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Montgomery reduction (view source)

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Return (A)
 
=={{header|C++11l}}==
{{trans|Python}}
<lang cpp>#include<iostream>
#include<conio.h>
using namespace std;
typedef unsigned long ulong;
 
<syntaxhighlight lang="11l">T Montgomery
int ith_digit_finder(long long n, long b, long i){
BigInt m
/**
Int n
n = number whose digits we need to extract
BigInt rrm
b = radix in which the number if represented
 
i = the ith bit (ie, index of the bit that needs to be extracted)
F (m)
**/
while(i>0){ .m = m
.n/ =b; bits:length(m)
.rrm = i--;(BigInt(2) ^ (.n * 2)) % m
 
F reduce(t)
V a = t
L 0 .< .n
I (a % 2) == 1
a += .m
a I/= 2
I a >= .m
a -= .m
R a
 
V m = BigInt(‘750791094644726559640638407699’)
V x1 = BigInt(‘540019781128412936473322405310’)
V x2 = BigInt(‘515692107665463680305819378593’)
 
V mont = Montgomery(m)
V t1 = x1 * mont.rrm
V t2 = x2 * mont.rrm
 
V r1 = mont.reduce(t1)
V r2 = mont.reduce(t2)
V r = BigInt(2) ^ mont.n
 
print(‘b : 2’)
print(‘n : ’mont.n)
print(‘r : ’r)
print(‘m : ’mont.m)
print(‘t1: ’t1)
print(‘t2: ’t2)
print(‘r1: ’r1)
print(‘r2: ’r2)
print()
print(‘Original x1 : ’x1)
print(‘Recovered from r1 : ’mont.reduce(r1))
print(‘Original x2 : ’x2)
print(‘Recovered from r2 : ’mont.reduce(r2))
 
print("\nMontgomery computation of x1 ^ x2 mod m:")
V prod = mont.reduce(mont.rrm)
V base = mont.reduce(x1 * mont.rrm)
V ex = x2
L bits:length(ex) > 0
I (ex % 2) == 1
prod = mont.reduce(prod * base)
ex I/= 2
base = mont.reduce(base * base)
print(mont.reduce(prod))
print("\nAlternate computation of x1 ^ x2 mod m:")
print(pow(x1, x2, m))</syntaxhighlight>
 
{{out}}
<pre>
b : 2
n : 100
r : 1267650600228229401496703205376
m : 750791094644726559640638407699
t1: 323165824550862327179367294465482435542970161392400401329100
t2: 308607334419945011411837686695175944083084270671482464168730
r1: 440160025148131680164261562101
r2: 435362628198191204145287283255
 
Original x1 : 540019781128412936473322405310
Recovered from r1 : 540019781128412936473322405310
Original x2 : 515692107665463680305819378593
Recovered from r2 : 515692107665463680305819378593
 
Montgomery computation of x1 ^ x2 mod m:
151232511393500655853002423778
 
Alternate computation of x1 ^ x2 mod m:
151232511393500655853002423778
</pre>
 
=={{header|C}}==
<syntaxhighlight lang="c">#include <stdio.h>
#include <stdlib.h>
 
typedef unsigned long long ulong;
 
const int BASE = 2;
 
struct Montgomery {
ulong m;
ulong rrm;
int n;
};
 
int bitLength(ulong v) {
int result = 0;
while (v > 0) {
v >>= 1;
result++;
}
return (n%b)result;
}
 
ulong modPow(ulong b, ulong e, ulong n) {
long eeuclid(long m, long b, long *inverse){ /// eeuclid( modulus, num whose inv is to be found, variable to put inverse )
ulong result = 1;
/// Algorithm used from Stallings book
long A1 = 1, A2 = 0, A3 = m,
B1 = 0, B2 = 1, B3 = b,
T1, T2, T3, Q;
 
if (n == 1) {
cout<<endl<<"eeuclid() started"<<endl;
return 0;
}
 
b = b % while(1){n;
while if(B3e ==> 0) {
if (e % 2 == 1) *inverse = 0;{
result = (result * returnb) A3% n; // A3 = gcd(m,b)
}
e >>= 1;
b = (b * b) % n;
}
 
return result;
if(B3 == 1){
}
*inverse = B2; // B2 = b^-1 mod m
return B3; // A3 = gcd(m,b)
}
 
struct Montgomery makeMontgomery(ulong m) {
Q = A3/B3;
struct Montgomery result;
 
if (m == 0 || (m & T11) == A1 -0) Q*B1;{
fprintf(stderr, "m must be T2greater =than A2zero -and Q*B2odd");
T3 = A3 - Q*B3exit(1);
}
 
A1 = B1; A2 = B2; A3result.m = B3m;
result.n = bitLength(m);
B1 = T1; B2 = T2; B3 = T3;
result.rrm = (1ULL << (result.n * 2)) % m;
 
return }result;
cout<<endl<<"ending eeuclid() "<<endl;
}
 
ulong reduce(struct Montgomery mont, ulong t) {
long long mon_red(long m, long m_dash, long T, int n, long b = 2){
ulong a = t;
/**
mint = modulusi;
 
m_dash = m' = -m^-1 mod b
for (i = 0; i < mont.n; i++) {
T = number whose modular reduction is needed, the o/p of the function is TR^-1 mod m
n = number of bits in mif (2n(a is& the1) number== of1) bits in T){
a += mont.m;
b = radix used (for practical implementations, is equal to 2, which is the default value)
}
**/
a = a >> 1;
long long A,ui, temp, Ai; // Ai is the ith bit of A, need not be llong long probably
if( m_dash < 0 ) m_dash = m_dash + b;
A = T;
for(int i = 0; i<n; i++){
/// ui = ( (A%b)*m_dash ) % b; // step 2.1; A%b gives ai (MISTAKE -- A%b will always give the last digit of A if A is represented in base b); hence we need the function ith_digit_finder()
Ai = ith_digit_finder(A, b, i);
ui = ( ( Ai % b) * m_dash ) % b;
temp = ui*m*power(b, i);
A = A + temp;
}
 
A = A/power(b, n);
if (Aa >= mont.m) A = A - m;{
return A a -= mont.m;
}
return a;
}
 
void check(ulong x1, ulong x2, ulong m) {
int main(){
struct Montgomery mont = makeMontgomery(m);
long a, b, c, d=0, e, inverse = 0;
cout<<"mulong >>t1 "= x1 * mont.rrm;
cinulong >>t2 a= x2 * mont.rrm;
 
cout<<"T >> ";
ulong r1 = reduce(mont, t1);
cin>>b;
ulong r2 = reduce(mont, t2);
cout<<"Radix b >> ";
ulong r = 1ULL << mont.n;
cin>>c;
 
eeuclid(c, a, &d); // eeuclid( modulus, num whose inverse is to be found, address of variable which is to store inverse)
printf("b : %d\n", BASE);
e = mon_red(a, -d, b, length_finder(a, c), c);
printf("n : %d\n", mont.n);
cout<<"Montgomery domain representation = "<<e;
printf("r : %llu\n", r);
printf("m : %llu\n", mont.m);
printf("t1: %llu\n", t1);
printf("t2: %llu\n", t2);
printf("r1: %llu\n", r1);
printf("r2: %llu\n", r2);
printf("\n");
printf("Original x1 : %llu\n", x1);
printf("Recovered from r1 : %llu\n", reduce(mont, r1));
printf("Original x2 : %llu\n", x2);
printf("Recovered from r2 : %llu\n", reduce(mont, r2));
 
printf("\nMontgomery computation of x1 ^ x2 mod m :\n");
ulong prod = reduce(mont, mont.rrm);
ulong base = reduce(mont, x1 * mont.rrm);
ulong exp = x2;
while (bitLength(exp) > 0) {
if ((exp & 1) == 1) {
prod = reduce(mont, prod * base);
}
exp >>= 1;
base = reduce(mont, base * base);
}
printf("%llu\n", reduce(mont, prod));
printf("\nAlternate computation of x1 ^ x2 mod m :\n");
printf("%llu\n", modPow(x1, x2, m));
}
 
int main() {
check(41, 11, 1549);
 
return 0;
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
n : 11
r : 2048
m : 1549
t1: 47601
t2: 12771
r1: 322
r2: 842
 
Original x1 : 41
Recovered from r1 : 41
Original x2 : 11
Recovered from r2 : 11
 
Montgomery computation of x1 ^ x2 mod m :
678
 
Alternate computation of x1 ^ x2 mod m :
678</pre>
 
=={{header|C#|C sharp|C#}}==
{{trans|D}}
<langsyntaxhighlight lang="csharp">using System;
using System.Numerics;
 
Line 198 ⟶ 334:
}
}
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 219 ⟶ 355:
Alternate computation of x1 ^ x2 mod m :
151232511393500655853002423778</pre>
 
=={{header|C++}}==
<syntaxhighlight lang="cpp">#include<iostream>
#include<conio.h>
using namespace std;
typedef unsigned long ulong;
 
int ith_digit_finder(long long n, long b, long i){
/**
n = number whose digits we need to extract
b = radix in which the number if represented
i = the ith bit (ie, index of the bit that needs to be extracted)
**/
while(i>0){
n/=b;
i--;
}
return (n%b);
}
 
long eeuclid(long m, long b, long *inverse){ /// eeuclid( modulus, num whose inv is to be found, variable to put inverse )
/// Algorithm used from Stallings book
long A1 = 1, A2 = 0, A3 = m,
B1 = 0, B2 = 1, B3 = b,
T1, T2, T3, Q;
 
cout<<endl<<"eeuclid() started"<<endl;
 
while(1){
if(B3 == 0){
*inverse = 0;
return A3; // A3 = gcd(m,b)
}
 
if(B3 == 1){
*inverse = B2; // B2 = b^-1 mod m
return B3; // A3 = gcd(m,b)
}
 
Q = A3/B3;
 
T1 = A1 - Q*B1;
T2 = A2 - Q*B2;
T3 = A3 - Q*B3;
 
A1 = B1; A2 = B2; A3 = B3;
B1 = T1; B2 = T2; B3 = T3;
 
}
cout<<endl<<"ending eeuclid() "<<endl;
}
 
long long mon_red(long m, long m_dash, long T, int n, long b = 2){
/**
m = modulus
m_dash = m' = -m^-1 mod b
T = number whose modular reduction is needed, the o/p of the function is TR^-1 mod m
n = number of bits in m (2n is the number of bits in T)
b = radix used (for practical implementations, is equal to 2, which is the default value)
**/
long long A,ui, temp, Ai; // Ai is the ith bit of A, need not be llong long probably
if( m_dash < 0 ) m_dash = m_dash + b;
A = T;
for(int i = 0; i<n; i++){
/// ui = ( (A%b)*m_dash ) % b; // step 2.1; A%b gives ai (MISTAKE -- A%b will always give the last digit of A if A is represented in base b); hence we need the function ith_digit_finder()
Ai = ith_digit_finder(A, b, i);
ui = ( ( Ai % b) * m_dash ) % b;
temp = ui*m*power(b, i);
A = A + temp;
}
A = A/power(b, n);
if(A >= m) A = A - m;
return A;
}
 
int main(){
long a, b, c, d=0, e, inverse = 0;
cout<<"m >> ";
cin >> a;
cout<<"T >> ";
cin>>b;
cout<<"Radix b >> ";
cin>>c;
eeuclid(c, a, &d); // eeuclid( modulus, num whose inverse is to be found, address of variable which is to store inverse)
e = mon_red(a, -d, b, length_finder(a, c), c);
cout<<"Montgomery domain representation = "<<e;
return 0;
}</syntaxhighlight>
 
=={{header|D}}==
{{trans|Kotlin}}
<langsyntaxhighlight Dlang="d">import std.bigint;
import std.stdio;
 
Line 320 ⟶ 544:
writeln("\nAlternate computation of x1 ^ x2 mod m :");
writeln(x1.modPow(x2, m));
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 341 ⟶ 565:
Alternate computation of x1 ^ x2 mod m :
151232511393500655853002423778</pre>
 
=={{header|Factor}}==
{{trans|Sidef}}
{{works with|Factor|0.99 2020-08-14}}
<syntaxhighlight lang="factor">USING: io kernel locals math math.bitwise math.functions
prettyprint ;
 
: montgomery-reduce ( m a -- n )
over bit-length [ dup odd? [ over + ] when 2/ ] times
swap mod ;
 
CONSTANT: m 750791094644726559640638407699
CONSTANT: t1 323165824550862327179367294465482435542970161392400401329100
 
CONSTANT: r1 440160025148131680164261562101
CONSTANT: r2 435362628198191204145287283255
 
CONSTANT: x1 540019781128412936473322405310
CONSTANT: x2 515692107665463680305819378593
 
"Original x1: " write x1 .
"Recovered from r1: " write m r1 montgomery-reduce .
"Original x2: " write x2 .
"Recovered from r2: " write m r2 montgomery-reduce .
 
nl "Montgomery computation of x1^x2 mod m: " write
 
[let
m t1 x1 / montgomery-reduce :> prod!
m t1 montgomery-reduce :> base!
x2 :> exponent!
 
[ exponent zero? ] [
exponent odd?
[ m prod base * montgomery-reduce prod! ] when
m base sq montgomery-reduce base! exponent 2/ exponent!
] until
 
m prod montgomery-reduce .
"Library-based computation of x1^x2 mod m: " write
x1 x2 m ^mod .
]</syntaxhighlight>
{{out}}
<pre>
Original x1: 540019781128412936473322405310
Recovered from r1: 540019781128412936473322405310
Original x2: 515692107665463680305819378593
Recovered from r2: 515692107665463680305819378593
 
Montgomery computation of x1^x2 mod m: 151232511393500655853002423778
Library-based computation of x1^x2 mod m: 151232511393500655853002423778
</pre>
 
=={{header|Go}}==
<langsyntaxhighlight lang="go">package main
 
import (
Line 450 ⟶ 726:
fmt.Println("\nLibrary-based computation of x1 ^ x2 mod m:")
fmt.Println(new(big.Int).Exp(x1, x2, m))
}</langsyntaxhighlight>
{{out}}
<pre>
Line 476 ⟶ 752:
=={{header|Java}}==
{{trans|Kotlin}}
<langsyntaxhighlight lang="java">import java.math.BigInteger;
 
public class MontgomeryReduction {
Line 553 ⟶ 829:
System.out.println(x1.modPow(x2, m));
}
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 574 ⟶ 850:
Library-based computation of x1 ^ x2 mod m :
151232511393500655853002423778</pre>
 
=={{header|Julia}}==
{{trans|Python}}
<syntaxhighlight lang="julia">""" base 2 type Montgomery numbers """
struct Montgomery2
m::BigInt
n::Int64
rrm::BigInt
end
 
function Montgomery2(x::BigInt)
bitlen = length(string(x, base=2))
r = (x == 0) ? 0 : (BigInt(1) << (bitlen * 2)) % x
Montgomery2(x, bitlen, r)
end
Montgomery2(n) = Montgomery2(BigInt(n))
 
function reduce(mm::Montgomery2, t)
a = BigInt(t)
for i in 1:mm.n
if isodd(a)
a += mm.m
end
a >>= 1
end
return a >= mm.m ? a - mm.m : a
end
 
BASE(::Montgomery2) = 2
 
const mmm = Montgomery2(20)
 
function testmontgomery2()
m = big"750791094644726559640638407699"
x1 = big"540019781128412936473322405310"
x2 = big"515692107665463680305819378593"
 
mont = Montgomery2(m)
t1 = x1 * mont.rrm
t2 = x2 * mont.rrm
r1 = reduce(mont, t1)
r2 = reduce(mont, t2)
r = 1 << mont.n
println("b : ", BASE(mont))
println("n : ", mont.n)
println("r : ", r)
println("m : ", mont.m)
println("t1: ", t1)
println("t2: ", t2)
println("r1: ", r1)
println("r2: ", r2)
println()
println("Original x1 :", x1)
println("Recovered from r1 :", reduce(mont, r1))
println("Original x2 :", x2)
println("Recovered from r2 :", reduce(mont, r2))
println("\nMontgomery computation of x1 ^ x2 mod m:")
prod = reduce(mont, mont.rrm)
base = reduce(mont, x1 * mont.rrm)
pow = x2
while pow > 0
if isodd(pow)
prod = reduce(mont, prod * base)
end
pow >>= 1
base = reduce(mont, base * base)
end
println(reduce(mont, prod))
println("\nAlternate computation of x1 ^ x2 mod m :")
println(powermod(x1, x2, m))
end
 
testmontgomery2()
</syntaxhighlight>{{out}}
<pre>
b : 2
n : 100
r : 0
m : 750791094644726559640638407699
t1: 323165824550862327179367294465482435542970161392400401329100
t2: 308607334419945011411837686695175944083084270671482464168730
r1: 440160025148131680164261562101
r2: 435362628198191204145287283255
 
Original x1 :540019781128412936473322405310
Recovered from r1 :540019781128412936473322405310
Original x2 :515692107665463680305819378593
Recovered from r2 :515692107665463680305819378593
 
Montgomery computation of x1 ^ x2 mod m:
151232511393500655853002423778
 
Alternate computation of x1 ^ x2 mod m :
151232511393500655853002423778
</pre>
 
=={{header|Kotlin}}==
{{trans|Go}}
<langsyntaxhighlight lang="scala">// version 1.1.3
 
import java.math.BigInteger
Line 649 ⟶ 1,020:
println("\nLibrary-based computation of x1 ^ x2 mod m :")
println(x1.modPow(x2, m))
}</langsyntaxhighlight>
 
{{out}}
Line 673 ⟶ 1,044:
151232511393500655853002423778
</pre>
 
=={{header|Nim}}==
{{trans|D}}
{{libheader|bignum}}
<syntaxhighlight lang="nim">import bignum
 
# Missing functions in "bignum".
 
template isOdd(val: Int): bool =
## Needed as bignum "odd" function crashes.
(val and 1) != 0
 
func exp(x, y, m: Int): Int =
## Missing "exp" function in "bignum".
if m == 1: return newInt(0)
result = newInt(1)
var x = x mod m
var y = y
while y > 0:
if y.isOdd:
result = (result * x) mod m
y = y shr 1
x = (x * x) mod m
 
 
type Montgomery = object
m: Int # Modulus; must be odd.
n: int # m.bitLen().
rrm: Int # (1<<2n) mod m.
 
const Base = 2
 
 
func initMontgomery(m: Int): Montgomery =
## Initialize a Mongtgomery object.
doAssert m > 0 and m.isOdd, "argument must be positive and odd."
result.m = m
result.n = m.bitLen
result.rrm = newInt(1) shl culong(result.n * 2) mod m
 
 
func reduce(mont: Montgomery; t: Int): Int =
## Montgomery reduction algorithm.
result = t
for i in 0..<mont.n:
if result.isOdd: inc result, mont.m
result = result shr 1
if result >= mont.m: dec result, mont.m
 
 
when isMainModule:
 
let
m = newInt("750791094644726559640638407699")
x1 = newInt("540019781128412936473322405310")
x2 = newInt("515692107665463680305819378593")
 
mont = initMontgomery(m)
t1 = x1 * mont.rrm
t2 = x2 * mont.rrm
 
r1 = mont.reduce(t1)
r2 = mont.reduce(t2)
r = newInt(1) shl culong(mont.n)
 
echo "b: ", Base
echo "n: ", mont.n
echo "r: ", r
echo "m: ", mont.m
echo "t1: ", t1
echo "t2: ", t2
echo "r1: ", r1
echo "r2: ", r2
echo()
echo "Original x1: ", x1
echo "Recovered from r1: ", mont.reduce(r1)
echo "Original x2: ", x2
echo "Recovered from r2: ", mont.reduce(r2)
 
echo "\nMontgomery computation of x1^x2 mod m:"
var
prod = mont.reduce(mont.rrm)
base = mont.reduce(x1 * mont.rrm)
e = x2
while e > 0:
if e.isOdd: prod = mont.reduce(prod * base)
e = e shr 1
base = mont.reduce(base * base)
echo mont.reduce(prod)
echo "\nAlternate computation of x1^x2 mod m:"
echo x1.exp(x2, m)</syntaxhighlight>
 
{{out}}
<pre>b: 2
n: 100
r: 1267650600228229401496703205376
m: 750791094644726559640638407699
t1: 323165824550862327179367294465482435542970161392400401329100
t2: 308607334419945011411837686695175944083084270671482464168730
r1: 440160025148131680164261562101
r2: 435362628198191204145287283255
 
Original x1: 540019781128412936473322405310
Recovered from r1: 540019781128412936473322405310
Original x2: 515692107665463680305819378593
Recovered from r2: 515692107665463680305819378593
 
Montgomery computation of x1^x2 mod m:
151232511393500655853002423778
 
Alternate computation of x1^x2 mod m:
151232511393500655853002423778</pre>
 
=={{header|Perl}}==
{{trans|Perl 6Raku}}
{{libheader|ntheory}}
<langsyntaxhighlight lang="perl">use bigint;
use ntheory qw(powmod);
 
Line 720 ⟶ 1,203:
 
print montgomery_reduce($m, $prod) . "\n";
printf "Built-in op computation x1**x2 mod m: %s\n", powmod($x1, $x2, $m);</langsyntaxhighlight>
{{out}}
<pre>Original x1: 540019781128412936473322405310
Line 729 ⟶ 1,212:
Montgomery computation x1**x2 mod m: 151232511393500655853002423778
Built-in op computation x1**x2 mod m: 151232511393500655853002423778</pre>
 
=={{header|Perl 6}}==
{{works with|Rakudo|2018.03}}
{{trans|Sidef}}
 
<lang perl6>sub montgomery-reduce($m, $a is copy) {
for 0..$m.msb {
$a += $m if $a +& 1;
$a +>= 1
}
$a % $m
}
 
my $m = 750791094644726559640638407699;
my $t1 = 323165824550862327179367294465482435542970161392400401329100;
 
my $r1 = 440160025148131680164261562101;
my $r2 = 435362628198191204145287283255;
 
my $x1 = 540019781128412936473322405310;
my $x2 = 515692107665463680305819378593;
 
say "Original x1: ", $x1;
say "Recovered from r1: ", montgomery-reduce($m, $r1);
say "Original x2: ", $x2;
say "Recovered from r2: ", montgomery-reduce($m, $r2);
 
print "\nMontgomery computation x1**x2 mod m: ";
my $prod = montgomery-reduce($m, $t1/$x1);
my $base = montgomery-reduce($m, $t1);
 
for $x2, {$_ +> 1} ... 0 -> $exponent {
$prod = montgomery-reduce($m, $prod * $base) if $exponent +& 1;
$base = montgomery-reduce($m, $base * $base);
}
 
say montgomery-reduce($m, $prod);
say "Built-in op computation x1**x2 mod m: ", $x1.expmod($x2, $m);</lang>
{{out}}
<pre>Original x1: 540019781128412936473322405310
Recovered from r1: 540019781128412936473322405310
Original x2: 515692107665463680305819378593
Recovered from r2: 515692107665463680305819378593
 
Montgomery computation x1**x2 mod m: 151232511393500655853002423778
Built-in op computation x1**x2 mod m: 151232511393500655853002423778
</pre>
 
=={{header|Phix}}==
{{trans|D}}
{{libheader|Phix/mpfr}}
<lang Phix>include builtins\bigint.e
<!--<syntaxhighlight lang="phix">(phixonline)-->
 
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
function bitLength(bigint v)
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
if bi_sign(v)=-1 then v = bi_neg(v) end if
integer result = 0
<span style="color: #008080;">enum</span> <span style="color: #000000;">BASE</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">BITLEN</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">MODULUS</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">RRM</span>
while bi_compare(v,0)!=0 do
{v,?} = bi_div3(v, 2)
<span style="color: #008080;">function</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">mpz</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
result += 1
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">BITLEN</span><span style="color: #0000FF;">]</span>
end while
<span style="color: #004080;">mpz</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">MODULUS</span><span style="color: #0000FF;">],</span>
return result
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
end function
<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: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_odd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">mpz_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_fdiv_q_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)>=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #7060A8;">mpz_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</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;">r</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">Montgomery</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
enum BASE, BITLEN, MODULUS, RRM
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_sign</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #7060A8;">crash</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"must be positive"</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
 
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #7060A8;">mpz_odd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #7060A8;">crash</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"must be odd"</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
function reduce(sequence mont, bigint a)
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_sizeinbase</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
integer n = mont[BITLEN]
<span style="color: #004080;">mpz</span> <span style="color: #000000;">rrm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
bigint m = mont[MODULUS]
<span style="color: #7060A8;">mpz_powm_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rrm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rrm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
for i=1 to n do
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">-- BASE</span>
if bi_mod(a,2)=BI_ONE then -- odd
<span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">-- BITLEN</span>
a = bi_add(a,m)
<span style="color: #000000;">m</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">-- MODULUS</span>
end if
<span style="color: #000000;">rrm</span> <span style="color: #000080;font-style:italic;">-- 1&lt;&lt;(n*2) % m</span>
{a,?} = bi_div3(a,2)
<span style="color: #0000FF;">}</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
if bi_compare(a,m)>=0 then a = bi_sub(a,m) end if
return a
<span style="color: #004080;">mpz</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"750791094644726559640638407699"</span><span style="color: #0000FF;">),</span>
end function
<span style="color: #000000;">x1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"540019781128412936473322405310"</span><span style="color: #0000FF;">),</span>
 
<span style="color: #000000;">x2</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"515692107665463680305819378593"</span><span style="color: #0000FF;">),</span>
function Montgomery(bigint m)
<span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(),</span>
if bi_sign(m)=-1 then crash("must be positive") end if
<span style="color: #000000;">t2</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
if bi_mod(m,2)!=BI_ONE then crash("must be odd") end if
integer n = bitLength(m)
<span style="color: #004080;">sequence</span> <span style="color: #000000;">mont</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">Montgomery</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
bigint rrm = bi_mod(bi_shl(1,n*2),m)
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">RRM</span><span style="color: #0000FF;">])</span>
return {2, -- BASE
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">RRM</span><span style="color: #0000FF;">])</span>
n, -- BITLEN
<span style="color: #004080;">mpz</span> <span style="color: #000000;">r1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t1</span><span style="color: #0000FF;">),</span>
m, -- MODULUS
<span style="color: #000000;">r2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t2</span><span style="color: #0000FF;">),</span>
rrm -- 1<<(n*2) % m
<span style="color: #000000;">r</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;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">BITLEN</span><span style="color: #0000FF;">])</span>
end function
<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;">"b : %d\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">BASE</span><span style="color: #0000FF;">]})</span>
bigint m = bi_new("750791094644726559640638407699"),
<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;">"n : %d\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">BITLEN</span><span style="color: #0000FF;">]})</span>
x1 = bi_new("540019781128412936473322405310"),
<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;">"r : %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)})</span>
x2 = bi_new("515692107665463680305819378593");
<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;">"m : %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">MODULUS</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;">"t1: %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t1</span><span style="color: #0000FF;">)})</span>
sequence mont = Montgomery(m)
<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;">"t2: %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t2</span><span style="color: #0000FF;">)})</span>
bigint t1 = bi_mul(x1,mont[RRM]),
<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;">"r1: %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r1</span><span style="color: #0000FF;">)})</span>
t2 = bi_mul(x2,mont[RRM]),
<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;">"r2: %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">)})</span>
r1 = reduce(mont,t1),
<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;">"\n"</span><span style="color: #0000FF;">)</span>
r2 = reduce(mont,t2),
<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;">"Original x1 : %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">)})</span>
r = bi_shl(1,mont[BITLEN])
<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;">"Recovered from r1 : %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r1</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;">"Original x2 : %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">)})</span>
printf(1,"b : %d\n", {mont[BASE]})
<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;">"Recovered from r2 : %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">))})</span>
printf(1,"n : %d\n", {mont[BITLEN]})
printf(1,"r : %s\n", {bi_sprint(r)})
<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;">"\nMontgomery computation of x1 ^ x2 mod m :"</span><span style="color: #0000FF;">)</span>
printf(1,"m : %s\n", {bi_sprint(mont[MODULUS])})
<span style="color: #004080;">mpz</span> <span style="color: #000000;">prod</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">RRM</span><span style="color: #0000FF;">])</span>
printf(1,"t1: %s\n", {bi_sprint(t1)})
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">[</span><span style="color: #000000;">RRM</span><span style="color: #0000FF;">])</span>
printf(1,"t2: %s\n", {bi_sprint(t2)})
<span style="color: #004080;">mpz</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">),</span>
printf(1,"r1: %s\n", {bi_sprint(r1)})
<span style="color: #000000;">expn</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">)</span>
printf(1,"r2: %s\n", {bi_sprint(r2)})
printf(1,"\n")
<span style="color: #008080;">while</span> <span style="color: #7060A8;">mpz_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">expn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
printf(1,"Original x1 : %s\n", {bi_sprint(x1)})
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_odd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">expn</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
printf(1,"Recovered from r1 : %s\n", {bi_sprint(reduce(mont,r1))})
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">prod</span><span style="color: #0000FF;">,</span><span style="color: #000000;">prod</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
printf(1,"Original x2 : %s\n", {bi_sprint(x2)})
<span style="color: #000000;">prod</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">prod</span><span style="color: #0000FF;">)</span>
printf(1,"Recovered from r2 : %s\n", {bi_sprint(reduce(mont,r2))})
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_fdiv_q_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">expn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">expn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
printf(1,"\nMontgomery computation of x1 ^ x2 mod m :")
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
bigint prod = reduce(mont,mont[RRM]),
<span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
base = reduce(mont,bi_mul(x1,mont[RRM])),
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
expn = x2;
<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;">reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mont</span><span style="color: #0000FF;">,</span><span style="color: #000000;">prod</span><span style="color: #0000FF;">))})</span>
while bitLength(expn)>0 do
<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;">" alternate computation of x1 ^ x2 mod m :"</span><span style="color: #0000FF;">)</span>
if bi_mod(expn,2)=BI_ONE then -- odd
<span style="color: #7060A8;">mpz_powm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
prod = reduce(mont,bi_mul(prod,base))
<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;">r</span><span style="color: #0000FF;">)})</span>
end if
<!--</syntaxhighlight>-->
{expn,?} = bi_div3(expn,2)
base = reduce(mont,bi_mul(base,base))
end while
printf(1,"%s\n",{bi_sprint(reduce(mont,prod))})
printf(1,"\n alternate computation of x1 ^ x2 mod m :");
printf(1,"%s\n",{bi_sprint(bi_mod_exp(x1, x2, m))})</lang>
{{out}}
<pre>
Line 878 ⟶ 1,318:
 
=={{header|PicoLisp}}==
<langsyntaxhighlight PicoLisplang="picolisp">(de **Mod (X Y N)
(let M 1
(loop
Line 930 ⟶ 1,370:
Base (reduce (* Base Base)) ) )
(prinl (reduce Prod))
(prinl "Montgomery computation of x1 \^ x2 mod m : " (**Mod X1 X2 M)) )</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 950 ⟶ 1,390:
 
=={{header|Python}}==
{{todo|python|Update the output}}
{{trans|D}}
<syntaxhighlight lang ="python">class Montgomery:
class Montgomery:
BASE = 2
 
Line 982 ⟶ 1,424:
r = 1 << mont.n
 
print(
print "b : ", Montgomery.BASE
f"b: {Montgomery.BASE}\n"
print "n : ", mont.n
f"n: {mont.n}\n"
print "r : ", r
print "m : f",r: mont.m{r}\n"
f"m: {mont.m}\n"
print "t1: ", t1
f"t1: {t1}\n"
print "t2: ", t2
f"t2: {t2}\n"
print "r1: ", r1
f"r1: {r1}\n"
print "r2: ", r2
f"r2: {r2}\n"
print
print f"Original x1 :", {x1}\n"
print f"Recovered from r1 :", {mont.reduce(r1)}\n"
print f"Original x2 :", {x2}\n"
print f"Recovered from r2 :", {mont.reduce(r2)}\n"
)
 
print ("\nMontgomeryMontgomery computation of x1 ^ x2 mod m:")
prod = mont.reduce(mont.rrm)
base = mont.reduce(x1 * mont.rrm)
Line 1,005 ⟶ 1,448:
exp = exp >> 1
base = mont.reduce(base * base)
print (mont.reduce(prod))
print (f"\nAlternate computation of x1 ^ x2 mod m : {pow(x1, x2, m)}")
</syntaxhighlight>
print pow(x1, x2, m)</lang>
{{out}}
<pre>b : 2
b : 2
n : 100
r : 1267650600228229401496703205376
Line 1,027 ⟶ 1,471:
 
Alternate computation of x1 ^ x2 mod m :
151232511393500655853002423778</pre>
</pre>
 
=={{header|Quackery}}==
 
{{trans|Factor}}
 
<code>**mod</code> is defined at [[Modular exponentiation#Quackery]].
 
<syntaxhighlight lang="Quackery"> [ 0 swap [ dup while dip 1+ 1 >> again ] drop ] is bits ( n --> n )
 
[ 1 & ] is odd ( n --> b )
 
[ over bits times [ dup odd if [ over + ] 1 >> ] swap mod ] is monred ( n n --> n )
 
[ 750791094644726559640638407699 ] is m ( --> n )
[ 323165824550862327179367294465482435542970161392400401329100 ] is t1 ( --> n )
 
[ 440160025148131680164261562101 ] is r1 ( --> n )
[ 435362628198191204145287283255 ] is r2 ( --> n )
 
[ 540019781128412936473322405310 ] is x1 ( --> n )
[ 515692107665463680305819378593 ] is x2 ( --> n )
 
[ unrot dip
[ dip dup
over t1 rot / monred temp put
t1 monred ]
[ dup 0 != while
dup odd if
[ over temp take *
m swap monred
temp put ]
dip [ dup * m swap monred ]
1 >> again ]
2drop temp take monred ] is **mon ( n n n --> n )
 
say "Original x1: " x1 echo cr
say "Recovered from r1: " m r1 monred echo cr
cr
say "Original x2: " x2 echo cr
say "Recovered from r2: " m r2 monred echo cr
cr
say "Montgomery computation of x1^x2 mod m: " x1 x2 m **mon echo cr
say "Modular exponentiation of x1^x2 mod m: " x1 x2 m **mod echo cr</syntaxhighlight>
 
{{out}}
 
<pre>Original x1: 540019781128412936473322405310
Recovered from r1: 540019781128412936473322405310
 
Original x2: 515692107665463680305819378593
Recovered from r2: 515692107665463680305819378593
 
Montgomery computation of x1^x2 mod m: 151232511393500655853002423778
Modular exponentiation of x1^x2 mod m: 151232511393500655853002423778</pre>
 
=={{header|Racket}}==
 
<langsyntaxhighlight lang="racket">#lang typed/racket
(require math/number-theory)
 
Line 1,079 ⟶ 1,578:
(define mr (montgomery-reduce-fn m b))
(check-equal? (mr R1 n) x1)
(check-equal? (mr R2 n) x2)))</langsyntaxhighlight>
 
Tests, which are courtesy of #Go implementation, all pass.
 
=={{header|Raku}}==
(formerly Perl 6)
{{works with|Rakudo|2018.03}}
{{trans|Sidef}}
 
<syntaxhighlight lang="raku" line>sub montgomery-reduce($m, $a is copy) {
for 0..$m.msb {
$a += $m if $a +& 1;
$a +>= 1
}
$a % $m
}
 
my $m = 750791094644726559640638407699;
my $t1 = 323165824550862327179367294465482435542970161392400401329100;
 
my $r1 = 440160025148131680164261562101;
my $r2 = 435362628198191204145287283255;
 
my $x1 = 540019781128412936473322405310;
my $x2 = 515692107665463680305819378593;
 
say "Original x1: ", $x1;
say "Recovered from r1: ", montgomery-reduce($m, $r1);
say "Original x2: ", $x2;
say "Recovered from r2: ", montgomery-reduce($m, $r2);
 
print "\nMontgomery computation x1**x2 mod m: ";
my $prod = montgomery-reduce($m, $t1/$x1);
my $base = montgomery-reduce($m, $t1);
 
for $x2, {$_ +> 1} ... 0 -> $exponent {
$prod = montgomery-reduce($m, $prod * $base) if $exponent +& 1;
$base = montgomery-reduce($m, $base * $base);
}
 
say montgomery-reduce($m, $prod);
say "Built-in op computation x1**x2 mod m: ", $x1.expmod($x2, $m);</syntaxhighlight>
{{out}}
<pre>Original x1: 540019781128412936473322405310
Recovered from r1: 540019781128412936473322405310
Original x2: 515692107665463680305819378593
Recovered from r2: 515692107665463680305819378593
 
Montgomery computation x1**x2 mod m: 151232511393500655853002423778
Built-in op computation x1**x2 mod m: 151232511393500655853002423778
</pre>
 
=={{header|Sidef}}==
{{trans|zkl}}
<langsyntaxhighlight lang="ruby">func montgomeryReduce(m, a) {
{
a += m if a.is_odd
Line 1,118 ⟶ 1,665:
 
say(montgomeryReduce(m, prod))
say("Library-based computation of x1^x2 mod m: ", x1.powmod(x2, m))</langsyntaxhighlight>
{{out}}
<pre>
Line 1,132 ⟶ 1,679:
=={{header|Tcl}}==
{{in progress|lang=Tcl|day=25|month=06|year=2012}}
<langsyntaxhighlight lang="tcl">package require Tcl 8.5
 
proc montgomeryReduction {m mDash T n {b 2}} {
Line 1,146 ⟶ 1,693:
set A [expr {$A / ($b ** $n)}]
return [expr {$A >= $m ? $A - $m : $A}]
}</langsyntaxhighlight>
<!-- Not quite sure how to demonstrate this working; examples above aren't very clear… -->
 
=={{header|Visual Basic .NET}}==
{{trans|C#}}
<syntaxhighlight lang="vbnet">Imports System.Numerics
Imports System.Runtime.CompilerServices
 
Module Module1
 
<Extension()>
Function BitLength(v As BigInteger) As Integer
If v < 0 Then
v *= -1
End If
 
Dim result = 0
While v > 0
v >>= 1
result += 1
End While
Return result
End Function
 
Structure Montgomery
Shared ReadOnly BASE = 2
Dim m As BigInteger
Dim rrm As BigInteger
Dim n As Integer
 
Sub New(m As BigInteger)
If m < 0 OrElse m.IsEven Then
Throw New ArgumentException()
End If
 
Me.m = m
n = m.BitLength
rrm = (BigInteger.One << (n * 2)) Mod m
End Sub
 
Function Reduce(t As BigInteger) As BigInteger
Dim a = t
For i = 1 To n
If Not a.IsEven Then
a += m
End If
a >>= 1
Next
If a >= m Then
a -= m
End If
Return a
End Function
End Structure
 
Sub Main()
Dim m = BigInteger.Parse("750791094644726559640638407699")
Dim x1 = BigInteger.Parse("540019781128412936473322405310")
Dim x2 = BigInteger.Parse("515692107665463680305819378593")
 
Dim mont As New Montgomery(m)
Dim t1 = x1 * mont.rrm
Dim t2 = x2 * mont.rrm
 
Dim r1 = mont.Reduce(t1)
Dim r2 = mont.Reduce(t2)
Dim r = BigInteger.One << mont.n
 
Console.WriteLine("b : {0}", Montgomery.BASE)
Console.WriteLine("n : {0}", mont.n)
Console.WriteLine("r : {0}", r)
Console.WriteLine("m : {0}", mont.m)
Console.WriteLine("t1: {0}", t1)
Console.WriteLine("t2: {0}", t2)
Console.WriteLine("r1: {0}", r1)
Console.WriteLine("r2: {0}", r2)
Console.WriteLine()
Console.WriteLine("Original x1 : {0}", x1)
Console.WriteLine("Recovered from r1 : {0}", mont.Reduce(r1))
Console.WriteLine("Original x2 : {0}", x2)
Console.WriteLine("Recovered from r2 : {0}", mont.Reduce(r2))
 
Console.WriteLine()
Console.WriteLine("Montgomery computation of x1 ^ x2 mod m :")
Dim prod = mont.Reduce(mont.rrm)
Dim base = mont.Reduce(x1 * mont.rrm)
Dim exp = x2
While exp.BitLength > 0
If Not exp.IsEven Then
prod = mont.Reduce(prod * base)
End If
exp >>= 1
base = mont.Reduce(base * base)
End While
Console.WriteLine(mont.Reduce(prod))
Console.WriteLine()
Console.WriteLine("Alternate computation of x1 ^ x2 mod m :")
Console.WriteLine(BigInteger.ModPow(x1, x2, m))
End Sub
 
End Module</syntaxhighlight>
{{out}}
<pre>b : 2
n : 100
r : 1267650600228229401496703205376
m : 750791094644726559640638407699
t1: 323165824550862327179367294465482435542970161392400401329100
t2: 308607334419945011411837686695175944083084270671482464168730
r1: 440160025148131680164261562101
r2: 435362628198191204145287283255
 
Original x1 : 540019781128412936473322405310
Recovered from r1 : 540019781128412936473322405310
Original x2 : 515692107665463680305819378593
Recovered from r2 : 515692107665463680305819378593
 
Montgomery computation of x1 ^ x2 mod m :
151232511393500655853002423778
 
Alternate computation of x1 ^ x2 mod m :
151232511393500655853002423778</pre>
 
=={{header|Wren}}==
{{trans|Kotlin}}
{{libheader|Wren-big}}
<syntaxhighlight lang="wren">import "./big" for BigInt
 
class Montgomery {
static base { 2 }
 
construct new(m) {
if (m <= BigInt.zero || !m.testBit(0)) {
Fiber.abort("Argument must be a positive, odd big integer.")
}
_m = m
_n = m.bitLength.toSmall
_rrm = (BigInt.one << (n*2)) % m
}
 
m { _m }
n { _n }
rrm { _rrm }
 
reduce(t) {
var a = t.copy()
for (i in 0..._n) {
if (a.testBit(0)) a = a + _m
a = a >> 1
}
if (a >= _m) a = a - _m
return a
}
}
 
var m = BigInt.new("750791094644726559640638407699")
var x1 = BigInt.new("540019781128412936473322405310")
var x2 = BigInt.new("515692107665463680305819378593")
 
var mont = Montgomery.new(m)
var t1 = x1 * mont.rrm
var t2 = x2 * mont.rrm
 
var r1 = mont.reduce(t1)
var r2 = mont.reduce(t2)
var r = BigInt.one << (mont.n)
 
System.print("b : %(Montgomery.base)")
System.print("n : %(mont.n)")
System.print("r : %(r)")
System.print("m : %(mont.m)")
System.print("t1: %(t1)")
System.print("t2: %(t2)")
System.print("r1: %(r1)")
System.print("r2: %(r2)")
System.print()
System.print("Original x1 : %(x1)")
System.print("Recovered from r1 : %(mont.reduce(r1))")
System.print("Original x2 : %(x2)")
System.print("Recovered from r2 : %(mont.reduce(r2))")
 
System.print("\nMontgomery computation of x1 ^ x2 mod m :")
var prod = mont.reduce(mont.rrm)
var base = mont.reduce(x1 * mont.rrm)
var exp = x2
while (exp.bitLength > 0) {
if (exp.testBit(0)) prod = mont.reduce(prod * base)
exp = exp >> 1
base = mont.reduce(base * base)
}
System.print(mont.reduce(prod))
System.print("\nLibrary-based computation of x1 ^ x2 mod m :")
System.print(x1.modPow(x2, m))</syntaxhighlight>
 
{{out}}
<pre>
b : 2
n : 100
r : 1267650600228229401496703205376
m : 750791094644726559640638407699
t1: 323165824550862327179367294465482435542970161392400401329100
t2: 308607334419945011411837686695175944083084270671482464168730
r1: 440160025148131680164261562101
r2: 435362628198191204145287283255
 
Original x1 : 540019781128412936473322405310
Recovered from r1 : 540019781128412936473322405310
Original x2 : 515692107665463680305819378593
Recovered from r2 : 515692107665463680305819378593
 
Montgomery computation of x1 ^ x2 mod m :
151232511393500655853002423778
 
Library-based computation of x1 ^ x2 mod m :
151232511393500655853002423778
</pre>
 
=={{header|zkl}}==
{{Trans|Go}}
Uses GMP (GNU Multi Precision library).
<langsyntaxhighlight lang="zkl">var [const] BN=Import("zklBigNum"); // libGMP
 
fcn montgomeryReduce(modulus,T){
Line 1,163 ⟶ 1,923:
if(a>=modulus) a.sub(modulus);
a
}</langsyntaxhighlight>
<langsyntaxhighlight lang="zkl"> // magic numbers from the Go solution
//b:= 2;
//n:= 100;
Line 1,198 ⟶ 1,958:
}
println(montgomeryReduce(m,prod));
println("Library-based computation of x1 ^ x2 mod m: ",x1.powm(x2,m));</langsyntaxhighlight>
{{out}}
<pre>
PureFox
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