Multiplicative order: Difference between revisions
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{{task|Discrete math}}
;Example:
The multiplicative order of 37 relative to 1000 is 100 because 37^100 is 1 (modulo 1000), and no number smaller than 100 would do.
One possible algorithm that is efficient also for large numbers is the following: By the [[wp:Chinese_Remainder_Theorem|Chinese Remainder Theorem]], it's enough to calculate the multiplicative order for each prime exponent ''p^k'' of ''m'', and
combine the results with the ''[[least common multiple]]'' operation.
Now the order of ''a'' with regard to ''p^k'' must divide ''Φ(p^k)''. Call this number ''t'', and determine it's factors ''q^e''. Since each multiple of the order will also yield 1 when used as exponent for ''a'', it's enough to find the least d such that ''(q^d)*(t/(q^e))'' yields 1 when used as exponent.
;Task:
Implement a routine to calculate the multiplicative order along these lines. You may assume that routines to determine the factorization into prime powers are available in some library.
----
An algorithm for the multiplicative order can be found in Bach & Shallit, <i>Algorithmic Number Theory, Volume I: Efficient Algorithms</i>, The MIT Press, 1996:
<p>Exercise 5.8, page 115:</p>
<p>Suppose you are given a prime<tt> p </tt>and a complete factorization
of<tt> p-1</tt>
element<tt> a </tt>in<tt> (Z/(p))<sup>*</sup> </tt>using<tt> O((lg p)<sup>4</sup>/(lg lg p)) </tt>bit
operations.</p>
<p>Solution, page 337:</p>
<p>Let the prime factorization of<tt> p-1 </tt> be<tt> q1<sup>e1</sup>q2<sup>e2</sup>...qk<sup>ek</sup></tt> .<tt> </tt>We use the following observation:
if<tt> x^((p-1)/qi<sup>fi</sup>) = 1 (mod p)</tt> ,<tt> </tt>
and<tt> fi=ei </tt>or<tt> x^((p-1)/qi<sup>fi+1</sup>) != 1 (mod p)</tt> ,<tt> </tt>then<tt> qi<sup>ei-fi</sup>||ord<sub>p</sub> x</tt>. (This follows by combining Exercises 5.1 and 2.10.)
Hence it suffices to find, for each<tt> i</tt> ,<tt> </tt>the exponent<tt> fi </tt> such that the condition above holds.</p>
<p>This can be done as follows: first compute<tt> q1<sup>e1</sup>, q2<sup>e2</sup>, ... ,
qk<sup>ek</sup></tt> .<tt> </tt> This can be done using<tt> O((lg p)<sup>2</sup>) </tt>bit operations. Next, compute<tt> y1=(p-1)/q1<sup>e1</sup>, ... , yk=(p-1)/qk<sup>ek</sup></tt> .<tt> </tt>
This can be done using<tt> O((lg p)<sup>2</sup>) </tt>bit operations.
compute<tt> x1=a<sup>y1</sup>(mod p), ... , xk=a<sup>yk</sup>(mod p) </tt>.<tt> </tt>
This can be done using<tt> O(k(lg p)<sup>3</sup>) </tt>bit operations, and<tt> k=O((lg p)/(lg lg p)) </tt>by Theorem 8.8.10.
Finally, for each<tt> i</tt> ,<tt> </tt>repeatedly raise<tt> xi </tt>to the<tt> qi</tt>-th power<tt> (mod p) </tt>(as many as<tt> ei-1 </tt> times), checking to see when 1 is obtained.
This can be done using<tt> O((lg p)<sup>3</sup>) </tt>steps.
The total cost is dominated by<tt> O(k(lg p)<sup>3</sup>)</tt> ,<tt> </tt>which is<tt> O((lg p)<sup>4</sup>/(lg lg p))</tt>.
<br><br>
=={{header|11l}}==
{{trans|D}}
<syntaxhighlight lang="11l">T PExp
BigInt prime
Int exp
F (prime, exp)
.prime = prime
.exp = exp
F isqrt(self)
V b = self
L
V a = b
b = (self I/ a + a) I/ 2
I b >= a
R a
F factor(BigInt n)
[PExp] pf
V nn = n
V b = 0
L ((nn % 2) == 0)
nn I/= 2
b++
I b > 0
pf [+]= PExp(BigInt(2), b)
V s = isqrt(nn)
V d = BigInt(3)
L nn > 1
I d > s
d = nn
V e = 0
L
V (div, rem) = divmod(nn, d)
I bits:length(rem) > 0
L.break
nn = div
e++
I e > 0
pf [+]= PExp(d, e)
s = isqrt(nn)
d += 2
R pf
F moBachShallit58(BigInt a, BigInt n; pf)
V n1 = n - 1
V mo = BigInt(1)
L(pe) pf
V y = n1 I/ pow(pe.prime, BigInt(pe.exp))
V o = 0
V x = pow(a, y, n)
L x > 1
x = pow(x, pe.prime, n)
o++
V o1 = pow(pe.prime, BigInt(o))
o1 I/= gcd(mo, o1)
mo *= o1
R mo
F moTest(a, n)
I bits:length(a) < 100
print(‘ord(’a‘)’, end' ‘’)
E
print(‘ord([big])’, end' ‘’)
print(‘ mod ’n‘ = ’moBachShallit58(a, n, factor(n - 1)))
moTest(37, 3343)
moTest(pow(BigInt(10), 100) + 1, 7919)
moTest(pow(BigInt(10), 1000) + 1, 15485863)
moTest(pow(BigInt(10), 10000) - 1, BigInt(22801763489))
moTest(1511678068, 7379191741)
moTest(BigInt(‘3047753288’), BigInt(‘2257683301’))</syntaxhighlight>
{{out}}
<pre>
ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
</pre>
=={{header|Ada}}==
Instead of assuming a library call to factorize the modulus, we assume the caller of our Find_Order function has already factorized it. The Multiplicative_Order package is specified as follows ("multiplicative_order.ads").
<syntaxhighlight lang="ada">package Multiplicative_Order is
type Positive_Array is array (Positive range <>) of Positive;
function Find_Order(Element, Modulus: Positive) return Positive;
-- naive algorithm
-- returns the smallest I such that (Element**I) mod Modulus = 1
function Find_Order(Element: Positive;
Coprime_Factors: Positive_Array) return Positive;
-- faster algorithm for the same task
-- computes the order of all Coprime_Factors(I)
-- and returns their least common multiple
-- this gives the same result as Find_Order(Element, Modulus)
-- with Modulus being the product of all the Coprime_Factors(I)
--
-- preconditions: (1) 1 = GCD(Coprime_Factors(I), Coprime_Factors(J))
-- for all pairs I, J with I /= J
-- (2) 1 < Coprime_Factors(I) for all I
end Multiplicative_Order;</syntaxhighlight>
Here is the implementation ("multiplicative_order.adb"):
<syntaxhighlight lang="ada">package body Multiplicative_Order is
function Find_Order(Element, Modulus: Positive) return Positive is
function Power(Exp, Pow, M: Positive) return Positive is
-- computes Exp**Pow mod M;
-- note that Ada's native integer exponentiation "**" may overflow on
-- computing Exp**Pow before ever computing the "mod M" part
Result: Positive := 1;
E: Positive := Exp;
P: Natural := Pow;
begin
while P > 0 loop
if P mod 2 = 1 then
Result := (Result * E) mod M;
end if;
E := (E * E) mod M;
P := P / 2;
end loop;
return Result;
end Power;
begin -- Find_Order(Element, Modulus)
for I in 1 .. Modulus loop
if Power(Element, I, Modulus) = 1 then
return Positive(I);
end if;
end loop;
raise Program_Error with
Positive'Image(Element) &" is not coprime to" &Positive'Image(Modulus);
end Find_Order;
function Find_Order(Element: Positive;
Coprime_Factors: Positive_Array) return Positive is
function GCD (A, B : Positive) return Integer is
M : Natural := A;
N : Natural := B;
T : Natural;
begin
while N /= 0 loop
T := M;
M := N;
N ;:= T mod N;
end loop;
return M;
end GCD; -- from http://rosettacode.org/wiki/Least_common_multiple#Ada
function LCM (A, B : Natural) return Integer is
begin
if A = 0 or B = 0 then
return 0;
end if;
return abs (A * B) / Gcd (A, B);
end LCM; -- from http://rosettacode.org/wiki/Least_common_multiple#Ada
Result : Positive := 1;
begin -- Find_Order(Element, Coprime_Factors)
for I in Coprime_Factors'Range loop
Result := LCM(Result, Find_Order(Element, Coprime_Factors(I)));
end loop;
return Result;
end Find_Order;
end Multiplicative_Order;</syntaxhighlight>
This is a sample program using the Multiplicative_Order package:
<syntaxhighlight lang="ada">with Ada.Text_IO, Multiplicative_Order;
procedure Main is
package IIO is new Ada.Text_IO.Integer_IO(Integer);
use Multiplicative_Order;
begin
IIO.Put(Find_Order(3,10));
IIO.Put(Find_Order(37,1000));
IIO.Put(Find_Order(37,10_000));
IIO.Put(Find_Order(37, 3343));
IIO.Put(Find_Order(37, 3344));
-- IIO.Put(Find_Order( 2,1000));
--would raise Program_Error, because there is no I with 2**I=1 mod 1000
Ada.Text_IO.New_Line;
IIO.Put(Find_Order(3, (2,5))); -- 3 * 5 = 10
IIO.Put(Find_Order(37, (8, 125))); -- 8 * 125 = 1000
IIO.Put(Find_Order(37, (16, 625))); -- 16 * 625 = 10_000
IIO.Put(Find_Order(37, (1 => 3343))); -- 1-element-array: 3343 is a prime
IIO.Put(Find_Order(37, (11, 19, 16))); -- 11 * 19 * 16 = 3344
-- this violates the precondition, because 8 and 2 are not coprime
-- it gives an incorrect result
IIO.Put(Find_Order(37, (11, 19, 8, 2)));
end Main;</syntaxhighlight>
The output from the sample program:
<pre>
4 100 500 1114 20
4 100 500 1114 20 10
</pre>
=={{header|ALGOL 68}}==
{{trans|python}}
{{works with|ALGOL 68|Standard - with preludes manually inserted}}
{{works with|ALGOL 68G|Any - tested with release mk15-0.8b.fc9.i386}}
<!-- {{does not work with|ELLA ALGOL 68|Any (with appropriate job cards AND formatted transput statements removed) - tested with release 1.8.8d.fc9.i386 - ELLA has no FORMATted transput, also it generates a call to undefined C LONG externals }} -->
<syntaxhighlight lang="algol68">MODE LOOPINT = INT;
MODE POWMODSTRUCT = LONG INT;
PR READ "prelude/pow_mod.a68" PR;
MODE SORTSTRUCT = LONG INT;
PR READ "prelude/sort.a68" PR;
MODE GCDSTRUCT = LONG INT;
PR READ "prelude/gcd.a68" PR;
PR READ "prelude/iterator.a68" PR;
PROC is prime = (LONG INT p)BOOL:
( p > 1 |#ANDF# ALL((YIELDBOOL yield)VOID: factored(p, (LONG INT f, LONG INT e)VOID: yield(f = p))) | FALSE );
FLEX[4]LONG INT prime list := (2,3,5,7);
OP +:= = (REF FLEX[]LONG INT lhs, LONG INT rhs)VOID: (
[UPB lhs +1] LONG INT next lhs;
next lhs[:UPB lhs] := lhs;
lhs := next lhs;
lhs[UPB lhs] := rhs
);
PROC primes = (PROC (LONG INT)VOID yield)VOID: (
LONG INT p;
FOR p index TO UPB prime list DO
p:= prime list[p index];
yield(p)
OD;
DO
p +:= 2;
WHILE NOT is prime(p) DO
p +:= 2
OD;
prime list +:= p;
yield(p)
OD
);
PROC factored = (LONG INT in a, PROC (LONG INT,LONG INT)VOID yield)VOID: (
LONG INT a := in a;
# FOR p IN # primes( # DO #
(LONG INT p)VOID:(
LONG INT j := 0;
WHILE a MOD p = 0 DO
a := a % p;
j +:= 1
OD;
IF j > 0 THEN yield (p,j) FI;
IF a < p*p THEN done FI
)
# ) OD # );
done:
IF a > 1 THEN yield (a,1) FI
);
PROC mult0rdr1 = (LONG INT a, p, e)LONG INT: (
LONG INT m := p ** SHORTEN e;
LONG INT t := (p-1)*(p**SHORTEN (e-1)); # = Phi(p**e) where p prime #
LONG INT q;
FLEX[0]LONG INT qs := (1);
# FOR f0,f1 IN # factored(t # DO #,
(LONG INT f0,f1)VOID: (
FLEX[SHORTEN((f1+1)*UPB qs)]LONG INT next qs;
FOR j TO SHORTEN f1 + 1 DO
FOR q index TO UPB qs DO
q := qs[q index];
next qs[(j-1)*UPB qs+q index] := q * f0**(j-1)
OD
OD;
qs := next qs
)
# OD # );
VOID(in place shell sort(qs));
FOR q index TO UPB qs DO
q := qs[q index];
IF pow mod(a,q,m)=1 THEN done FI
OD;
done:
q
);
PROC reduce = (PROC (LONG INT,LONG INT)LONG INT diadic, FORLONGINT iterator, LONG INT initial value)LONG INT: (
LONG INT out := initial value;
# FOR next IN # iterator( # DO #
(LONG INT next)VOID:
out := diadic(out, next)
# OD # );
out
);
PROC mult order = (LONG INT a, LONG INT m)LONG INT: (
PROC mofs = (YIELDLONGINT yield)VOID:(
# FOR p, count IN # factored(m, # DO #
(LONG INT p, LONG INT count)VOID:
yield(mult0rdr1(a,p,count))
)
# OD # );
reduce(lcm, mofs, 1)
);
main:(
FORMAT d = $g(-0)$;
printf((d, mult order(37, 1000), $l$)); # 100 #
LONG INT b := LENG 10**20-1;
printf((d, mult order(2, b), $l$)); # 3748806900 #
printf((d, mult order(17,b), $l$)); # 1499522760 #
b := 100001;
printf((d, mult order(54,b), $l$));
printf((d, pow mod( 54, mult order(54,b),b), $l$));
IF ANY( (YIELDBOOL yield)VOID: FOR r FROM 2 TO SHORTEN mult order(54,b)-1 DO yield(1=pow mod(54,r, b)) OD )
THEN
printf(($g$, "Exists a power r < 9090 where pow mod(54,r,b) = 1", $l$))
ELSE
printf(($g$, "Everything checks.", $l$))
FI
)</syntaxhighlight>
Output:
<pre>
100
3748806900
1499522760
9090
1
Everything checks.
</pre>
=={{header|C}}==
Uses prime/factor functions from [[Factors of an integer#Prime factoring]]. This implementation is not robust because of integer overflows. To properly deal with even moderately large numbers, an arbitrary precision integer package is a must.
<syntaxhighlight lang="c">ulong mpow(ulong a, ulong p, ulong m)
{
ulong r = 1;
while (p) {
if ((1 & p)) r = r * a % m;
a = a * a % m;
p >>= 1;
}
return r;
}
ulong ipow(ulong a, ulong p) {
ulong r = 1;
while (p) {
if ((1 & p)) r = r * a;
a *= a;
p >>= 1;
}
return r;
}
ulong gcd(ulong m, ulong n)
{
ulong t;
while (m) { t = m; m = n % m; n = t; }
return n;
}
ulong lcm(ulong m, ulong n)
{
ulong g = gcd(m, n);
return m / g * n;
}
ulong multi_order_p(ulong a, ulong p, ulong e)
{
ulong fac[10000];
ulong m = ipow(p, e);
ulong t = m / p * (p - 1);
int i, len = get_factors(t, fac);
for (i = 0; i < len; i++)
if (mpow(a, fac[i], m) == 1)
return fac[i];
return 0;
}
ulong multi_order(ulong a, ulong m)
{
prime_factor pf[100];
int i, len = get_prime_factors(m, pf);
ulong res = 1;
for (i = 0; i < len; i++)
res = lcm(res, multi_order_p(a, pf[i].p, pf[i].e));
return res;
}
int main()
{
sieve();
printf("The multiplicative order of %d related to %d is %lu \n", 37, 1000, multi_order(37, 1000));
printf("The multiplicative order of %d related to %d is %lu \n", 54, 100001, multi_order(54, 100001));
return 0;
}</syntaxhighlight>
=={{header|C sharp|C#}}==
{{trans|Java}}
<syntaxhighlight lang="csharp">using System;
using System.Collections.Generic;
using System.Numerics;
using System.Threading;
namespace MultiplicativeOrder {
// Taken from https://stackoverflow.com/a/33918233
public static class PrimeExtensions {
// Random generator (thread safe)
private static ThreadLocal<Random> s_Gen = new ThreadLocal<Random>(
() => {
return new Random();
}
);
// Random generator (thread safe)
private static Random Gen {
get {
return s_Gen.Value;
}
}
public static bool IsProbablyPrime(this BigInteger value, int witnesses = 10) {
if (value <= 1)
return false;
if (witnesses <= 0)
witnesses = 10;
BigInteger d = value - 1;
int s = 0;
while (d % 2 == 0) {
d /= 2;
s += 1;
}
byte[] bytes = new byte[value.ToByteArray().LongLength];
BigInteger a;
for (int i = 0; i < witnesses; i++) {
do {
Gen.NextBytes(bytes);
a = new BigInteger(bytes);
}
while (a < 2 || a >= value - 2);
BigInteger x = BigInteger.ModPow(a, d, value);
if (x == 1 || x == value - 1)
continue;
for (int r = 1; r < s; r++) {
x = BigInteger.ModPow(x, 2, value);
if (x == 1)
return false;
if (x == value - 1)
break;
}
if (x != value - 1)
return false;
}
return true;
}
}
static class Helper {
public static BigInteger Sqrt(this BigInteger self) {
BigInteger b = self;
while (true) {
BigInteger a = b;
b = self / a + a >> 1;
if (b >= a) return a;
}
}
public static long BitLength(this BigInteger self) {
BigInteger bi = self;
long bitlength = 0;
while (bi != 0) {
bitlength++;
bi >>= 1;
}
return bitlength;
}
public static bool BitTest(this BigInteger self, int pos) {
byte[] arr = self.ToByteArray();
int idx = pos / 8;
int mod = pos % 8;
if (idx >= arr.Length) {
return false;
}
return (arr[idx] & (1 << mod)) > 0;
}
}
class PExp {
public PExp(BigInteger prime, int exp) {
Prime = prime;
Exp = exp;
}
public BigInteger Prime { get; }
public int Exp { get; }
}
class Program {
static void MoTest(BigInteger a, BigInteger n) {
if (!n.IsProbablyPrime(20)) {
Console.WriteLine("Not computed. Modulus must be prime for this algorithm.");
return;
}
if (a.BitLength() < 100) {
Console.Write("ord({0})", a);
} else {
Console.Write("ord([big])");
}
if (n.BitLength() < 100) {
Console.Write(" mod {0} ", n);
} else {
Console.Write(" mod [big] ");
}
BigInteger mob = MoBachShallit58(a, n, Factor(n - 1));
Console.WriteLine("= {0}", mob);
}
static BigInteger MoBachShallit58(BigInteger a, BigInteger n, List<PExp> pf) {
BigInteger n1 = n - 1;
BigInteger mo = 1;
foreach (PExp pe in pf) {
BigInteger y = n1 / BigInteger.Pow(pe.Prime, pe.Exp);
int o = 0;
BigInteger x = BigInteger.ModPow(a, y, BigInteger.Abs(n));
while (x > 1) {
x = BigInteger.ModPow(x, pe.Prime, BigInteger.Abs(n));
o++;
}
BigInteger o1 = BigInteger.Pow(pe.Prime, o);
o1 = o1 / BigInteger.GreatestCommonDivisor(mo, o1);
mo = mo * o1;
}
return mo;
}
static List<PExp> Factor(BigInteger n) {
List<PExp> pf = new List<PExp>();
BigInteger nn = n;
int e = 0;
while (!nn.BitTest(e)) e++;
if (e > 0) {
nn = nn >> e;
pf.Add(new PExp(2, e));
}
BigInteger s = nn.Sqrt();
BigInteger d = 3;
while (nn > 1) {
if (d > s) d = nn;
e = 0;
while (true) {
BigInteger div = BigInteger.DivRem(nn, d, out BigInteger rem);
if (rem.BitLength() > 0) break;
nn = div;
e++;
}
if (e > 0) {
pf.Add(new PExp(d, e));
s = nn.Sqrt();
}
d = d + 2;
}
return pf;
}
static void Main(string[] args) {
MoTest(37, 3343);
MoTest(BigInteger.Pow(10, 100) + 1, 7919);
MoTest(BigInteger.Pow(10, 1000) + 1, 15485863);
MoTest(BigInteger.Pow(10, 10000) - 1, 22801763489);
MoTest(1511678068, 7379191741);
MoTest(3047753288, 2257683301);
}
}
}</syntaxhighlight>
{{out}}
<pre>ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425</pre>
=={{header|C++}}==
{{trans|C}}
<syntaxhighlight lang="cpp">#include <algorithm>
#include <bitset>
#include <iostream>
#include <vector>
typedef unsigned long ulong;
std::vector<ulong> primes;
typedef struct {
ulong p, e;
} prime_factor; /* prime, exponent */
void sieve() {
/* 65536 = 2^16, so we can factor all 32 bit ints */
constexpr int SIZE = 1 << 16;
std::bitset<SIZE> bits;
bits.flip(); // set all bits
bits.reset(0);
bits.reset(1);
for (int i = 0; i < 256; i++) {
if (bits.test(i)) {
for (int j = i * i; j < SIZE; j += i) {
bits.reset(j);
}
}
}
/* collect primes into a list. slightly faster this way if dealing with large numbers */
for (int i = 0; i < SIZE; i++) {
if (bits.test(i)) {
primes.push_back(i);
}
}
}
auto get_prime_factors(ulong n) {
std::vector<prime_factor> lst;
ulong e, p;
for (ulong i = 0; i < primes.size(); i++) {
p = primes[i];
if (p * p > n) break;
for (e = 0; !(n % p); n /= p, e++);
if (e) {
lst.push_back({ p, e });
}
}
if (n != 1) {
lst.push_back({ n, 1 });
}
return lst;
}
auto get_factors(ulong n) {
auto f = get_prime_factors(n);
std::vector<ulong> lst{ 1 };
size_t len2 = 1;
/* L = (1); L = (L, L * p**(1 .. e)) forall((p, e)) */
for (size_t i = 0; i < f.size(); i++, len2 = lst.size()) {
for (ulong j = 0, p = f[i].p; j < f[i].e; j++, p *= f[i].p) {
for (size_t k = 0; k < len2; k++) {
lst.push_back(lst[k] * p);
}
}
}
std::sort(lst.begin(), lst.end());
return lst;
}
ulong mpow(ulong a, ulong p, ulong m) {
ulong r = 1;
while (p) {
if (p & 1) {
r = r * a % m;
}
a = a * a % m;
p >>= 1;
}
return r;
}
ulong ipow(ulong a, ulong p) {
ulong r = 1;
while (p) {
if (p & 1) r *= a;
a *= a;
p >>= 1;
}
return r;
}
ulong gcd(ulong m, ulong n) {
ulong t;
while (m) {
t = m;
m = n % m;
n = t;
}
return n;
}
ulong lcm(ulong m, ulong n) {
ulong g = gcd(m, n);
return m / g * n;
}
ulong multi_order_p(ulong a, ulong p, ulong e) {
ulong m = ipow(p, e);
ulong t = m / p * (p - 1);
auto fac = get_factors(t);
for (size_t i = 0; i < fac.size(); i++) {
if (mpow(a, fac[i], m) == 1) {
return fac[i];
}
}
return 0;
}
ulong multi_order(ulong a, ulong m) {
auto pf = get_prime_factors(m);
ulong res = 1;
for (size_t i = 0; i < pf.size(); i++) {
res = lcm(res, multi_order_p(a, pf[i].p, pf[i].e));
}
return res;
}
int main() {
sieve();
printf("%lu\n", multi_order(37, 1000)); // expect 100
printf("%lu\n", multi_order(54, 100001)); // expect 9090
return 0;
}</syntaxhighlight>
{{out}}
<pre>100
9090</pre>
=={{header|Clojure}}==
Translation of Julie, then revised to be more clojure idiomatic. It references some external modules for factoring and integer exponentiation.
<syntaxhighlight lang="clojure">(defn gcd [a b]
(if (zero? b)
a
(recur b (mod a b))))
(defn lcm [a b]
(/ (* a b) (gcd a b)))
(def NaN (Math/log -1))
(defn ord' [a [p e]]
(let [m (imath/expt p e)
t (* (quot m p) (dec p))]
(loop [dv (factor/divisors t)]
(let [d (first dv)]
(if (= (mmath/expm a d m) 1)
d
(recur (next dv)))))))
(defn ord [a n]
(if (not= (gcd a n) 1)
NaN
(->>
(factor/factorize n)
(map (partial ord' a))
(reduce lcm))))
</syntaxhighlight>
{{out}}
<pre>
user=> (ord 37 1000)
100
</pre>
=={{header|D}}==
{{trans|Java}}
<syntaxhighlight lang="d">import std.bigint;
import std.random;
import std.stdio;
struct PExp {
BigInt prime;
int exp;
}
BigInt gcd(BigInt x, BigInt y) {
if (y == 0) {
return x;
}
return gcd(y, x % y);
}
/// https://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method
BigInt modPow(BigInt b, BigInt e, BigInt n) {
if (n == 1) return BigInt(0);
BigInt result = 1;
b = b % n;
while (e > 0) {
if (e % 2 == 1) {
result = (result * b) % n;
}
e >>= 1;
b = (b*b) % n;
}
return result;
}
BigInt pow(long b, long e) {
return pow(BigInt(b), BigInt(e));
}
BigInt pow(BigInt b, BigInt e) {
if (e == 0) {
return BigInt(1);
}
BigInt result = 1;
while (e > 1) {
if (e % 2 == 0) {
b *= b;
e /= 2;
} else {
result *= b;
b *= b;
e = (e - 1) / 2;
}
}
return b * result;
}
BigInt sqrt(BigInt self) {
BigInt b = self;
while (true) {
BigInt a = b;
b = self / a + a >> 1;
if (b >= a) return a;
}
}
long bitLength(BigInt self) {
BigInt bi = self;
long length;
while (bi != 0) {
length++;
bi >>= 1;
}
return length;
}
PExp[] factor(BigInt n) {
PExp[] pf;
BigInt nn = n;
int b = 0;
int e = 1;
while ((nn & e) == 0) {
e <<= 1;
b++;
}
if (b > 0) {
nn = nn >> b;
pf ~= PExp(BigInt(2), b);
}
BigInt s = nn.sqrt();
BigInt d = 3;
while (nn > 1) {
if (d > s) d = nn;
e = 0;
while (true) {
BigInt div, rem;
nn.divMod(d, div, rem);
if (rem.bitLength > 0) break;
nn = div;
e++;
}
if (e > 0) {
pf ~= PExp(d, e);
s = nn.sqrt();
}
d += 2;
}
return pf;
}
BigInt moBachShallit58(BigInt a, BigInt n, PExp[] pf) {
BigInt n1 = n - 1;
BigInt mo = 1;
foreach(pe; pf) {
BigInt y = n1 / pe.prime.pow(BigInt(pe.exp));
int o = 0;
BigInt x = a.modPow(y, n);
while (x > 1) {
x = x.modPow(pe.prime, n);
o++;
}
BigInt o1 = pe.prime.pow(BigInt(o));
o1 = o1 / gcd(mo, o1);
mo = mo * o1;
}
return mo;
}
void moTest(ulong a, ulong n) {
moTest(BigInt(a), n);
}
void moTest(BigInt a, ulong n) {
// Commented out because the implementations tried all failed for the -2 and -3 tests.
// if (!n.isProbablePrime()) {
// writeln("Not computed. Modulus must be prime for this algorithm.");
// return;
// }
if (a.bitLength < 100) {
write("ord(", a, ")");
} else {
write("ord([big])");
}
write(" mod ", n, " ");
BigInt nn = n;
BigInt mob = moBachShallit58(a, nn, factor(nn - 1));
writeln("= ", mob);
}
void main() {
moTest(37, 3343);
moTest(pow(10, 100) + 1, 7919);
moTest(pow(10, 1000) + 1, 15485863);
moTest(pow(10, 10000) - 1, 22801763489);
moTest(1511678068, 7379191741);
moTest(3047753288, 2257683301);
}</syntaxhighlight>
{{out}}
<pre>ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425</pre>
=={{header|EchoLisp}}==
<syntaxhighlight lang="scheme">
(require 'bigint)
;; factor-exp returns a list ((p k) ..) : a = p1^k1 * p2^k2 ..
(define (factor-exp a)
(map (lambda (g) (list (first g) (length g)))
(group* (prime-factors a))))
;; copied from Ruby
(define (_mult_order a p k (x))
(define pk (expt p k))
(define t (* (1- p) (expt p (1- k))))
(define r 1)
(for [((q e) (factor-exp t))]
(set! x (powmod a (/ t (expt q e)) pk))
(while (!= x 1)
(*= r q)
(set! x (powmod x q pk))))
r)
(define (order a m)
"multiplicative order : (order a m) → n : a^n = 1 (mod m)"
(assert (= 1 (gcd a m)) "a and m must be coprimes")
(define mopks (for/list [((p k) (factor-exp m))] (_mult_order a p k)))
(for/fold (n 1) ((mopk mopks)) (lcm n mopk)))
;; results
order 37 1000)
→ 100
(order (+ (expt 10 100) 1) 7919)
→ 3959
(order (+ (expt 10 1000) 1) 15485863)
→ 15485862
</syntaxhighlight>
=={{header|Factor}}==
{{works with|Factor|0.99 2020-01-23}}
<syntaxhighlight lang="factor">USING: kernel math math.functions math.primes.factors sequences ;
: (ord) ( a pair -- n )
first2 dupd ^ swap dupd [ /i ] keep 1 - * divisors
[ swap ^mod 1 = ] 2with find nip ;
: ord ( a n -- m )
2dup gcd nip 1 =
[ group-factors [ (ord) ] with [ lcm ] map-reduce ]
[ 2drop 0/0. ] if ;</syntaxhighlight>
{{out}}
<pre>
IN: scratchpad 37 1000 ord .
100
IN: scratchpad 10 100 ^ 1 + 7919 ord .
3959
</pre>
=={{header|Go}}==
<syntaxhighlight lang="go">package main
import (
"fmt"
"math/big"
)
func main() {
moTest(big.NewInt(37), big.NewInt(3343))
b := big.NewInt(100)
moTest(b.Add(b.Exp(ten, b, nil), one), big.NewInt(7919))
moTest(b.Add(b.Exp(ten, b.SetInt64(1000), nil), one), big.NewInt(15485863))
moTest(b.Sub(b.Exp(ten, b.SetInt64(10000), nil), one),
big.NewInt(22801763489))
moTest(big.NewInt(1511678068), big.NewInt(7379191741))
moTest(big.NewInt(3047753288), big.NewInt(2257683301))
}
func moTest(a, n *big.Int) {
if a.BitLen() < 100 {
fmt.Printf("ord(%v)", a)
} else {
fmt.Print("ord([big])")
}
if n.BitLen() < 100 {
fmt.Printf(" mod %v ", n)
} else {
fmt.Print(" mod [big] ")
}
if !n.ProbablyPrime(20) {
fmt.Println("not computed. modulus must be prime for this algorithm.")
return
}
fmt.Println("=", moBachShallit58(a, n, factor(new(big.Int).Sub(n, one))))
}
var one = big.NewInt(1)
var two = big.NewInt(2)
var ten = big.NewInt(10)
func moBachShallit58(a, n *big.Int, pf []pExp) *big.Int {
n1 := new(big.Int).Sub(n, one)
var x, y, o1, g big.Int
mo := big.NewInt(1)
for _, pe := range pf {
y.Quo(n1, y.Exp(pe.prime, big.NewInt(pe.exp), nil))
var o int64
for x.Exp(a, &y, n); x.Cmp(one) > 0; o++ {
x.Exp(&x, pe.prime, n)
}
o1.Exp(pe.prime, o1.SetInt64(o), nil)
mo.Mul(mo, o1.Quo(&o1, g.GCD(nil, nil, mo, &o1)))
}
return mo
}
type pExp struct {
prime *big.Int
exp int64
}
func factor(n *big.Int) (pf []pExp) {
var e int64
for ; n.Bit(int(e)) == 0; e++ {
}
if e > 0 {
n.Rsh(n, uint(e))
pf = []pExp{{big.NewInt(2), e}}
}
s := sqrt(n)
q, r := new(big.Int), new(big.Int)
for d := big.NewInt(3); n.Cmp(one) > 0; d.Add(d, two) {
if d.Cmp(s) > 0 {
d.Set(n)
}
for e = 0; ; e++ {
q.QuoRem(n, d, r)
if r.BitLen() > 0 {
break
}
n.Set(q)
}
if e > 0 {
pf = append(pf, pExp{new(big.Int).Set(d), e})
s = sqrt(n)
}
}
return
}
func sqrt(n *big.Int) *big.Int {
a := new(big.Int)
for b := new(big.Int).Set(n); ; {
a.Set(b)
b.Rsh(b.Add(b.Quo(n, a), a), 1)
if b.Cmp(a) >= 0 {
return a
}
}
return a.SetInt64(0)
}</syntaxhighlight>
{{out}}
<pre>
ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
</pre>
=={{header|Haskell}}==
Assuming a function
<syntaxhighlight lang="haskell">powerMod
:: (Integral a, Integral b)
=> a -> a -> b -> a
powerMod m _ 0 = 1
powerMod m x n
| n > 0 = f x_ (n - 1) x_
where
x_ = x `rem` m
f _ 0 y = y
f a d y = g a d
where
g b i
| even i = g (b * b `rem` m) (i `quot` 2)
| otherwise = f b (i - 1) (b * y `rem` m)
powerMod m _ _ = error "powerMod: negative exponent"</syntaxhighlight>
to efficiently calculate powers modulo some <code>Integral</code>, we get
<syntaxhighlight lang="haskell">import Data.List (foldl1') --'
foldl1_ = foldl1' --'
multOrder a m
| gcd a m /= 1 = error "Arguments not coprime"
| otherwise = foldl1_ lcm $ map (multOrder_ a) $ primeFacsExp m
multOrder_ a (p, k) = r
where
pk = p ^ k
t = (p - 1) * p ^ (k - 1) -- totient \Phi(p^k)
r = product $ map find_qd $ primeFacsExp t
find_qd (q, e) = q ^ d
where
x = powerMod pk a (t `div` (q ^ e))
d = length $ takeWhile (/= 1) $ iterate (\y -> powerMod pk y q) x</syntaxhighlight>
=={{header|J}}==
The dyadic verb ''mo'' converts its arguments to exact numbers ''a'' and ''m'', executes ''mopk'' on the factorization of ''m'', and combines the result with the ''least common multiple'' operation.
assert. 1=a+.m
*./ a mopk"1 |: __ q: m
)</syntaxhighlight>
The dyadic verb ''mopk'' expects a pair of prime and exponent
in the second argument. It sets up a verb ''pm'' to calculate
powers module ''p^k''. Then calculates ''Φ(p^k)'' as ''t'',
factorizes it again into ''q'' and ''e'', and calculates
''a^(t/(q^e))'' as ''x''. Now, it finds the least ''d'' such that subsequent application of ''pm'' yields ''1''. Finally, it combines the
exponents ''q^d'' into a product.
<syntaxhighlight lang="j">mopk=: 4 : 0
a=. x: x
'p k'=. x: y
pm=. (p^k)&|@^
t=. (p-1)*p^k-1 NB. totient
'q e'=. __ q: t
x=. a pm t%q^e
d=. (1<x)+x (pm i. 1:)&> (e-1) */\@$&.> q
*/q^d
)</syntaxhighlight>
For example:
<syntaxhighlight lang="j"> 37 mo 1000
=={{header|
{{trans|Kotlin}}
<syntaxhighlight lang="java">import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;
public class MultiplicativeOrder {
private static final BigInteger ONE = BigInteger.ONE;
private static final BigInteger TWO = BigInteger.valueOf(2);
private static final BigInteger THREE = BigInteger.valueOf(3);
private static final BigInteger TEN = BigInteger.TEN;
private static class PExp {
BigInteger prime;
long exp;
PExp(BigInteger prime, long exp) {
this.prime = prime;
this.exp = exp;
}
}
private static void moTest(BigInteger a, BigInteger n) {
if (!n.isProbablePrime(20)) {
System.out.println("Not computed. Modulus must be prime for this algorithm.");
if (a.bitLength() < 100) System.out.printf("ord(%s)", a);
else System.out.print("ord([big])");
if (n.bitLength() <
else System.out.print(" mod [big] ");
BigInteger mob = moBachShallit58(a, n, factor(n.subtract(ONE)));
System.out.println("= " + mob);
}
private static BigInteger moBachShallit58(BigInteger a, BigInteger n, List<PExp> pf) {
BigInteger n1 = n.subtract(ONE);
BigInteger mo = ONE;
for (PExp pe : pf) {
BigInteger y = n1.divide(pe.prime.pow((int) pe.exp));
long o = 0;
BigInteger x = a.modPow(y, n.abs());
while (x.compareTo(ONE) > 0) {
x = x.modPow(pe.prime, n.abs());
o++;
}
BigInteger o1 = BigInteger.valueOf(o);
o1 = pe.prime.pow(o1.intValue());
o1 = o1.divide(mo.gcd(o1));
mo = mo.multiply(o1);
}
return mo;
}
private static List<PExp> factor(BigInteger n) {
List<PExp> pf = new ArrayList<>();
BigInteger nn = n;
Long e = 0L;
while (!nn.testBit(e.intValue())) e++;
if (e > 0L) {
nn = nn.shiftRight(e.intValue());
pf.add(new PExp(TWO, e));
}
BigInteger s = sqrt(nn);
BigInteger d = THREE;
while (nn.compareTo(ONE) > 0) {
if (d.compareTo(s) > 0) d = nn;
e = 0L;
while (true) {
BigInteger[] qr = nn.divideAndRemainder(d);
if (qr[1].bitLength() > 0) break;
nn = qr[0];
e++;
}
if (e > 0L) {
pf.add(new PExp(d, e));
s = sqrt(nn);
}
d = d.add(TWO);
}
return pf;
}
private static BigInteger sqrt(BigInteger n) {
BigInteger b = n;
while (true) {
BigInteger a = b;
b = n.divide(a).add(a).shiftRight(1);
if (b.compareTo(a) >= 0) return a;
}
}
public static void main(String[] args) {
moTest(BigInteger.valueOf(37), BigInteger.valueOf(3343));
BigInteger b = TEN.pow(100).add(ONE);
moTest(b, BigInteger.valueOf(7919));
b = TEN.pow(1000).add(ONE);
moTest(b, BigInteger.valueOf(15485863));
b = TEN.pow(10000).subtract(ONE);
moTest(b, BigInteger.valueOf(22801763489L));
moTest(BigInteger.valueOf(1511678068), BigInteger.valueOf(7379191741L));
moTest(BigInteger.valueOf(3047753288L), BigInteger.valueOf(2257683301L));
}
}</syntaxhighlight>
{{out}}
<pre>ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425</pre>
=={{header|jq}}==
'''Adapted from [[#Wren|Wren]]'''
'''Works with gojq, the Go implementation of jq'''
The Go implementation of jq supports unbounded-precision integer
arithmetic and so is suitable for the specified tasks. The program
given here also runs using the C implementation of jq but falters for
large integers.
<syntaxhighlight lang="jq">
# Part 1: Library functions
### Counting and integer arithmetic
def count(s): reduce s as $x (0; .+1);
# If $j is 0, then an error condition is raised;
# otherwise, assuming infinite-precision integer arithmetic,
# if the input and $j are integers, then the result will be an integer.
def idivide($j):
(. - (. % $j)) / $j ;
def idivide($i; $j):
$i | idivide($j);
# Emit [dividend, mod]
def divmod($j):
(. % $j) as $mod
| [(. - $mod) / $j, $mod] ;
# input should be a non-negative integer for accuracy
# but may be any non-negative finite number
def isqrt:
def irt:
. as $x
| 1 | until(. > $x; . * 4) as $q
| {$q, $x, r: 0}
| until( .q <= 1;
.q |= idivide(4)
| .t = .x - .r - .q
| .r |= idivide(2)
| if .t >= 0
then .x = .t
| .r += .q
else .
end)
| .r ;
if type == "number" and (isinfinite|not) and (isnan|not) and . >= 0
then irt
else "isqrt requires a non-negative integer for accuracy" | error
end ;
# It is assumed that $n >= 0
def power($n):
. as $in
| reduce range(0;$n) as $i (1; .* $in);
# For syntactic convenience
def power($in; $n): $in | power($n);
def gcd(a; b):
# subfunction expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd: if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[a,b] | rgcd;
### Bit arrays and streams
def rightshift($n):
reduce range(0;$n) as $i (.; idivide(2));
# Convert the input integer to a stream of 0s and 1s, least significant bit first
def bitwise:
recurse( if . >= 2 then idivide(2) else empty end) | . % 2;
def bitLength: count(bitwise);
def firstBit:
if . == 0 then empty
else first( foreach bitwise as $b (-1; .+1; if $b == 1 then . else empty end))
end;
# Return true if the $i-th least-significant bit is 1, and false otherwise
def testBit($i):
(nth($i; bitwise) // 0) == 1;
# Part 2: "modulo" functions
# The multiplicative inverse of . modulo $n
def modInv($n):
. as $in
| { r: $n,
newR: length, # abs
t: 0,
newT: 1 }
| until (.newR != 0.;
idivide(.r; .newR) as $q
| .lastT = .t
| .lastR = .r
| .t = .newT
| .r = .newR
| .newT = .lastT - $q*.newT
| .newR = .lastR - $q*.newR )
| if .r != 1
then "\($in) and \($n) are not co-prime." | error
else if (.t < 0) then .t += $n end
| if ($in < 0) then - .t else .t end
end;
# Return . to the power $exp modulo $mod
def modPow($exp; $mod):
def isOdd: . % 2 == 1;
if $mod == 0 then "Cannot take modPow with modulus 0." | error
else {r: 1, base: (. % $mod), $exp}
| if .exp < 0
then .exp *= -1
| .base |= modInv($mod)
end
| until ((.exp == 0) or .emit;
if .base == 0 then .emit = 0
else if (.exp | isOdd) then .r = (.r * .base) % $mod end
| .exp |= idivide(2)
| .base |= (.*.) % $mod
end )
| (.emit // .r)
end ;
# Part 3: Multiplicative order
def moBachShallit58($a; $n; $pf):
{n1: ($n - 1),
mo: 1 }
| reduce $pf[] as $pe (.;
(.n1 | idivide($pe.prime | power($pe.exp))) as $y
| .o = 0
| .x = ($a | modPow($y; ($n|length)))
| until (.x <= 1;
.x |= modPow($pe.prime; ($n|length) )
| .o += 1 )
| .o1 = .o
| .o1 = power($pe.prime;.o1)
| .o1 = idivide(.o1; gcd(.mo; .o1) )
| .mo = .mo * .o1 )
| .mo ;
def factor($n):
{ pf: [],
nn: $n,
e: ($n | firstBit)}
| if .e > 0
then .e as $e
| .nn |= rightshift($e)
| .pf = [{prime: 2, exp: .e}]
end
| (.nn | isqrt) as $s
| .d = 3
| until (.nn <= 1;
if .d > $s then .d = .nn end
| .e = 0
| .done = null
| until( .done;
.d as $d
| (.nn | divmod($d)) as $dm
| if $dm[1] > 0
then .done = true
else .nn = $dm[0]
| .e += 1
end )
| if .e > 0
then .pf += [{prime: .d, exp: .e}]
|.s = (.nn|isqrt)
end
| .d += 2
)
| .pf ;
# $n should be prime
def moTest($a; $n):
if ($a|bitLength) < 100 then "ord(\($a)) " else "ord([big]) " end +
if ($n|bitLength) < 100 then "mod \($n) " else "mod [big] " end +
"= \(moBachShallit58($a; $n; factor($n - 1)))" ;
moTest(37; 3343),
moTest(1 + power(10;100); 7919),
moTest(1 + power(10;100); 15485863),
moTest(power(10;10000) - 1; 22801763489),
moTest(1511678068; 7379191741),
moTest(3047753288; 2257683301)
</syntaxhighlight>
{{output}}
<pre>
ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
</pre>
=={{header|Julia}}==
(Uses the <code>factors</code> function from [[Factors of an integer#Julia]].)
<syntaxhighlight lang="julia">using Primes
function factors(n)
f = [one(n)]
for (p,e) in factor(n)
f = reduce(vcat, [f*p^j for j in 1:e], init=f)
end
return length(f) == 1 ? [one(n), n] : sort!(f)
end
function multorder(a, m)
gcd(a,m) == 1 || error("$a and $m are not coprime")
res = one(m)
for (p,e) in factor(m)
m = p^e
t = div(m, p) * (p-1)
for f in factors(t)
if powermod(a, f, m) == 1
res = lcm(res, f)
break
end
end
end
res
end</syntaxhighlight>
Example output (using <code>big</code> to employ arbitrary-precision arithmetic where needed):
<pre>
julia> multorder(37, 1000)
100
julia> multorder(big(10)^100 + 1, 7919)
3959
julia> multorder(big(10)^1000 + 1, 15485863)
15485862
julia> multorder(big(10)^10000 - 1, 22801763489)
22801763488
</pre>
=={{header|Kotlin}}==
{{trans|Go}}
<syntaxhighlight lang="scala">// version 1.2.10
import java.math.BigInteger
val bigOne = BigInteger.ONE
val bigTwo = 2.toBigInteger()
val bigThree = 3.toBigInteger()
val bigTen = BigInteger.TEN
class PExp(val prime: BigInteger, val exp: Long)
fun moTest(a: BigInteger, n: BigInteger) {
if (!n.isProbablePrime(20)) {
println("Not computed. Modulus must be prime for this algorithm.")
return
}
if (a.bitLength() < 100) print("ord($a)") else print("ord([big])")
if (n.bitLength() < 100) print(" mod $n ") else print(" mod [big] ")
val mob = moBachShallit58(a, n, factor(n - bigOne))
println("= $mob")
}
fun moBachShallit58(a: BigInteger, n: BigInteger, pf: List<PExp>): BigInteger {
val n1 = n - bigOne
var mo = bigOne
for (pe in pf) {
val y = n1 / pe.prime.pow(pe.exp.toInt())
var o = 0L
var x = a.modPow(y, n.abs())
while (x > bigOne) {
x = x.modPow(pe.prime, n.abs())
o++
}
var o1 = o.toBigInteger()
o1 = pe.prime.pow(o1.toInt())
o1 /= mo.gcd(o1)
mo *= o1
}
return mo
}
fun factor(n: BigInteger): List<PExp> {
val pf = mutableListOf<PExp>()
var nn = n
var e = 0L
while (!nn.testBit(e.toInt())) e++
if (e > 0L) {
nn = nn shr e.toInt()
pf.add(PExp(bigTwo, e))
}
var s = bigSqrt(nn)
var d = bigThree
while (nn > bigOne) {
if (d > s) d = nn
e = 0L
while (true) {
val (q, r) = nn.divideAndRemainder(d)
if (r.bitLength() > 0) break
nn = q
e++
}
if (e > 0L) {
pf.add(PExp(d, e))
s = bigSqrt(nn)
}
d += bigTwo
}
return pf
}
fun bigSqrt(n: BigInteger): BigInteger {
var b = n
while (true) {
val a = b
b = (n / a + a) shr 1
if (b >= a) return a
}
}
fun main(args: Array<String>) {
moTest(37.toBigInteger(), 3343.toBigInteger())
var b = bigTen.pow(100) + bigOne
moTest(b, 7919.toBigInteger())
b = bigTen.pow(1000) + bigOne
moTest(b, BigInteger("15485863"))
b = bigTen.pow(10000) - bigOne
moTest(b, BigInteger("22801763489"))
moTest(BigInteger("1511678068"), BigInteger("7379191741"))
moTest(BigInteger("3047753288"), BigInteger("2257683301"))
}</syntaxhighlight>
{{out}}
<pre>
ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
</pre>
=={{header|Maple}}==
<syntaxhighlight lang="maple">numtheory:-order( a, n )</syntaxhighlight>
For example,
<syntaxhighlight lang="maple">> numtheory:-order( 37, 1000 );
100</syntaxhighlight>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
In Mathematica this is really easy, as this function is built-in:
MultiplicativeOrder[k,n] gives the multiplicative order of k modulo n, defined as the smallest integer m such that k^m == 1 mod n.<br>
MultiplicativeOrder[k,n,{r1,r2,...}] gives the generalized multiplicative order of k modulo n, defined as the smallest integer m such that k^m==ri mod n for some i.<BR>
Examples:
<syntaxhighlight lang="mathematica">MultiplicativeOrder[37, 1000]
MultiplicativeOrder[10^100 + 1, 7919] (*10^3th prime number Prime[1000]*)
MultiplicativeOrder[10^1000 + 1, 15485863] (*10^6th prime number*)
MultiplicativeOrder[10^10000 - 1, 22801763489] (*10^9th prime number*)
MultiplicativeOrder[13, 1 + 10^80]
MultiplicativeOrder[11, 1 + 10^100]</syntaxhighlight>
gives back:
<pre>100
3959
15485862
22801763488
109609547199756140150989321269669269476675495992554276140800
2583496112724752500580158969425549088007844580826869433740066152289289764829816356800</pre>
=={{header|Maxima}}==
<syntaxhighlight lang="maxima">zn_order(37, 1000);
/* 100 */
zn_order(10^100 + 1, 7919);
/* 3959 */
zn_order(10^1000 + 1, 15485863);
/* 15485862 */
zn_order(10^10000 - 1, 22801763489);
/* 22801763488 */
zn_order(13, 1 + 10^80);
/* 109609547199756140150989321269669269476675495992554276140800 */
zn_order(11, 1 + 10^100);
/* 2583496112724752500580158969425549088007844580826869433740066152289289764829816356800 */</syntaxhighlight>
=={{header|Nim}}==
{{trans|Kotlin}}
{{libheader|bignum}}
<syntaxhighlight lang="nim">import strformat
import bignum
type PExp = tuple[prime: Int; exp: uint]
let
one = newInt(1)
two = newInt(2)
ten = newInt(10)
func sqrt(n: Int): Int =
var s = n
while true:
result = s
s = (n div result + result) shr 1
if s >= result: break
proc factor(n: Int): seq[PExp] =
var n = n
var e = 0u
while n.bit(e) == 0: inc e
if e != 0:
n = n shr e
result.add (two, e)
var s = sqrt(n)
var d = newInt(3)
while n > one:
if d > s: d = n
e = 0u
while true:
let (q, r) = divMod(n, d)
if not r.isZero: break
n = q
inc e
if e != 0:
result.add (d.clone, e)
s = sqrt(n)
inc d, two
proc moBachShallit58(a, n: Int; pf: seq[PExp]): Int =
let n = abs(n)
let n1 = n - one
result = newInt(1)
for pe in pf:
let y = n1 div pe.prime.pow(pe.exp)
var o = 0u
var x = a.exp(y.toInt.uint, n)
while x > one:
x = x.exp(pe.prime.toInt.uint, n)
inc o
var o1 = pe.prime.pow(o)
o1 = o1 div gcd(result, o1)
result *= o1
proc moTest(a, n: Int) =
if n.probablyPrime(25) == 0:
echo "Not computed. Modulus must be prime for this algorithm."
return
stdout.write if a.bitLen < 100: &"ord({a})" else: "ord([big])"
stdout.write if n.bitlen < 100: &" mod {n}" else: " mod [big]"
let mob = moBachShallit58(a, n, factor(n - one))
echo &" = {mob}"
when isMainModule:
moTest(newInt(37), newInt(3343))
var b = ten.pow(100) + one
motest(b, newInt(7919))
b = ten.pow(1000) + one
moTest(b, newInt("15485863"))
b = ten.pow(10000) - one
moTest(b, newInt("22801763489"))
moTest(newInt("1511678068"), newInt("7379191741"))
moTest(newInt("3047753288"), newInt("2257683301"))</syntaxhighlight>
{{out}}
<pre>ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425</pre>
=={{header|PARI/GP}}==
<syntaxhighlight lang="parigp">znorder(Mod(a,n))</syntaxhighlight>
=={{header|Perl}}==
Using modules:
{{libheader|ntheory}}
<syntaxhighlight lang="perl">use ntheory qw/znorder/;
say znorder(54, 100001);
use bigint; say znorder(11, 1 + 10**100);</syntaxhighlight>
or
<syntaxhighlight lang="perl">use Math::Pari qw/znorder Mod/;
say znorder(Mod(54, 100001));
say znorder(Mod(11, 1 + Math::Pari::PARI(10)**100));</syntaxhighlight>
=={{header|Phix}}==
{{trans|Ruby}}
{{libheader|Phix/mpfr}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">multi_order</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">sequence</span> <span style="color: #000000;">p_and_k</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">5</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p_and_k</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">string</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">ps</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p_and_k</span>
<span style="color: #7060A8;">mpz_set_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ps</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_sub_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">else</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p_and_k</span>
<span style="color: #7060A8;">mpz_set_d</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_pow_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_pow_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_sub_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pz</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">pf</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_prime_factors</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pf</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">string</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #7060A8;">mpz_set_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">else</span>
<span style="color: #0000FF;">{</span><span style="color: #004080;">integer</span> <span style="color: #000000;">qi</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">ei</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #7060A8;">mpz_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qi</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_pow_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ei</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">mpz_fdiv_q</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_powm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pk</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">guard</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_powm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pk</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">guard</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">guard</span><span style="color: #0000FF;">></span><span style="color: #000000;">100</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- (increase if rqd)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_free</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">multiplicative_order</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">ri</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">mpz_gcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #008000;">"(a,m) not coprime"</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">pf</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_prime_factors</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10000</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (increase if rqd)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pf</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">multi_order</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">mpz_lcm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">shorta</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">lr</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">lr</span><span style="color: #0000FF;">></span><span style="color: #000000;">80</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">6</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">6</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"..."</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" (%d digits)"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">lr</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">multiplicative_order</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"ord(%s) mod %s = %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">shorta</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span><span style="color: #000000;">shorta</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span><span style="color: #000000;">res</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #000080;font-style:italic;">-- (ugh)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">p10</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- init to 10^e+i</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">mpz_ui_pow_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_add_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">37</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">37</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10000</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">37</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3343</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">37</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3344</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p10</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">7919</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p10</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">15485863</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p10</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10000</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">22801763489</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1511678068</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">7379191741</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3047753288</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2257683301</span><span style="color: #0000FF;">))</span>
<span style="color: #0000FF;">?</span><span style="color: #008000;">"==="</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p10</span><span style="color: #0000FF;">(</span><span style="color: #000000;">20</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">17</span><span style="color: #0000FF;">),</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">mo_test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">54</span><span style="color: #0000FF;">),</span><span style="color: #000000;">i</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100001</span><span style="color: #0000FF;">))</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">s9090</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">multiplicative_order</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">54</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100001</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">s9090</span><span style="color: #0000FF;">!=</span><span style="color: #008000;">"9090"</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">m54</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">54</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">m100001</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100001</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_powm_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m54</span><span style="color: #0000FF;">,</span><span style="color: #000000;">9090</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m100001</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">))</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">error</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">9090</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_powm_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m54</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m100001</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"mpz_powm_ui(54,%d,100001) gives 1!\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">error</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
<span style="color: #008080;">exit</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">error</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Everything checks. (%s)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
ord(3) mod 10 = 4
ord(37) mod 1000 = 100
ord(37) mod 10000 = 500
ord(37) mod 3343 = 1114
ord(37) mod 3344 = 20
ord(2) mod 1000 = (a,m) not coprime
ord(10000...00001 (101 digits)) mod 7919 = 3959
ord(10000...00001 (1001 digits)) mod 15485863 = 15485862
ord(99999...99999 (10000 digits)) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
"==="
ord(2) mod 99999999999999999999 = 3748806900
ord(17) mod 99999999999999999999 = 1499522760
ord(54) mod 100001 = 9090
1
Everything checks. (0.2s)
</pre>
=={{header|Python}}==
<syntaxhighlight lang="python">def gcd(a, b):
while b != 0:
a, b = b, a % b
return a
def lcm(a, b):
return (a*b) / gcd(a, b)
def isPrime(p):
return (p > 1) and all(f == p for f,e in factored(p))
primeList = [2,3,5,7]
def primes():
for p in primeList:
yield p
while 1:
p += 2
while not isPrime(p):
p += 2
primeList.append(p)
yield p
def factored( a):
for p in primes():
j = 0
while a%p == 0:
a /= p
j += 1
if j > 0:
yield (p,j)
if a < p*p: break
if a > 1:
yield (a,1)
def multOrdr1(a,(p,e) ):
m = p**e
t = (p-1)*(p**(e-1)) # = Phi(p**e) where p prime
qs = [1,]
for f in factored(t):
qs = [ q * f[0]**j for j in range(1+f[1]) for q in qs ]
qs.sort()
for q in qs:
if pow( a, q, m )==1: break
return q
def multOrder(a,m):
assert gcd(a,m) == 1
mofs = (multOrdr1(a,r) for r in factored(m))
return reduce(lcm, mofs, 1)
if __name__ == "__main__":
print multOrder(37, 1000) # 100
b = 10**20-1
print multOrder(2, b) # 3748806900
print multOrder(17,b) # 1499522760
b = 100001
print multOrder(54,b)
print pow( 54, multOrder(54,b),b)
if any( (1==pow(54,r, b)) for r in range(1,multOrder(54,b))):
print 'Exists a power r < 9090 where pow(54,r,b)==1'
else:
print 'Everything checks.'</syntaxhighlight>
=={{header|Racket}}==
The Racket function unit-group-order from racket/math computes the multiplicative order
of an element a in Zn. An implementation of the algorithm in the tast description is
shown below.
<syntaxhighlight lang="racket">
#lang racket
(require math)
(define (order a n)
(unless (coprime? a n) (error 'order "arguments must be coprime"))
(for/fold ([o 1]) ([r (factorize n)])
(lcm o (order1 a r))))
(define (order1 a p&e)
(match-define (list p e) p&e)
(define m (expt p e))
(define t (* (- p 1) (expt p (- e 1))))
(define qs
(for/fold ([qs '(1)]) ([f (factorize t)])
(match f [(list f0 f1)
(for*/list ([q qs] [j (in-range (+ 1 f1))])
(* q (expt f0 j)))])))
(for/or ([q (sort qs <)] #:when (= (modular-expt a q m) 1)) q))
(order 37 1000)
(order (+ (expt 10 100) 1) 7919)
(order (+ (expt 10 1000) 1) 15485863)
(order (- (expt 10 10000) 1) 22801763489)
(order 13 (+ 1 (expt 10 80)))
</syntaxhighlight>
Output:
<syntaxhighlight lang="racket">
100
3959
15485862
22801763488
109609547199756140150989321269669269476675495992554276140800
</syntaxhighlight>
=={{header|Raku}}==
(formerly Perl 6)
<syntaxhighlight lang="raku" line>use Prime::Factor;
sub mo-prime($a, $p, $e) {
my $m = $p ** $e;
my $t = ($p - 1) * ($p ** ($e - 1)); # = Phi($p**$e) where $p prime
my @qs = 1;
for prime-factors($t).Bag -> $f {
@qs = flat @qs.map(-> $q { (0..$f.value).map(-> $j { $q * $f.key ** $j }) });
}
@qs.sort.first: -> $q { expmod( $a, $q, $m ) == 1 };
}
sub mo($a, $m) {
$a gcd $m == 1 or die "$a and $m are not relatively prime";
[lcm] flat 1, prime-factors($m).Bag.map: { mo-prime($a, .key, .value) };
}
multi MAIN('test') {
use Test;
for (10, 21, 25, 150, 1231, 123141, 34131) -> $n {
is ([*] prime-factors($n).Bag.map( { .key ** .value } )), $n, "$n factors correctly";
}
is mo(37, 1000), 100, 'mo(37,1000) == 100';
my $b = 10**20-1;
is mo(2, $b), 3748806900, 'mo(2,10**20-1) == 3748806900';
is mo(17, $b), 1499522760, 'mo(17,10**20-1) == 1499522760';
$b = 100001;
is mo(54, $b), 9090, 'mo(54,100001) == 9090';
}</syntaxhighlight>
{{out}}
<pre>ok 1 - 10 factors correctly
ok 2 - 21 factors correctly
ok 3 - 25 factors correctly
ok 4 - 150 factors correctly
ok 5 - 1231 factors correctly
ok 6 - 123141 factors correctly
ok 7 - 34131 factors correctly
ok 8 - mo(37,1000) == 100
ok 9 - mo(2,10**20-1) == 3748806900
ok 10 - mo(17,10**20-1) == 1499522760
ok 11 - mo(54,100001) == 9090</pre>
=={{header|REXX}}==
<syntaxhighlight lang="rexx">/*REXX pgm computes multiplicative order of a minimum integer N such that a^n mod m≡1*/
wa= 0; wm= 0 /* ═a═ ══m══ */ /*maximum widths of the A and M values.*/
@.=.; @.1= 3 10
@.2= 37 1000
@.3= 37 10000
@.4= 37 3343
@.5= 37 3344
@.6= 2 1000
pad= left('', 9)
d= 500 /*use 500 decimal digits for a starter.*/
do w=1 for 2 /*when W≡1, find max widths of A and M.*/
do j=1 while @.j\==.; parse var @.j a . 1 r m , n
if w==1 then do; wa= max(wa, length(a) ); wm= max(wm, length(m) ); iterate
end
if m//a==0 then n= ' [solution not possible]' /*test co─prime for A and B. */
numeric digits d /*start with 100 decimal digits. */
if n=='' then do n= 2; p= r * a /*compute product──may have an exponent*/
parse var p 'E' _ /*try to extract the exponent from P. */
if _\=='' then do; numeric digits _+d /*bump the decimal digs.*/
p=r*a /*recalculate integer P.*/
end
if p//m==1 then leave /*now, perform the nitty─gritty modulo.*/
r= p /*assign product to R for next multiply*/
end /*n*/ /* [↑] // is really ÷ remainder.*/
say pad 'a=' right(a,wa) pad "m=" right(m,wm) pad 'multiplicative order:' n
end /*j*/
end /*w*/ /*stick a fork in it, we're all done. */</syntaxhighlight>
{{out|output|.}}
<pre>
a= 3 m= 10 multiplicative order: 4
a= 37 m= 1000 multiplicative order: 100
a= 37 m= 10000 multiplicative order: 500
a= 37 m= 3343 multiplicative order: 1114
a= 37 m= 3344 multiplicative order: 20
a= 2 m= 1000 multiplicative order: [solution not possible]
</pre>
=={{header|Ruby}}==
<syntaxhighlight lang="ruby">require 'prime'
def powerMod(b, p, m)
p.to_s(2).each_char.inject(1) do |result, bit|
result = (result * result) % m
bit=='1' ? (result * b) % m : result
end
end
def multOrder_(a, p, k)
pk = p ** k
t = (p - 1) * p ** (k - 1)
r = 1
for q, e in t.prime_division
x = powerMod(a, t / q**e, pk)
while x != 1
r *= q
x = powerMod(x, q, pk)
end
end
r
end
def multOrder(a, m)
m.prime_division.inject(1) do |result, f|
result.lcm(multOrder_(a, *f))
end
end
puts multOrder(37, 1000)
b = 10**20-1
puts multOrder(2, b)
puts multOrder(17,b)
b = 100001
puts multOrder(54,b)
puts powerMod(54, multOrder(54,b), b)
if (1...multOrder(54,b)).any? {|r| powerMod(54, r, b) == 1}
puts 'Exists a power r < 9090 where powerMod(54,r,b)==1'
else
puts 'Everything checks.'
end</syntaxhighlight>
{{out}}
<pre>
100
3748806900
1499522760
9090
1
Everything checks.
</pre>
=={{header|Seed7}}==
<syntaxhighlight lang="seed7">$ include "seed7_05.s7i";
include "bigint.s7i";
const type: oneFactor is new struct
var bigInteger: prime is 0_;
var integer: exp is 0;
end struct;
const func oneFactor: oneFactor (in bigInteger: prime, in integer: exp) is func
result
var oneFactor: aFactor is oneFactor.value;
begin
aFactor.prime := prime;
aFactor.exp := exp;
end func;
const func array oneFactor: factor (in var bigInteger: n) is func
result
var array oneFactor: pf is 0 times oneFactor.value;
local
var integer: e is 0;
var bigInteger: d is 0_;
var bigInteger: s is 0_;
begin
e := lowestSetBit(n);
if e > 0 then
n >>:= e;
pf := [] (oneFactor(2_, e));
end if;
s := sqrt(n);
d := 3_;
while n > 1_ do
if d > s then
d := n;
end if;
e := 0;
while n rem d = 0_ do
n := n div d;
incr(e);
end while;
if e > 0 then
pf &:= oneFactor(d, e);
s := sqrt(n);
end if;
d +:= 2_;
end while;
end func;
const func bigInteger: moBachShallit58(in bigInteger: a, in bigInteger: n, in array oneFactor: pf) is func
result
var bigInteger: mo is 0_;
local
var bigInteger: n1 is 0_;
var oneFactor: pe is oneFactor.value;
var bigInteger: x is 0_;
var bigInteger: y is 0_;
var integer: o is 0;
var bigInteger: o1 is 0_;
begin
n1 := n - 1_;
mo := 1_;
for pe range pf do
y := n1 div pe.prime ** pe.exp;
x := modPow(a, y, n);
o := 0;
while x > 1_ do
x := modPow(x, pe.prime, n);
incr(o);
end while;
o1 := pe.prime ** o;
mo *:= o1 div gcd(mo, o1);
end for;
end func;
const func boolean: isProbablyPrime (in bigInteger: primeCandidate, in var integer: count) is func
result
var boolean: isProbablyPrime is TRUE;
local
var bigInteger: aRandomNumber is 0_;
begin
while isProbablyPrime and count > 0 do
aRandomNumber := rand(1_, pred(primeCandidate));
isProbablyPrime := modPow(aRandomNumber, pred(primeCandidate), primeCandidate) = 1_;
decr(count);
end while;
# writeln(count);
end func;
const proc: moTest (in bigInteger: a, in bigInteger: n) is func
begin
if bitLength(a) < 100 then
write("ord(" <& a <& ")");
else
write("ord([big])");
end if;
if bitLength(n) < 100 then
write(" mod " <& n <& " ");
else
write(" mod [big] ");
end if;
if not isProbablyPrime(n, 20) then
writeln("not computed. modulus must be prime for this algorithm.")
else
writeln("= " <& moBachShallit58(a, n, factor(n - 1_)));
end if;
end func;
const proc: main is func
local
var bigInteger: b is 100_;
begin
moTest(37_, 3343_);
moTest(10_ ** 100 + 1_, 7919_);
moTest(10_ ** 1000 + 1_, 15485863_);
moTest(10_ ** 10000 - 1_, 22801763489_);
moTest(1511678068_, 7379191741_);
moTest(3047753288_, 2257683301_);
end func;</syntaxhighlight>
{{out}}
<pre>
ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
</pre>
=={{header|Sidef}}==
Built-in:
<syntaxhighlight lang="ruby">say 37.znorder(1000) #=> 100
say 54.znorder(100001) #=> 9090</syntaxhighlight>
{{trans|Raku}}
<syntaxhighlight lang="ruby">func mo_prime(a, p, e) {
var m = p**e
var t = (p-1)*(p**(e-1))
var qs = [1]
for f in (t.factor_exp) {
qs.map! {|q|
0..f[1] -> map {|j| q * f[0]**j }...
}
}
qs.sort.first_by {|q| powmod(a, q, m) == 1 }
}
func mo(a, m) {
gcd(a, m) == 1 || die "#{a} and #{m} are not relatively prime"
Math.lcm(1, m.factor_exp.map {|r| mo_prime(a, r...) }...)
}
say mo(37, 1000)
say mo(54, 100001)
with (10**20 - 1) {|b|
say mo(2, b)
say mo(17, b)
}</syntaxhighlight>
{{out}}
<pre>
100
9090
3748806900
1499522760
</pre>
=={{header|Tcl}}==
{{trans|Python}}
{{tcllib|struct::list}}
<syntaxhighlight lang="tcl">package require Tcl 8.5
package require struct::list
proc multOrder {a m} {
assert {[gcd $a $m] == 1}
set mofs [list]
dict for {p e} [factor_num $m] {
lappend mofs [multOrdr1 $a $p $e]
}
return [struct::list fold $mofs 1 lcm]
}
proc multOrdr1 {a p e} {
set m [expr {$p ** $e}]
set t [expr {($p - 1) * ($p ** ($e - 1))}]
set qs [dict create 1 ""]
dict for {f0 f1} [factor_num $t] {
dict for {q -} $qs {
foreach j [range [expr {1 + $f1}]] {
dict set qs [expr {$q * $f0 ** $j}] ""
}
}
}
dict for {q -} $qs {
if {pypow($a, $q, $m) == 1} break
}
return $q
}
####################################################
# utility procs
proc assert {condition {message "Assertion failed!"}} {
if { ! [uplevel 1 [list expr $condition]]} {
return -code error $message
}
}
proc gcd {a b} {
while {$b != 0} {
lassign [list $b [expr {$a % $b}]] a b
}
return $a
}
proc lcm {a b} {
expr {$a * $b / [gcd $a $b]}
}
proc factor_num {num} {
primes::restart
set factors [dict create]
for {set i [primes::get_next_prime]} {$i <= $num} {} {
if {$num % $i == 0} {
dict incr factors $i
set num [expr {$num / $i}]
continue
} elseif {$i*$i > $num} {
dict incr factors $num
break
} else {
set i [primes::get_next_prime]
}
}
return $factors
}
####################################################
# a range command akin to Python's
proc range args {
foreach {start stop step} [switch -exact -- [llength $args] {
1 {concat 0 $args 1}
2 {concat $args 1}
3 {concat $args }
default {error {wrong # of args: should be "range ?start? stop ?step?"}}
}] break
if {$step == 0} {error "cannot create a range when step == 0"}
set range [list]
while {$step > 0 ? $start < $stop : $stop < $start} {
lappend range $start
incr start $step
}
return $range
}
# python's pow()
proc ::tcl::mathfunc::pypow {x y {z ""}} {
expr {$z eq "" ? $x ** $y : ($x ** $y) % $z}
}
####################################################
# prime number generator
# ref http://wiki.tcl.tk/5996
####################################################
namespace eval primes {}
proc primes::reset {} {
variable list [list]
variable current_index end
}
namespace eval primes {reset}
proc primes::restart {} {
variable list
variable current_index
if {[llength $list] > 0} {
set current_index 0
}
}
proc primes::is_prime {candidate} {
variable list
foreach prime $list {
if {$candidate % $prime == 0} {
return false
}
if {$prime * $prime > $candidate} {
return true
}
}
while true {
set largest [get_next_prime]
if {$largest * $largest >= $candidate} {
return [is_prime $candidate]
}
}
}
proc primes::get_next_prime {} {
variable list
variable current_index
if {$current_index ne "end"} {
set p [lindex $list $current_index]
if {[incr current_index] == [llength $list]} {
set current_index end
}
return $p
}
switch -exact -- [llength $list] {
0 {set candidate 2}
1 {set candidate 3}
default {
set candidate [lindex $list end]
while true {
incr candidate 2
if {[is_prime $candidate]} break
}
}
}
lappend list $candidate
return $candidate
}
####################################################
puts [multOrder 37 1000] ;# 100
set b [expr {10**20 - 1}]
puts [multOrder 2 $b] ;# 3748806900
puts [multOrder 17 $b] ;# 1499522760
set a 54
set m 100001
puts [set n [multOrder $a $m]] ;# 9090
puts [expr {pypow($a, $n, $m)}] ;# 1
set lambda {{a n m} {expr {pypow($a, $n, $m) == 1}}}
foreach r [lreverse [range 1 $n]] {
if {[apply $lambda $a $r $m]} {
error "Oops, $n is not the smallest: {$a $r $m} satisfies $lambda"
}
if {$r % 1000 == 0} {puts "$r ..."}
}
puts "OK, $n is the smallest n such that {$a $n $m} satisfies $lambda"</syntaxhighlight>
=={{header|Visual Basic .NET}}==
{{trans|C#}}
<syntaxhighlight lang="vbnet">Imports System.Numerics
Imports System.Runtime.CompilerServices
Imports System.Threading
Module Module1
Private s_gen As New ThreadLocal(Of Random)(Function() New Random())
Private Function Gen()
Return s_gen.Value
End Function
<Extension()>
Public Function IsProbablyPrime(value As BigInteger, Optional witnesses As Integer = 10) As Boolean
If value <= 1 Then
Return False
End If
If witnesses <= 0 Then
witnesses = 10
End If
Dim d = value - 1
Dim s = 0
While d Mod 2 = 0
d /= 2
s += 1
End While
Dim bytes(value.ToByteArray.LongLength - 1) As Byte
Dim a As BigInteger
For i = 1 To witnesses
Do
Gen.NextBytes(bytes)
a = New BigInteger(bytes)
Loop While a < 2 OrElse a >= value - 2
Dim x = BigInteger.ModPow(a, d, value)
If x = 1 OrElse x = value - 1 Then
Continue For
End If
For r = 1 To s - 1
x = BigInteger.ModPow(x, 2, value)
If x = 1 Then
Return False
End If
If x = value - 1 Then
Exit For
End If
Next
If x <> value - 1 Then
Return False
End If
Next
Return True
End Function
<Extension()>
Function Sqrt(self As BigInteger) As BigInteger
Dim b = self
While True
Dim a = b
b = self / a + a >> 1
If b >= a Then
Return a
End If
End While
Throw New Exception("Should not have happened")
End Function
<Extension()>
Function BitLength(self As BigInteger) As Long
Dim bi = self
Dim len = 0L
While bi <> 0
len += 1
bi >>= 1
End While
Return len
End Function
<Extension()>
Function BitTest(self As BigInteger, pos As Integer) As Boolean
Dim arr = self.ToByteArray
Dim i = pos \ 8
Dim m = pos Mod 8
If i >= arr.Length Then
Return False
End If
Return (arr(i) And (1 << m)) > 0
End Function
Class PExp
Sub New(p As BigInteger, e As Integer)
Prime = p
Exp = e
End Sub
Public ReadOnly Property Prime As BigInteger
Public ReadOnly Property Exp As Integer
End Class
Function MoBachShallit58(a As BigInteger, n As BigInteger, pf As List(Of PExp)) As BigInteger
Dim n1 = n - 1
Dim mo As BigInteger = 1
For Each pe In pf
Dim y = n1 / BigInteger.Pow(pe.Prime, pe.Exp)
Dim o = 0
Dim x = BigInteger.ModPow(a, y, BigInteger.Abs(n))
While x > 1
x = BigInteger.ModPow(x, pe.Prime, BigInteger.Abs(n))
o += 1
End While
Dim o1 = BigInteger.Pow(pe.Prime, o)
o1 /= BigInteger.GreatestCommonDivisor(mo, o1)
mo *= o1
Next
Return mo
End Function
Function Factor(n As BigInteger) As List(Of PExp)
Dim pf As New List(Of PExp)
Dim nn = n
Dim e = 0
While Not nn.BitTest(e)
e += 1
End While
If e > 0 Then
nn >>= e
pf.Add(New PExp(2, e))
End If
Dim s = nn.Sqrt
Dim d As BigInteger = 3
While nn > 1
If d > s Then
d = nn
End If
e = 0
While True
Dim remainder As New BigInteger
Dim div = BigInteger.DivRem(nn, d, remainder)
If remainder.BitLength > 0 Then
Exit While
End If
nn = div
e += 1
End While
If e > 0 Then
pf.Add(New PExp(d, e))
s = nn.Sqrt
End If
d += 2
End While
Return pf
End Function
Sub MoTest(a As BigInteger, n As BigInteger)
If Not n.IsProbablyPrime(20) Then
Console.WriteLine("Not computed. Modulus must be prime for this algorithm.")
Return
End If
If a.BitLength < 100 Then
Console.Write("ord({0})", a)
Else
Console.Write("ord([big])")
End If
If n.BitLength < 100 Then
Console.Write(" mod {0}", n)
Else
Console.Write(" mod [big]")
End If
Dim mob = MoBachShallit58(a, n, Factor(n - 1))
Console.WriteLine(" = {0}", mob)
End Sub
Sub Main()
MoTest(37, 3343)
MoTest(BigInteger.Pow(10, 100) + 1, 7919)
MoTest(BigInteger.Pow(10, 1000) + 1, 15485863)
MoTest(BigInteger.Pow(10, 10000) - 1, 22801763489)
MoTest(1511678068, 7379191741)
MoTest(3047753288, 2257683301)
End Sub
End Module</syntaxhighlight>
{{out}}
<pre>ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425</pre>
=={{header|Wren}}==
{{trans|Kotlin}}
{{libheader|Wren-big}}
<syntaxhighlight lang="wren">import "./big" for BigInt
class PExp {
construct new(prime, exp) {
_prime = prime
_exp = exp
}
prime { _prime }
exp { _exp }
}
var moBachShallit58 = Fn.new { |a, n, pf|
var n1 = n - BigInt.one
var mo = BigInt.one
for (pe in pf) {
var y = n1 / pe.prime.pow(pe.exp)
var o = 0
var x = a.modPow(y, n.abs)
while (x > BigInt.one) {
x = x.modPow(pe.prime, n.abs)
o = o + 1
}
var o1 = BigInt.new(o)
o1 = pe.prime.pow(o1)
o1 = o1 / BigInt.gcd(mo, o1)
mo = mo * o1
}
return mo
}
var factor = Fn.new { |n|
var pf = []
var nn = n.copy()
var e = 0
while (!nn.testBit(e)) e = e + 1
if (e > 0) {
nn = nn >> e
pf.add(PExp.new(BigInt.two, e))
}
var s = nn.isqrt
var d = BigInt.three
while (nn > BigInt.one) {
if (d > s) d = nn
e = 0
while (true) {
var dm = nn.divMod(d)
if (dm[1].bitLength > 0) break
nn = dm[0]
e = e + 1
}
if (e > 0) {
pf.add(PExp.new(d, e))
s = nn.isqrt
}
d = d + BigInt.two
}
return pf
}
var moTest = Fn.new { |a, n|
if (!n.isProbablePrime(10)) {
System.print("Not computed. Modulus must be prime for this algorithm.")
return
}
System.write((a.bitLength < 100) ? "ord(%(a))" : "ord([big])")
System.write((n.bitLength < 100) ? " mod %(n) " : "mod([big])")
var mob = moBachShallit58.call(a, n, factor.call(n - BigInt.one))
System.print("= %(mob)")
}
moTest.call(BigInt.new(37), BigInt.new(3343))
var b = BigInt.ten.pow(100) + BigInt.one
moTest.call(b, BigInt.new(7919))
b = BigInt.ten.pow(1000) + BigInt.one
moTest.call(b, BigInt.new(15485863))
b = BigInt.ten.pow(10000) - BigInt.one
moTest.call(b, BigInt.new(22801763489))
moTest.call(BigInt.new(1511678068), BigInt.new(7379191741))
moTest.call(BigInt.new(3047753288), BigInt.new(2257683301))</syntaxhighlight>
{{out}}
<pre>
ord(37) mod 3343 = 1114
ord([big]) mod 7919 = 3959
ord([big]) mod 15485863 = 15485862
ord([big]) mod 22801763489 = 22801763488
ord(1511678068) mod 7379191741 = 614932645
ord(3047753288) mod 2257683301 = 62713425
</pre>
=={{header|zkl}}==
{{trans|Python}}
Using [[Extensible prime generator#zkl]] and the GMP library for lcm (least common multiple), pow and powm ((n^e)%m)
It would probably be nice to memoize the prime numbers but that isn't necessary for this task.
<syntaxhighlight lang="zkl">var BN =Import("zklBigNum");
var Sieve=Import("sieve");
// factor n into powers of primes
// eg 9090 == 2^1 * 3^2 * 5^1 * 101^1
fcn factor2PP(n){ // lazy factors using lazy primes --> (prime,power) ...
Utils.Generator(fcn(a){
primes:=Utils.Generator(Sieve.postponed_sieve);
foreach p in (primes){
e:=0; while(a%p == 0){ a /= p; e+=1; }
if (e) vm.yield(p,e);
if (a<p*p) break;
}
if (a>1) vm.yield(a,1);
},n)
}
fcn _multOrdr1(a,p,e){
m:=p.pow(e);
t:=m/p*(p - 1);
qs:=L(BN(1));
foreach p2,e2 in (factor2PP(t)){
qs=[[(e,q); [0..e2]; qs; '{ q*BN(p2).pow(e) }]];
}
qs.filter1('wrap(q){ a.powm(q,m)==1 });
}
fcn multiOrder(a,m){
if (m.gcd(a)!=1) throw(Exception.ValueError("Not co-prime"));
res:=BN(1);
foreach p,e in (factor2PP(m)){
res = res.lcm(_multOrdr1(BN(a),BN(p),e));
}
return(res);
}</syntaxhighlight>
<syntaxhighlight lang="zkl">multiOrder(37,1000).println();
b:=BN(10).pow(20)-1;
multiOrder(2,b).println();
multiOrder(17,b).println();
b=0d10_0001;
[BN(1)..multiOrder(54,b)-1].filter1('wrap(r,b54){b54.powm(r,b)==1},BN(54)) :
if (_) println("Exists a power r < 9090 where (54^r)%b)==1");
else println("Everything checks.");</syntaxhighlight>
{{out}}
<pre>
100
3748806900
1499522760
Everything checks.
</pre>
| |||