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Ingy döt Net 2023-07-01 11:58:00 -04:00
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---
from: http://rosettacode.org/wiki/Multiplicative_order
note: Discrete math

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The '''multiplicative order''' of ''a'' relative to ''m'' is the least positive integer ''n'' such that ''a^n'' is 1 (modulo ''m'').
;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>. &nbsp; Show how to compute the order of an
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>. &nbsp; (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. Now, using the binary method,
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>

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Task/Multiplicative-order/11l/multiplicative-order.11l Normal file
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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 bit_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 bit_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’))

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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
)

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Task/Multiplicative-order/Ada/multiplicative-order-1.ada Normal file
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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;

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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;

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Task/Multiplicative-order/Ada/multiplicative-order-3.ada Normal file
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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;

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#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;
}

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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);
}
}
}

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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("%lu\n", multi_order(37, 1000));
printf("%lu\n", multi_order(54, 100001));
return 0;
}

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Task/Multiplicative-order/Clojure/multiplicative-order.clj Normal file
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(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))))

156
Task/Multiplicative-order/D/multiplicative-order.d Normal file
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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);
}

32
Task/Multiplicative-order/EchoLisp/multiplicative-order.l Normal file
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(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

11
Task/Multiplicative-order/Factor/multiplicative-order.factor Normal file
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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 ;

102
Task/Multiplicative-order/Go/multiplicative-order.go Normal file
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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)
}

15
Task/Multiplicative-order/Haskell/multiplicative-order-1.hs Normal file
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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"

17
Task/Multiplicative-order/Haskell/multiplicative-order-2.hs Normal file
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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

6
Task/Multiplicative-order/J/multiplicative-order-1.j Normal file
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mo=: 4 : 0
a=. x: x
m=. x: y
assert. 1=a+.m
*./ a mopk"1 |: __ q: m
)

10
Task/Multiplicative-order/J/multiplicative-order-2.j Normal file
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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
)

4
Task/Multiplicative-order/J/multiplicative-order-3.j Normal file
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37 mo 1000
100
2 mo _1+10^80x
190174169488577769580266953193403101748804183400400

106
Task/Multiplicative-order/Java/multiplicative-order.java Normal file
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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.");
return;
}
if (a.bitLength() < 100) System.out.printf("ord(%s)", a);
else System.out.print("ord([big])");
if (n.bitLength() < 100) System.out.printf(" mod %s ", n);
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));
}
}

25
Task/Multiplicative-order/Julia/multiplicative-order.julia Normal file
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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

94
Task/Multiplicative-order/Kotlin/multiplicative-order.kotlin Normal file
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// 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"))
}

1
Task/Multiplicative-order/Maple/multiplicative-order-1.maple Normal file
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numtheory:-order( a, n )

2
Task/Multiplicative-order/Maple/multiplicative-order-2.maple Normal file
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> numtheory:-order( 37, 1000 );
100

6
Task/Multiplicative-order/Mathematica/multiplicative-order.math Normal file
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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]

17
Task/Multiplicative-order/Maxima/multiplicative-order.maxima Normal file
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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 */

84
Task/Multiplicative-order/Nim/multiplicative-order.nim Normal file
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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"))

1
Task/Multiplicative-order/PARI-GP/multiplicative-order.parigp Normal file
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znorder(Mod(a,n))

3
Task/Multiplicative-order/Perl/multiplicative-order-1.pl Normal file
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use ntheory qw/znorder/;
say znorder(54, 100001);
use bigint; say znorder(11, 1 + 10**100);

3
Task/Multiplicative-order/Perl/multiplicative-order-2.pl Normal file
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use Math::Pari qw/znorder Mod/;
say znorder(Mod(54, 100001));
say znorder(Mod(11, 1 + Math::Pari::PARI(10)**100));

115
Task/Multiplicative-order/Phix/multiplicative-order.phix Normal file
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(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>
<!--

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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.'

27
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/*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. */

25
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#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)))

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100
3959
15485862
22801763488
109609547199756140150989321269669269476675495992554276140800

32
Task/Multiplicative-order/Raku/multiplicative-order.raku Normal file
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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';
}

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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

118
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$ 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;

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Task/Multiplicative-order/Sidef/multiplicative-order-1.sidef Normal file
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say 37.znorder(1000) #=> 100
say 54.znorder(100001) #=> 9090

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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)
}

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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"

191
Task/Multiplicative-order/Visual-Basic-.NET/multiplicative-order.vb Normal file
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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

83
Task/Multiplicative-order/Wren/multiplicative-order.wren Normal file
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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))

35
Task/Multiplicative-order/Zkl/multiplicative-order-1.zkl Normal file
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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);
}

9
Task/Multiplicative-order/Zkl/multiplicative-order-2.zkl Normal file
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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.");