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Montgomery reduction: Difference between revisions

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Montgomery reduction (view source)

Revision as of 10:50, 4 January 2024

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Line 16:
A ← A - m
Return (A)
 
=={{header|11l}}==
{{trans|Python}}
 
<syntaxhighlight lang="11l">T Montgomery
BigInt m
Int n
BigInt rrm
 
F (m)
.m = m
.n = bits:length(m)
.rrm = (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}}==
<langsyntaxhighlight lang="c">#include <stdio.h>
#include <stdlib.h>
 
Line 134 ⟶ 219:
 
return 0;
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 158 ⟶ 243:
=={{header|C sharp|C#}}==
{{trans|D}}
<langsyntaxhighlight lang="csharp">using System;
using System.Numerics;
 
Line 249 ⟶ 334:
}
}
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 272 ⟶ 357:
 
=={{header|C++}}==
<langsyntaxhighlight lang="cpp">#include<iostream>
#include<conio.h>
using namespace std;
Line 357 ⟶ 442:
cout<<"Montgomery domain representation = "<<e;
return 0;
}</langsyntaxhighlight>
 
=={{header|D}}==
{{trans|Kotlin}}
<langsyntaxhighlight Dlang="d">import std.bigint;
import std.stdio;
 
Line 459 ⟶ 544:
writeln("\nAlternate computation of x1 ^ x2 mod m :");
writeln(x1.modPow(x2, m));
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 484 ⟶ 569:
{{trans|Sidef}}
{{works with|Factor|0.99 2020-08-14}}
<langsyntaxhighlight lang="factor">USING: io kernel locals math math.bitwise math.functions
prettyprint ;
 
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"Library-based computation of x1^x2 mod m: " write
x1 x2 m ^mod .
]</langsyntaxhighlight>
{{out}}
<pre>
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=={{header|Go}}==
<langsyntaxhighlight lang="go">package main
 
import (
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fmt.Println("\nLibrary-based computation of x1 ^ x2 mod m:")
fmt.Println(new(big.Int).Exp(x1, x2, m))
}</langsyntaxhighlight>
{{out}}
<pre>
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=={{header|Java}}==
{{trans|Kotlin}}
<langsyntaxhighlight lang="java">import java.math.BigInteger;
 
public class MontgomeryReduction {
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System.out.println(x1.modPow(x2, m));
}
}</langsyntaxhighlight>
{{out}}
<pre>b : 2
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=={{header|Julia}}==
{{trans|Python}}
<langsyntaxhighlight lang="julia">""" base 2 type Montgomery numbers """
struct Montgomery2
m::BigInt
Line 838 ⟶ 923:
 
testmontgomery2()
</langsyntaxhighlight>{{out}}
<pre>
b : 2
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=={{header|Kotlin}}==
{{trans|Go}}
<langsyntaxhighlight lang="scala">// version 1.1.3
 
import java.math.BigInteger
Line 935 ⟶ 1,020:
println("\nLibrary-based computation of x1 ^ x2 mod m :")
println(x1.modPow(x2, m))
}</langsyntaxhighlight>
 
{{out}}
Line 959 ⟶ 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|Raku}}
{{libheader|ntheory}}
<langsyntaxhighlight lang="perl">use bigint;
use ntheory qw(powmod);
 
Line 1,006 ⟶ 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 1,019 ⟶ 1,216:
{{trans|D}}
{{libheader|Phix/mpfr}}
<!--<langsyntaxhighlight Phixlang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
Line 1,099 ⟶ 1,296:
<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>
<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>
<!--</langsyntaxhighlight>-->
{{out}}
<pre>
Line 1,121 ⟶ 1,318:
 
=={{header|PicoLisp}}==
<langsyntaxhighlight PicoLisplang="picolisp">(de **Mod (X Y N)
(let M 1
(loop
Line 1,173 ⟶ 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 1,193 ⟶ 1,390:
 
=={{header|Python}}==
{{todo|python|Update the output}}
{{trans|D}}
<syntaxhighlight lang ="python">class Montgomery:
class Montgomery:
BASE = 2
 
Line 1,225 ⟶ 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 1,248 ⟶ 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 1,270 ⟶ 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 1,322 ⟶ 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.
Line 1,331 ⟶ 1,587:
{{trans|Sidef}}
 
<syntaxhighlight lang="raku" perl6line>sub montgomery-reduce($m, $a is copy) {
for 0..$m.msb {
$a += $m if $a +& 1;
Line 1,363 ⟶ 1,619:
 
say montgomery-reduce($m, $prod);
say "Built-in op computation x1**x2 mod m: ", $x1.expmod($x2, $m);</langsyntaxhighlight>
{{out}}
<pre>Original x1: 540019781128412936473322405310
Line 1,376 ⟶ 1,632:
=={{header|Sidef}}==
{{trans|zkl}}
<langsyntaxhighlight lang="ruby">func montgomeryReduce(m, a) {
{
a += m if a.is_odd
Line 1,409 ⟶ 1,665:
 
say(montgomeryReduce(m, prod))
say("Library-based computation of x1^x2 mod m: ", x1.powmod(x2, m))</langsyntaxhighlight>
{{out}}
<pre>
Line 1,423 ⟶ 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 1,437 ⟶ 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#}}
<langsyntaxhighlight lang="vbnet">Imports System.Numerics
Imports System.Runtime.CompilerServices
 
Line 1,537 ⟶ 1,793:
End Sub
 
End Module</langsyntaxhighlight>
{{out}}
<pre>b : 2
Line 1,562 ⟶ 1,818:
{{trans|Kotlin}}
{{libheader|Wren-big}}
<langsyntaxhighlight ecmascriptlang="wren">import "./big" for BigInt
 
class Montgomery {
Line 1,628 ⟶ 1,884:
System.print(mont.reduce(prod))
System.print("\nLibrary-based computation of x1 ^ x2 mod m :")
System.print(x1.modPow(x2, m))</langsyntaxhighlight>
 
{{out}}
Line 1,656 ⟶ 1,912:
{{Trans|Go}}
Uses GMP (GNU Multi Precision library).
<langsyntaxhighlight lang="zkl">var [const] BN=Import("zklBigNum"); // libGMP
 
fcn montgomeryReduce(modulus,T){
Line 1,667 ⟶ 1,923:
if(a>=modulus) a.sub(modulus);
a
}</langsyntaxhighlight>
<langsyntaxhighlight lang="zkl"> // magic numbers from the Go solution
//b:= 2;
//n:= 100;
Line 1,702 ⟶ 1,958:
}
println(montgomeryReduce(m,prod));
println("Library-based computation of x1 ^ x2 mod m: ",x1.powm(x2,m));</langsyntaxhighlight>
{{out}}
<pre>
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