMainModule:
const primeNumber1000 = 7919 # source: https://en.wikipedia.org/wiki/List_of_prime_numbers var i = 0u32 numberPrimes = 0 while true: if isPrime(i): if numberPrimes == 999: break numberPrimes += 1 i += 1 assert i == primeNumber1000 assert isPrime(2u32) assert isPrime(31u32) assert isPrime(37u32) assert isPrime(1123u32) assert isPrime(492366587u32) assert isPrime(1645333507u32)
</lang>
Oz
This naive implementation of a Miller-Rabin method is adapted from the Mercury and Prolog versions on this page.
<lang Oz> %--------------------------------------------------------------------------% % module: Primality % file: Primality.oz % version: 17 DEC 2014 @ 6:50AM %--------------------------------------------------------------------------%
declare %--------------------------------------------------------------------------%
fun {IsPrime N} % main interface of module if N < 2 then false elseif N < 4 then true elseif (N mod 2) == 0 then false elseif N < 341330071728321 then {IsMRprime N {DetWit N}} else {IsMRprime N {ProbWit N 20}} end end %--------------------------------------------------------------------------%
fun {DetWit N} % deterministic witnesses if N < 1373653 then [2 3] elseif N < 9080191 then [31 73] elseif N < 25326001 then [2 3 5] elseif N < 3215031751 then [2 3 5 7] elseif N < 4759123141 then [2 7 61] elseif N < 1122004669633 then [2 13 23 1662803] elseif N < 2152302898747 then [2 3 5 7 11] elseif N < 3474749660383 then [2 3 5 7 11 13] elseif N < 341550071728321 then [2 3 5 7 11 13 17] else nil end end %--------------------------------------------------------------------------%
fun {ProbWit N K} % probabilistic witnesses local A B C in A = 6364136223846793005 B = 1442695040888963407 C = N - 2 {RWloop N A B C K nil} end end
fun {RWloop N A B C K L} local N1 in N1 = (((N * A) + B) mod C) + 1 if K == 0 then L else {RWloop N1 A B C (K - 1) N1|L} end end end %--------------------------------------------------------------------------%
fun {IsMRprime N As} % the Miller-Rabin algorithm local D S T Ts in {FindDS N} = D|S {OuterLoop N As D S} end end
fun {OuterLoop N As D S} local A At Base C in As = A|At Base = {Powm A D N} C = {InnerLoop Base N 0 S} if {Not C} then false elseif {And C (At == nil)} then true else {OuterLoop N At D S} end end end
fun {InnerLoop Base N Loop S} local NextBase NextLoop in NextBase = (Base * Base) mod N NextLoop = Loop + 1 if {And (Loop == 0) (Base == 1)} then true elseif Base == (N - 1) then true elseif NextLoop == S then false else {InnerLoop NextBase N NextLoop S} end end end %--------------------------------------------------------------------------%
fun {FindDS N} {FindDS1 (N - 1) 0} end
fun {FindDS1 D S} if (D mod 2 == 0) then {FindDS1 (D div 2) (S + 1)} else D|S end end %--------------------------------------------------------------------------%
fun {Powm A D N} % returns (A ^ D) mod N if D == 0 then 1 elseif (D mod 2) == 0 then {Pow {Powm A (D div 2) N} 2} mod N else (A * {Powm A (D - 1) N}) mod N end end %--------------------------------------------------------------------------% % end_module Primality
</lang>
PARI/GP
Built-in
<lang parigp>MR(n,k)=ispseudoprime(n,k);</lang>
Custom
<lang parigp>sprp(n,b)={ my(s = valuation(n-1, 2), d = Mod(b, n)^(n >> s)); if (d == 1, return(1)); for(i=1,s-1, if (d == -1, return(1)); d = d^2; ); d == -1 };
MR(n,k)={
for(i=1,k, if(!sprp(n,random(n-2)+2), return(0)) ); 1
};</lang>
Deterministic version
A basic deterministic test can be obtained by an appeal to the ERH (as proposed by Gary Miller) and a result of Eric Bach (improving on Joseph Oesterlé). Calculations of Jan Feitsma can be used to speed calculations below 264 (by a factor of about 250). <lang parigp>A006945=[9, 2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383, 341550071728321, 341550071728321, 3825123056546413051]; Miller(n)={
if (n%2 == 0, return(n == 2)); \\ Handle even numbers if (n < 3, return(0)); \\ Handle 0, 1, and negative numbers
if (n < 1<<64,
\\ Feitsma
for(i=1,#A006945,
if (n < A006945[i], return(1));
if(!sprp(n, prime(i)), return(0));
);
sprp(n,31)&sprp(n,37)
,
\\ Miller + Bach
for(b=2,2*log(n)^2,
if(!sprp(n, b), return(0))
);
1
)
};</lang>
Perl
Custom
<lang perl>use bigint try => 'GMP';
sub is_prime {
my ($n, $k) = @_; return 1 if $n == 2; return 0 if $n < 2 or $n % 2 == 0;
my $d = $n - 1; my $s = 0;
while (!($d % 2)) {
$d /= 2;
$s++;
}
LOOP: for (1 .. $k) {
my $a = 2 + int(rand($n - 2));
my $x = $a->bmodpow($d, $n);
next if $x == 1 or $x == $n - 1;
for (1 .. $s - 1) {
$x = ($x * $x) % $n;
return 0 if $x == 1;
next LOOP if $x == $n - 1;
}
return 0;
}
return 1;
}
print join ", ", grep { is_prime $_, 10 } (1 .. 1000);</lang>
Modules
While normally one would use is_prob_prime, is_prime, or is_provable_prime, which will do a BPSW test and possibly more, we can use just the Miller-Rabin test if desired. For large values we can use an object (e.g. bigint, Math::GMP, Math::Pari, etc.) or just a numeric string. <lang perl>use ntheory qw/is_strong_pseudoprime miller_rabin_random/; sub is_prime_mr {
my $n = shift; # If 32-bit, we can do this with 3 bases. return is_strong_pseudoprime($n, 2, 7, 61) if ($n >> 32) == 0; # If 64-bit, 7 is all we need. return is_strong_pseudoprime($n, 2, 325, 9375, 28178, 450775, 9780504, 1795265022) if ($n >> 64) == 0; # Otherwise, perform a number of random base tests, and the result is a probable prime test. return miller_rabin_random($n, 20);
}</lang> Math::Primality also has this functionality, though its function takes only one base and requires the input number to be less than the base. <lang perl>use Math::Primality qw/is_strong_pseudoprime/; sub is_prime_mr {
my $n = shift;
return 0 if $n < 2;
for (2,3,5,7,11,13,17,19,23,29,31,37) {
return 0 unless $n <= $_ || is_strong_pseudoprime($n,$_);
}
1;
} for (1..100) { say if is_prime_mr($_) }</lang> Math::Pari can be used in a fashion similar to the Pari/GP custom function. The builtin accessed using a second argument to ispseudoprime was added to a later version of Pari (the Perl module uses version 2.1.7) so is not accessible directly from Perl.
Perl 6
<lang perl6># the expmod-function from: http://rosettacode.org/wiki/Modular_exponentiation sub expmod(Int $a is copy, Int $b is copy, $n) { my $c = 1; repeat while $b div= 2 { ($c *= $a) %= $n if $b % 2; ($a *= $a) %= $n; } $c; }
subset PrimeCandidate of Int where { $_ > 2 and $_ % 2 };
my Bool multi sub is_prime(Int $n, Int $k) { return False; } my Bool multi sub is_prime(2, Int $k) { return True; } my Bool multi sub is_prime(PrimeCandidate $n, Int $k) { my Int $d = $n - 1; my Int $s = 0;
while $d %% 2 { $d div= 2; $s++; }
for (2 ..^ $n).pick($k) -> $a { my $x = expmod($a, $d, $n);
# one could just write "next if $x == 1 | $n - 1" # but this takes much more time in current rakudo/nom next if $x == 1 or $x == $n - 1;
for 1 ..^ $s { $x = $x ** 2 mod $n; return False if $x == 1; last if $x == $n - 1; } return False if $x !== $n - 1; }
return True; }
say (1..1000).grep({ is_prime($_, 10) }).join(", "); </lang>
Phix
native, determinstic to 94,910,107
Native-types deterministic version, fails (false negative) at 94,910,107 on 32-bit [fully tested, ie from 1], and at 4,295,041,217 on 64-bit [only tested from 4,290,000,000] - those limits have now been hard-coded below. <lang Phix>function powermod(atom a, atom n, atom m) -- calculate a^n%mod
atom p = a, res = 1
while n do
if and_bits(n,1) then
res = mod(res*p,m)
end if
p = mod(p*p,m)
n = floor(n/2)
end while
return res
end function
function witness(atom n, atom s, atom d, sequence a) -- n-1 = 2^s * d with d odd by factoring powers of 2 from n-1
for i=1 to length(a) do
atom x = powermod(a[i], d, n), y, w=s
while w do
y = mod(x*x,n)
if y == 1 and x != 1 and x != n-1 then
return false
end if
x = y
w -= 1
end while
if y != 1 then
return false
end if
end for
return true;
end function
function is_prime_mr(atom n)
if (mod(n,2)==0 and n!=2)
or (n<2)
or (mod(n,3)==0 and n!=3) then
return false
elsif n<=3 then
return true
end if
atom d = floor(n/2)
atom s = 1;
while and_bits(d,1)=0 do
d /= 2
s += 1
end while
sequence a
if n < 1373653 then
a = {2, 3}
elsif n < 9080191 then
a = {31, 73}
elsif (machine_bits()=32 and n < 94910107)
or (machine_bits()=64 and n < 4295041217) then
a = {2, 7, 61}
else
puts(1,"limits exceeded\n")
return 0
end if
return witness(n, s, d, a)
end function
sequence tests = {999983,999809,999727,52633,60787,999999,999995,999991} for i=1 to length(tests) do
printf(1,"%d is %s\n",{tests[i],{"composite","prime"}[is_prime_mr(tests[i])+1]})
end for</lang>
- Output:
999983 is prime 999809 is prime 999727 is prime 52633 is composite 60787 is composite 999999 is composite 999995 is composite 999991 is composite
second version, deterministic to 341,550,071,728,321
The same using bigatoms, obviously significantly slower, deterministic up to 341,550,071,728,321
As noted on the Python submission and discussion page, some doubts may yet remain with this method.
<lang Phix>include bigatom.e
constant BA_ZERO = ba_new(0),
BA_ONE = ba_new(1),
BA_TWO = ba_new(2),
BA_THREE = ba_new(3)
function powermod(atom a, bigatom n, bigatom m) -- calculate a^n%mod
bigatom p = ba_new(a), res = ba_new(1)
while ba_compare(n,BA_ZERO) do
if ba_compare(ba_remainder(n,2),BA_ONE)=0 then
res = ba_remainder(ba_multiply(res,p),m)
end if
p = ba_remainder(ba_multiply(p,p),m)
n = ba_floor(ba_divide(n,2))
end while
return res
end function
function witness(bigatom n, bigatom s, bigatom d, sequence a) -- n-1 = 2^s * d with d odd by factoring powers of 2 from n-1
for i=1 to length(a) do
bigatom x = powermod(a[i], d, n), y, w=s
while ba_compare(w,BA_ZERO) do
y = ba_remainder(ba_multiply(x,x),n)
if ba_compare(y,1)=0
and ba_compare(x,1)!=0
and ba_compare(x,ba_sub(n,1))!=0 then
return false
end if
x = y
w = ba_sub(w,1)
end while
if ba_compare(y,BA_ONE)!=0 then
return false
end if
end for
return true
end function
function is_prime_mr(bigatom n)
if (ba_compare(ba_remainder(n,2),BA_ZERO)=0 and ba_compare(n,BA_TWO)!=0)
or (ba_compare(n,BA_TWO)<0)
or (ba_compare(ba_remainder(n,3),BA_ZERO)=0 and ba_compare(n,BA_THREE)!=0) then
return false
--bugfix 15/9/18: -- elsif ba_compare(n,BA_THREE)<0 then
elsif ba_compare(n,BA_THREE)<=0 then
return true
end if
bigatom d = ba_floor(ba_divide(n,2)), s = ba_new(1)
while ba_remainder(d,2)=0 do
d = ba_divide(d,2)
s = ba_add(s,1)
end while
sequence a
if ba_compare(n,ba_new(1373653))<0 then a = {2, 3}
elsif ba_compare(n,ba_new(9080191))<0 then a = {31, 73}
elsif ba_compare(n,ba_new(4759123141))<0 then a = {2, 7, 61}
elsif ba_compare(n,ba_new(1122004669633))<0 then a = {2, 13, 23, 1662803}
elsif ba_compare(n,ba_new(2152302898747))<0 then a = {2, 3, 5, 7, 11}
elsif ba_compare(n,ba_new(3474749660383))<0 then a = {2, 3, 5, 7, 11, 13}
else a = {2, 3, 5, 7, 11, 13, 17}
end if
return witness(n, s, d, a)
end function
atom t0 = time() ?is_prime_mr(ba_new("4547337172376300111955330758342147474062293202868155909489")) ?is_prime_mr(ba_new("4547337172376300111955330758342147474062293202868155909393")) printf(1,"%s\n",{elapsed(time()-t0)})</lang>
- Output:
1 0 34.81s
Unfortunately far too slow for any serious use <lang Phix>atom t0 = time() bigatom b = ba_new("9223372036854774808") integer c = 0 for i=4808 to 5808 do
if is_prime_mr(b) then
printf(1," %s",{ba_sprint(b)})
c += 1 if c=5 then printf(1,"\n") c=0 end if
end if
b = ba_add(b,1)
end for printf(1,"\n%s\n",{elapsed(time()-t0)})</lang>
- Output:
9223372036854774893 9223372036854774917 9223372036854774937 9223372036854774959 9223372036854775057 9223372036854775073 9223372036854775097 9223372036854775139 9223372036854775159 9223372036854775181 9223372036854775259 9223372036854775279 9223372036854775291 9223372036854775337 9223372036854775351 9223372036854775399 9223372036854775417 9223372036854775421 9223372036854775433 9223372036854775507 9223372036854775549 9223372036854775643 9223372036854775783 2 minutes and 29s
And anything much further than this gets measured in hours <lang Phix>atom t0 = time() for i=2 to 128 do
bigatom b = ba_sub(ba_power(2,tests[i]),1)
if is_prime_mr(b) then
printf(1,"2^%d-1 = prime\n",{tests[i]})
end if
end for printf(1,"\n%s\n",{elapsed(time()-t0)})</lang>
- Output:
2^3-1 = prime 2^5-1 = prime 2^7-1 = prime 2^13-1 = prime 2^17-1 = prime 2^19-1 = prime 2^31-1 = prime 2^61-1 = prime 2^89-1 = prime 2^107-1 = prime 2^127-1 = prime 42.61s
third version, this time actually using the pseudocode above
After (finally) using ba_mod_exp() based on FastExp() in the Liberty BASIC submission (and boy did this need it!), I thought I'd cracked it, but this is still marginally slower than the second version. <lang Phix>-- demo/rosetta/Miller_Rabin_primality_test.exw include bigatom.e
function ba_rand(object low, high) -- generate a random number between low and high (inclusive) -- low and high can be passed in as integer/string/bigatom -- (both low and high get given the ba_round(int) treatment)
low = ba_sub(ba_round(low),1)
high = ba_round(high) -- just in case...
bigatom hz = ba_sub(high,low) -- convert range to 0..hz
string hs = ba_sprint(hz) -- get length
integer l = length(hs)
string rs = repeat('9',l)
while 1 do
-- generate "000..." .. "999..." in blocks of up to 9
for p=1 to length(rs) by 9 do
integer cl = min(l-p+1,9)
string fmt = sprintf("%%0%dd",cl) -- "%01d".."%09d"
string chunk = sprintf(fmt,rand(power(10,cl))-1)
rs[p..p+cl-1] = chunk
end for
if length(rs)!=length(hs) then ?9/0 end if -- sanity
if rs<=hs then exit end if
end while
return ba_add(ba_new(rs),low)
end function
function ba_mod_exp(object base, exponent, modulus) -- base/exponent/modulus can be integer/string/bigatom
bigatom res
if ba_compare(exponent,1)=0 then
res = base
else
bool odd = (ba_compare(ba_mod(exponent,2),0)!=0)
if odd then
exponent = ba_sub(exponent,1)
end if
exponent = ba_divide(exponent,2)
res = ba_mod_exp(base,exponent,modulus)
res = ba_multiply(res,res)
if odd then
res = ba_multiply(res,base)
end if
end if
res = ba_mod(res,modulus)
return res
end function
constant COMPOSITE = 0,
PROBABLY_PRIME = 1
function Miller_Rabin(object n, integer k=10) -- n can be integer/string/bigatom
bigatom nm1 = ba_sub(n,1), d=nm1, x
integer s = 0
if ba_compare(n,2)=0 then
return PROBABLY_PRIME
elsif ba_mod(n,2)=0 or ba_compare(n,2)<0 then
return COMPOSITE
end if
while ba_mod(d,2)=0 do
d = ba_divide(d,2)
s += 1
end while
integer res = PROBABLY_PRIME
for i=1 to k do
printf(1,"working (%d/%d)...\r",{i,k})
x = ba_mod_exp(ba_rand(2,nm1),d,n)
if ba_compare(x,1)!=0
and ba_compare(x,nm1)!=0 then
for r=1 to s-1 do
-- x = ba_mod(ba_multiply(x,x),n) -- (far too slow!)
x = ba_mod_exp(x,2,n)
if ba_compare(x,1)=0 then res = COMPOSITE exit end if
if ba_compare(x,nm1)=0 then exit end if
end for
if ba_compare(x,nm1)!=0 then res = COMPOSITE end if
end if
if res=COMPOSITE then exit end if
end for
printf(1," \r")
return res
end function
?Miller_Rabin(17) --1 ?Miller_Rabin(21) --0 ?Miller_Rabin("4547337172376300111955330758342147474062293202868155909489") --1 ?Miller_Rabin("4547337172376300111955330758342147474062293202868155909393") --0</lang>
PHP
<lang php><?php function is_prime($n, $k) {
if ($n == 2)
return true;
if ($n < 2 || $n % 2 == 0)
return false;
$d = $n - 1; $s = 0;
while ($d % 2 == 0) {
$d /= 2;
$s++;
}
for ($i = 0; $i < $k; $i++) {
$a = rand(2, $n-1);
$x = bcpowmod($a, $d, $n);
if ($x == 1 || $x == $n-1)
continue;
for ($j = 1; $j < $s; $j++) {
$x = bcmod(bcmul($x, $x), $n);
if ($x == 1)
return false;
if ($x == $n-1)
continue 2;
}
return false;
}
return true;
}
for ($i = 1; $i <= 1000; $i++)
if (is_prime($i, 10))
echo "$i, ";
echo "\n"; ?></lang>
PicoLisp
<lang PicoLisp>(de longRand (N)
(use (R D)
(while (=0 (setq R (abs (rand)))))
(until (> R N)
(unless (=0 (setq D (abs (rand))))
(setq R (* R D)) ) )
(% R N) ) )
(de **Mod (X Y N)
(let M 1
(loop
(when (bit? 1 Y)
(setq M (% (* M X) N)) )
(T (=0 (setq Y (>> 1 Y)))
M )
(setq X (% (* X X) N)) ) ) )
(de _prim? (N D S)
(use (A X R)
(while (> 2 (setq A (longRand N))))
(setq R 0 X (**Mod A D N))
(loop
(T
(or
(and (=0 R) (= 1 X))
(= X (dec N)) )
T )
(T
(or
(and (> R 0) (= 1 X))
(>= (inc 'R) S) )
NIL )
(setq X (% (* X X) N)) ) ) )
(de prime? (N K)
(default K 50)
(and
(> N 1)
(bit? 1 N)
(let (D (dec N) S 0)
(until (bit? 1 D)
(setq
D (>> 1 D)
S (inc S) ) )
(do K
(NIL (_prim? N D S))
T ) ) ) )</lang>
- Output:
: (filter '((I) (prime? I)) (range 937 1000)) -> (937 941 947 953 967 971 977 983 991 997) : (prime? 4547337172376300111955330758342147474062293202868155909489) -> T : (prime? 4547337172376300111955330758342147474062293202868155909393) -> NIL
Prolog
This naive implementation of a Miller-Rabin method is adapted from the Erlang version on this page.
<lang prolog>:- module(primality, [is_prime/2]).
% is_prime/2 returns false if N is composite, true if N probably prime % implements a Miller-Rabin primality test and is deterministic for N < 3.415e+14, % and is probabilistic for larger N. Adapted from the Erlang version. is_prime(1, Ret) :- Ret = false, !. % 1 is non-prime is_prime(2, Ret) :- Ret = true, !. % 2 is prime is_prime(3, Ret) :- Ret = true, !. % 3 is prime is_prime(N, Ret) :- N > 3, (N mod 2 =:= 0), Ret = false, !. % even number > 3 is composite is_prime(N, Ret) :- N > 3, (N mod 2 =:= 1), % odd number > 3 N < 341550071728321, deterministic_witnesses(N, L),
is_mr_prime(N, L, Ret), !. % deterministic test
is_prime(N, Ret) :- random_witnesses(N, 100, [], Out), is_mr_prime(N, Out, Ret), !. % probabilistic test
% returns list of deterministic witnesses deterministic_witnesses(N, L) :- N < 1373653, L = [2, 3]. deterministic_witnesses(N, L) :- N < 9080191, L = [31, 73]. deterministic_witnesses(N, L) :- N < 25326001, L = [2, 3, 5]. deterministic_witnesses(N, L) :- N < 3215031751, L = [2, 3, 5, 7]. deterministic_witnesses(N, L) :- N < 4759123141, L = [2, 7, 61]. deterministic_witnesses(N, L) :- N < 1122004669633, L = [2, 13, 23, 1662803]. deterministic_witnesses(N, L) :- N < 2152302898747, L = [2, 3, 5, 7, 11]. deterministic_witnesses(N, L) :- N < 3474749660383, L = [2, 3, 5, 7, 11, 13]. deterministic_witnesses(N, L) :- N < 341550071728321, L = [2, 3, 5, 7, 11, 13, 17].
% random_witnesses/4 returns a list of K witnesses selected at random with range 2 -> N-2 random_witnesses(_, 0, T, T). random_witnesses(N, K, T, Out) :- G is N - 2, H is 1 + random(G), I is K - 1,
random_witnesses(N, I, [H | T], Out), !.
% find_ds/2 receives odd integer N and returns [D, S] s.t. N-1 = 2^S * D find_ds(N, L) :- A is N - 1,
find_ds(A, 0, L), !.
find_ds(D, S, L) :- D mod 2 =:= 0, P is D // 2, Q is S + 1, find_ds(P, Q, L), !. find_ds(D, S, L) :- L = [D, S].
is_mr_prime(N, As, Ret) :-
find_ds(N, L), L = [D | T], T = [S | _], outer_loop(N, As, D, S, Ret), !.
outer_loop(N, As, D, S, Ret) :-
As = [A | At], Base is powm(A, D, N), inner_loop(Base, N, 0, S, Result),
( Result == false -> Ret = false ; Result == true, At == [] -> Ret = true ; outer_loop(N, At, D, S, Ret) ).
inner_loop(Base, N, Loop, S, Result) :-
Next_Base is (Base * Base) mod N, Next_Loop is Loop + 1,
( Loop =:= 0, Base =:= 1 -> Result = true ; Base =:= N-1 -> Result = true ; Next_Loop =:= S -> Result = false ; inner_loop(Next_Base, N, Next_Loop, S, Result) ).</lang>
PureBasic
<lang PureBasic>Enumeration
#Composite #Probably_prime
EndEnumeration
Procedure Miller_Rabin(n, k)
Protected d=n-1, s, x, r
If n=2
ProcedureReturn #Probably_prime
ElseIf n%2=0 Or n<2
ProcedureReturn #Composite
EndIf
While d%2=0
d/2
s+1
Wend
While k>0
k-1
x=Int(Pow(2+Random(n-4),d))%n
If x=1 Or x=n-1: Continue: EndIf
For r=1 To s-1
x=(x*x)%n
If x=1: ProcedureReturn #Composite: EndIf
If x=n-1: Break: EndIf
Next
If x<>n-1: ProcedureReturn #Composite: EndIf
Wend
ProcedureReturn #Probably_prime
EndProcedure</lang>
Python
Python: Probably correct answers
This versions will give answers with a very small probability of being false. That probability being dependent number of trials (automatically set to 8).
<lang python>
import random
def is_Prime(n):
""" Miller-Rabin primality test.
A return value of False means n is certainly not prime. A return value of
True means n is very likely a prime.
"""
if n!=int(n):
return False
n=int(n)
#Miller-Rabin test for prime
if n==0 or n==1 or n==4 or n==6 or n==8 or n==9:
return False
if n==2 or n==3 or n==5 or n==7:
return True
s = 0
d = n-1
while d%2==0:
d>>=1
s+=1
assert(2**s * d == n-1)
def trial_composite(a):
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2**i * d, n) == n-1:
return False
return True
for i in range(8):#number of trials
a = random.randrange(2, n)
if trial_composite(a):
return False
return True </lang>
Python: Proved correct up to large N
This versions will give correct answers for n less than 341550071728321 and then reverting to the probabilistic form of the first solution. By selecting predetermined values for the a values to use instead of random values, the results can be shown to be deterministically correct below certain thresholds.
For 341550071728321 and beyond, I have followed the pattern in choosing a from the set of prime numbers.
While this uses the best sets known in 1993, there are better sets known, and at most 7 are needed for 64-bit numbers.
<lang python>def _try_composite(a, d, n, s):
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2**i * d, n) == n-1:
return False
return True # n is definitely composite
def is_prime(n, _precision_for_huge_n=16):
if n in _known_primes or n in (0, 1):
return True
if any((n % p) == 0 for p in _known_primes):
return False
d, s = n - 1, 0
while not d % 2:
d, s = d >> 1, s + 1
# Returns exact according to http://primes.utm.edu/prove/prove2_3.html
if n < 1373653:
return not any(_try_composite(a, d, n, s) for a in (2, 3))
if n < 25326001:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5))
if n < 118670087467:
if n == 3215031751:
return False
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7))
if n < 2152302898747:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7, 11))
if n < 3474749660383:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7, 11, 13))
if n < 341550071728321:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5, 7, 11, 13, 17))
# otherwise
return not any(_try_composite(a, d, n, s)
for a in _known_primes[:_precision_for_huge_n])
_known_primes = [2, 3] _known_primes += [x for x in range(5, 1000, 2) if is_prime(x)]</lang>
- Testing
Includes test values from other examples:
>>> is_prime(4547337172376300111955330758342147474062293202868155909489) True >>> is_prime(4547337172376300111955330758342147474062293202868155909393) False >>> [x for x in range(901, 1000) if is_prime(x)] [907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997] >>> is_prime(643808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153) True >>> is_prime(743808006803554439230129854961492699151386107534013432918073439524138264842370630061369715394739134090922937332590384720397133335969549256322620979036686633213903952966175107096769180017646161851573147596390153) False >>>
Racket
<lang Racket>#lang racket (define (miller-rabin-expmod base exp m)
(define (squaremod-with-check x)
(define (check-nontrivial-sqrt1 x square)
(if (and (= square 1)
(not (= x 1))
(not (= x (- m 1))))
0
square))
(check-nontrivial-sqrt1 x (remainder (expt x 2) m)))
(cond ((= exp 0) 1)
((even? exp) (squaremod-with-check
(miller-rabin-expmod base (/ exp 2) m)))
(else
(remainder (* base (miller-rabin-expmod base (- exp 1) m))
m))))
(define (miller-rabin-test n)
(define (try-it a)
(define (check-it x)
(and (not (= x 0)) (= x 1)))
(check-it (miller-rabin-expmod a (- n 1) n)))
(try-it (+ 1 (random (remainder (- n 1) 4294967087)))))
(define (fast-prime? n times)
(for/and ((i (in-range times))) (miller-rabin-test n)))
(define (prime? n(times 100))
(fast-prime? n times))
(prime? 4547337172376300111955330758342147474062293202868155909489) ;-> outputs true </lang>
REXX
With a K of 1, there seems to be a not uncommon number of failures, but
- with a K ≥ 2, the failures are seldom,
- with a K ≥ 3, the failures are rare as hen's teeth.
This would be in the same vein as: 3 is prime, 5 is prime, 7 is prime, all odd numbers are prime.
The K (above) is signified by the REXX variable times in the REXX program below.
To make the program small, the true prime generator was coded to be small, but not particularly fast.
<lang rexx>/*REXX program puts the Miller─Rabin primality test through its paces. */
parse arg limit times . /*obtain optional arguments from the CL*/
if limit== | limit=="," then limit=1000 /*Not specified? Then use the default.*/
if times== | times=="," then times= 10 /* " " " " " " */
numeric digits max(200, 2*limit) /*we're dealing with some ginormous #s.*/
tell= times<0 /*display primes only if times is neg.*/
times=abs(times); w=length(times) /*use absolute value of TIMES; get len.*/
call genP limit /*suspenders now, use a belt later ··· */
@MR= 'Miller─Rabin primality test' /*define a character literal for SAY. */
say "There are" # 'primes ≤' limit /*might as well display some stuff. */
say /* [↓] (skipping unity); show sep line*/
do a=2 to times; say copies('─', 89) /*(skipping unity) do range of TIMEs.*/
p=0 /*counter of primes for this pass. */
do z=1 for limit /*now, let's get busy and crank primes.*/
if \M_Rt(z, a) then iterate /*Not prime? Then try another number.*/
p=p + 1 /*well, we found another one, by gum! */
if tell then say z 'is prime according to' @MR "with K="a
if !.z then iterate
say '[K='a"] " z "isn't prime !" /*oopsy─doopsy and/or whoopsy─daisy !*/
end /*z*/
say ' for 1──►'limit", K="right(a,w)',' @MR "found" p 'primes {out of' #"}."
end /*a*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: parse arg high; @.=0; !.=0; @.1=2; @.2=3; !.2=1; !.3=1; #=2
do j=@.#+2 by 2 to high /*just examine odd integers from here. */
do k=2 while k*k<=j; if j//@.k==0 then iterate j; end /*k*/
#=#+1; @.#=j; !.j=1 /*bump prime counter; add prime to the */
end /*j*/; return /*@. array; define a prime in !. array.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ M_Rt: procedure; parse arg n,k; d=n-1; nL=d /*Miller─Rabin: A.K.A. Rabin─Miller.*/
if n==2 then return 1 /*special case of (the) even prime. */
if n<2 | n//2==0 then return 0 /*check for too low, or an even number.*/
s=-1; do while d//2==0; d=d%2; s=s+1 /*keep halving until a zero remainder.*/
end /*while*/
do k; ?=random(2, nL) /* [↓] perform the DO loop K times.*/
x=?**d // n /*X can get real gihugeic really fast.*/
if x==1 | x==nL then iterate /*First or penultimate? Try another pow*/
do s; x=x**2 // n /*compute new X ≡ X² modulus N. */
if x==1 then return 0 /*if unity, it's definitely not prime.*/
if x==nL then leave /*if N-1, then it could be prime. */
end /*r*/ /* [↑] // is REXX's division remainder*/
if x\==nL then return 0 /*nope, it ain't prime nohows, noway. */
end /*k*/ /*maybe it's prime, maybe it ain't ··· */
return 1 /*coulda/woulda/shoulda be prime; yup.*/</lang>
- output when using the input of: 10000 10
There are 1229 primes ≤ 10000
─────────────────────────────────────────────────────────────────────────────────────────
[K=2] 7201 isn't prime !
for 1──►10000, K= 2, Miller─Rabin primality test found 1230 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 3, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 4, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 5, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 6, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 7, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 8, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K= 9, Miller─Rabin primality test found 1229 primes {out of 1229}.
─────────────────────────────────────────────────────────────────────────────────────────
for 1──►10000, K=10, Miller─Rabin primality test found 1229 primes {out of 1229}.
Ring
<lang ring>
- Project : Miller–Rabin primality test
see "Input a number: " give n see "Input test: " give k
test = millerrabin(n,k) if test = 0
see "Probably Prime" + nl
else
see "Composite" + nl
ok
func millerrabin(n, k)
if n = 2
millerRabin = 0
return millerRabin
ok
if n % 2 = 0 or n < 2
millerRabin = 1
return millerRabin
ok
d = n - 1
s = 0
while d % 2 = 0
d = d / 2
s = s + 1
end
while k > 0
k = k - 1
base = 2 + floor((random(10)/10)*(n-3))
x = pow(base, d) % n
if x != 1 and x != n-1
for r=1 to s-1
x = (x * x) % n
if x = 1
millerRabin = 1
return millerRabin
ok
if x = n-1
exit
ok
next
if x != n-1
millerRabin = 1
return millerRabin
ok
ok
end
</lang> Output:
Input a number: 17 Input test: 8 Probably Prime
Ruby
<lang ruby> def mod_exp(n, e, mod)
fail ArgumentError, 'negative exponent' if e < 0 prod = 1 base = n % mod until e.zero? prod = (prod * base) % mod if e.odd? e >>= 1 base = (base * base) % mod end prod
end
def miller_rabin_prime?(n, g)
d = n - 1
s = 0
while d % 2 == 0
d /= 2
s += 1
end
g.times do
a = 2 + rand(n - 4)
x = mod_exp(a, d, n) # x = (a**d) % n
next if x == 1 || x == n - 1
for r in (1..s - 1)
x = mod_exp(x, 2, n) # x = (x**2) % n
return false if x == 1
break if x == n - 1
end
return false if x != n - 1
end
true # probably
end
p primes = (3..1000).step(2).find_all {|i| miller_rabin_prime?(i,10)} </lang>
- Output:
[3, 5, 7, 11, 13, 17, ..., 971, 977, 983, 991, 997]
The following larger examples all produce true: <lang ruby> puts miller_rabin_prime?(94366396730334173383107353049414959521528815310548187030165936229578960209523421808912459795329035203510284576187160076386643700441216547732914250578934261891510827140267043592007225160798348913639472564715055445201512461359359488795427875530231001298552452230535485049737222714000227878890892901228389026881,1000) puts miller_rabin_prime?(138028649176899647846076023812164793645371887571371559091892986639999096471811910222267538577825033963552683101137782650479906670021895135954212738694784814783986671046107023185842481502719762055887490765764329237651328922972514308635045190654896041748716218441926626988737664133219271115413563418353821396401,1000) puts miller_rabin_prime?(123301261697053560451930527879636974557474268923771832437126939266601921428796348203611050423256894847735769138870460373141723679005090549101566289920247264982095246187318303659027201708559916949810035265951104246512008259674244307851578647894027803356820480862664695522389066327012330793517771435385653616841,1000) puts miller_rabin_prime?(119432521682023078841121052226157857003721669633106050345198988740042219728400958282159638484144822421840470442893056822510584029066504295892189315912923804894933736660559950053226576719285711831138657839435060908151231090715952576998400120335346005544083959311246562842277496260598128781581003807229557518839,1000) puts miller_rabin_prime?(132082885240291678440073580124226578272473600569147812319294626601995619845059779715619475871419551319029519794232989255381829366374647864619189704922722431776563860747714706040922215308646535910589305924065089149684429555813953571007126408164577035854428632242206880193165045777949624510896312005014225526731,1000) puts miller_rabin_prime?(153410708946188157980279532372610756837706984448408515364579602515073276538040155990230789600191915021209039203172105094957316552912585741177975853552299222501069267567888742458519569317286299134843250075228359900070009684517875782331709619287588451883575354340318132216817231993558066067063143257425853927599,1000) puts miller_rabin_prime?(103130593592068072608023213244858971741946977638988649427937324034014356815504971087381663169829571046157738503075005527471064224791270584831779395959349442093395294980019731027051356344056416276026592333932610954020105156667883269888206386119513058400355612571198438511950152690467372712488391425876725831041,1000) </lang>
Run BASIC
This code has not been fully tested. Remove this comment after review.
<lang runbasic>input "Input a number:";n input "Input test:";k
test = millerRabin(n,k) if test = 0 then
print "Probably Prime" else print "Composite"
end if wait
' ---------------------------------------- ' Returns ' Composite = 1 ' Probably Prime = 0 ' ----------------------------------------
FUNCTION millerRabin(n, k) if n = 2 then millerRabin = 0 'probablyPrime goto [funEnd] end if
if n mod 2 = 0 or n < 2 then millerRabin = 1 'composite goto [funEnd] end if
d = n - 1 while d mod 2 = 0
d = d / 2 s = s + 1
wend
while k > 0
k = k - 1
base = 2 + int(rnd(1)*(n-3))
x = (base^d) mod n
if x <> 1 and x <> n-1 then
for r=1 To s-1
x =(x * x) mod n
if x=1 then
millerRabin = 1 ' composite
goto [funEnd]
end if
if x = n-1 then exit for
next r
if x<>n-1 then
millerRabin = 1 ' composite
goto [funEnd]
end if
end if
wend [funEnd] END FUNCTION</lang>
Scala
<lang scala>import scala.math.BigInt
object MillerRabinPrimalityTest extends App {
val (n, certainty )= (BigInt(args(0)), args(1).toInt)
println(s"$n is ${if (n.isProbablePrime(certainty)) "probably prime" else "composite"}")
}</lang>
Direct implementation of algorithm:
<lang scala> import scala.annotation.tailrec import scala.language.{implicitConversions, postfixOps} import scala.util.Random
object MillerRabin {
implicit def int2Bools(b: Int): Seq[Boolean] = 31 to 0 by -1 map isBitSet(b)
def isBitSet(byte: Int)(bit: Int): Boolean = ((byte >> bit) & 1) == 1
def mod(num: Int, denom: Int) = if (num % denom >= 0) num % denom else (num % denom) + denom
@tailrec
def isSimple(p: Int, s: Int): Boolean = {
if (s == 0) {
true
}
else if (witness(Random.nextInt(p - 1), p)) {
false
}
else {
isSimple(p, s - 1)
}
}
def witness(a: Int, p: Int): Boolean = {
val b: Seq[Boolean] = p - 1
b.foldLeft(1)((d, b) => if (mod(d * d, p) == 1 && d != 1 && d != p - 1) {
return true
} else {
b match {
case true => mod(mod(d*d, p)*a,p)
case false => mod(d*d, p)
}
}) != 1
}
}</lang>
Scheme
<lang scheme>#!r6rs (import (rnrs base (6))
(srfi :27 random-bits))
- Fast modular exponentiation.
(define (modexpt b e M)
(cond ((zero? e) 1) ((even? e) (modexpt (mod (* b b) M) (div e 2) M)) ((odd? e) (mod (* b (modexpt b (- e 1) M)) M))))
- Return s, d such that d is odd and 2^s * d = n.
(define (split n)
(let recur ((s 0) (d n))
(if (odd? d)
(values s d)
(recur (+ s 1) (div d 2)))))
- Test whether the number a proves that n is composite.
(define (composite-witness? n a)
(let*-values (((s d) (split (- n 1)))
((x) (modexpt a d n)))
(and (not (= x 1))
(not (= x (- n 1)))
(let try ((r (- s 1)))
(set! x (modexpt x 2 n))
(or (zero? r)
(= x 1)
(and (not (= x (- n 1)))
(try (- r 1))))))))
- Test whether n > 2 is a Miller-Rabin pseudoprime, k trials.
(define (pseudoprime? n k)
(or (zero? k)
(let ((a (+ 2 (random-integer (- n 2)))))
(and (not (composite-witness? n a))
(pseudoprime? n (- k 1))))))
- Test whether any integer is prime.
(define (prime? n)
(and (> n 1)
(or (= n 2)
(pseudoprime? n 50))))</lang>
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "bigint.s7i";
const func boolean: millerRabin (in bigInteger: n, in integer: k) is func
result
var boolean: probablyPrime is TRUE;
local
var bigInteger: d is 0_;
var integer: r is 0;
var integer: s is 0;
var bigInteger: a is 0_;
var bigInteger: x is 0_;
var integer: tests is 0;
begin
if n < 2_ or (n > 2_ and not odd(n)) then
probablyPrime := FALSE;
elsif n > 3_ then
d := pred(n);
s := lowestSetBit(d);
d >>:= s;
while tests < k and probablyPrime do
a := rand(2_, pred(n));
x := modPow(a, d, n);
if x <> 1_ and x <> pred(n) then
r := 1;
while r < s and x <> 1_ and x <> pred(n) do
x := modPow(x, 2_, n);
incr(r);
end while;
probablyPrime := x = pred(n);
end if;
incr(tests);
end while;
end if;
end func;
const proc: main is func
local
var bigInteger: number is 0_;
begin
for number range 2_ to 1000_ do
if millerRabin(number, 10) then
writeln(number);
end if;
end for;
end func;</lang>Original source: [3]
Sidef
<lang ruby>func is_prime(n, k) {
n == 2 && return true n <= 1 && return false n & 1 || return false
var d = n-1 var s = valuation(d, 2) d >>= s
k.times {
var a = irand(2, n-1)
var x = expmod(a, d, n)
next if (x ~~ [1, n-1])
(s-1).times {
x = expmod(x, 2, n)
return false if x==1
break if (x == n-1)
}
return false if (x != n-1)
}
return true
}
say {|n| is_prime(n, 10) }.grep(^1000).join(', ')</lang>
Smalltalk
Smalltalk handles big numbers naturally and trasparently (the parent class Integer has many subclasses, and a subclass is picked according to the size of the integer that must be handled) <lang smalltalk>Integer extend [
millerRabinTest: kl [ |k| k := kl.
self <= 3
ifTrue: [ ^true ]
ifFalse: [
(self even)
ifTrue: [ ^false ]
ifFalse: [ |d s|
d := self - 1.
s := 0.
[ (d rem: 2) == 0 ]
whileTrue: [
d := d / 2.
s := s + 1.
].
[ k:=k-1. k >= 0 ]
whileTrue: [ |a x r|
a := Random between: 2 and: (self - 2).
x := (a raisedTo: d) rem: self.
( x = 1 )
ifFalse: [ |r|
r := -1.
[ r := r + 1. (r < s) & (x ~= (self - 1)) ]
whileTrue: [
x := (x raisedTo: 2) rem: self
].
( x ~= (self - 1) ) ifTrue: [ ^false ]
]
].
^true
]
]
]
].</lang> <lang smalltalk>1 to: 1000 do: [ :n |
(n millerRabinTest: 10) ifTrue: [ n printNl ]
].</lang>
Standard ML
<lang sml>open LargeInt;
val mr_iterations = Int.toLarge 20; val rng = Random.rand (557216670, 13504100); (* arbitrary pair to seed RNG *)
fun expmod base 0 m = 1
| expmod base exp m =
if exp mod 2 = 0
then let val rt = expmod base (exp div 2) m;
val sq = (rt*rt) mod m
in if sq = 1
andalso rt <> 1 (* ignore the two *)
andalso rt <> (m-1) (* 'trivial' roots *)
then 0
else sq
end
else (base*(expmod base (exp-1) m)) mod m;
(* arbitrary precision random number [0,n) *) fun rand n =
let val base = Int.toLarge(valOf Int.maxInt)+1;
fun step r lim =
if lim < n then step (Int.toLarge(Random.randNat rng) + r*base) (lim*base)
else r mod n
in step 0 1 end;
fun miller_rabin n =
let fun trial n 0 = true
| trial n t = let val a = 1+rand(n-1)
in (expmod a (n-1) n) = 1
andalso trial n (t-1)
end
in trial n mr_iterations end;
fun trylist label lst = (label, ListPair.zip (lst, map miller_rabin lst));
trylist "test the first six Carmichael numbers"
[561, 1105, 1729, 2465, 2821, 6601];
trylist "test some known primes"
[7369, 7393, 7411, 27367, 27397, 27407];
(* find ten random 30 digit primes (according to Miller-Rabin) *) let fun findPrime trials = let val t = trials+1;
val n = 2*rand(500000000000000000000000000000)+1
in if miller_rabin n
then (n,t)
else findPrime t end
in List.tabulate (10, fn e => findPrime 0) end;</lang>
- Sample run:
...
val it =
("test the first six Carmichael numbers",
[(561,false),(1105,false),(1729,false),(2465,false),(2821,false),
(6601,false)]) : string * (int * bool) list
val it =
("test some known primes",
[(7369,true),(7393,true),(7411,true),(27367,true),(27397,true),
(27407,true)]) : string * (int * bool) list
[autoloading]
[autoloading done]
val it =
[(505776511533674858497882481471,8),(668742242620107711631417930007,111),
(831749124005136073184150011961,24),(159858916052323079037919394483,14),
(810857757001516064878680795563,43),(903375242242638088171051457359,6),
(506008872035764637556989600477,91),(105574439115200786396150347661,29),
(349239056313926786302179212509,7),(565349019043144709861293116613,126)]
: (int * int) list
Swift
<lang swift>import BigInt
private let numTrails = 5
func isPrime(_ n: BigInt) -> Bool {
guard n >= 2 else { fatalError() }
guard n != 2 else { return true }
guard n % 2 != 0 else { return false }
var s = 0 var d = n - 1
while true {
let (quo, rem) = (d / 2, d % 2)
guard rem != 1 else { break }
s += 1 d = quo }
func tryComposite(_ a: BigInt) -> Bool {
guard a.power(d, modulus: n) != 1 else { return false }
for i in 0..<s where a.power((2 as BigInt).power(i) * d, modulus: n) == n - 1 {
return false
}
return true }
for _ in 0..<numTrails where tryComposite(BigInt(BigUInt.randomInteger(lessThan: BigUInt(n)))) {
return false
}
return true
}</lang>
Tcl
Use Tcl 8.5 for large integer support <lang tcl>package require Tcl 8.5
proc miller_rabin {n k} {
if {$n <= 3} {return true}
if {$n % 2 == 0} {return false}
# write n - 1 as 2^s·d with d odd by factoring powers of 2 from n − 1
set d [expr {$n - 1}]
set s 0
while {$d % 2 == 0} {
set d [expr {$d / 2}]
incr s
}
while {$k > 0} {
incr k -1
set a [expr {2 + int(rand()*($n - 4))}]
set x [expr {($a ** $d) % $n}]
if {$x == 1 || $x == $n - 1} continue
for {set r 1} {$r < $s} {incr r} {
set x [expr {($x ** 2) % $n}]
if {$x == 1} {return false}
if {$x == $n - 1} break
}
if {$x != $n-1} {return false}
} return true
}
for {set i 1} {$i < 1000} {incr i} {
if {[miller_rabin $i 10]} {
puts $i
}
}</lang>
- Output:
1 2 3 5 7 11 ... 977 983 991 997
zkl
Using the Miller-Rabin primality test in GMP: <lang zkl>zkl: var BN=Import("zklBigNum"); zkl: BN("4547337172376300111955330758342147474062293202868155909489").probablyPrime() True zkl: BN("4547337172376300111955330758342147474062293202868155909393").probablyPrime() False</lang>
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