Modular exponentiation

From Rosetta Code
Revision as of 00:20, 22 February 2013 by rosettacode>Oak eagle (add code for Maple)
Task
Modular exponentiation
You are encouraged to solve this task according to the task description, using any language you may know.

Find the last 40 decimal digits of , where

A computer is too slow to find the entire value of . Instead, the program must use a fast algorithm for modular exponentiation: .

The algorithm must work for any integers where and .

Ada

Using the big integer implementation from a cryptographic library [1].

<lang Ada>with Ada.Text_IO, Ada.Command_Line, Crypto.Types.Big_Numbers;

procedure Mod_Exp is

  A: String :=
    "2988348162058574136915891421498819466320163312926952423791023078876139";
  B: String :=
    "2351399303373464486466122544523690094744975233415544072992656881240319";
  D: constant Positive := Positive'Max(Positive'Max(A'Length, B'Length), 40);
    -- the number of decimals to store A, B, and result
  Bits: constant Positive := (34*D)/10;
    -- (slightly more than) the number of bits to store A, B, and result
  package LN is new Crypto.Types.Big_Numbers (Bits + (32 - Bits mod 32));
    -- the actual number of bits has to be a multiple of 32
  use type LN.Big_Unsigned;
  function "+"(S: String) return LN.Big_Unsigned
    renames LN.Utils.To_Big_Unsigned;
  M: LN.Big_Unsigned := (+"10") ** (+"40");

begin

  Ada.Text_IO.Put("A**B (mod 10**40) = ");
  Ada.Text_IO.Put_Line(LN.Utils.To_String(LN.Mod_Utils.Pow((+A), (+B), M)));

end Mod_Exp;</lang>

Output:
A**B (mod 10**40) = 1527229998585248450016808958343740453059

Bracmat

Translation of: Icon_and_Unicon

<lang bracmat> ( ( mod-power

   =   base exponent modulus result
     .   !arg:(?base,?exponent,?modulus)
       & !exponent:~<0
       & 1:?result
       &   whl
         ' ( !exponent:>0
           &     ( (   mod$(!exponent.2):1
                     & mod$(!result*!base.!modulus):?result
                     & -1
                   | 0
                   )
                 + !exponent
                 )
               * 1/2
             : ?exponent
           & mod$(!base^2.!modulus):?base
           )
       & !result
   )
 & ( a
   = 2988348162058574136915891421498819466320163312926952423791023078876139
   )
 & ( b
   = 2351399303373464486466122544523690094744975233415544072992656881240319
   )
 & out$("last 40 digits = " mod-power$(!a,!b,10^40))
 )</lang>

Output:

last 40 digits =  1527229998585248450016808958343740453059

BBC BASIC

Uses the Huge Integer Math & Encryption library. <lang bbcbasic> INSTALL @lib$+"HIMELIB"

     PROC_himeinit("")
     
     PROC_hiputdec(1, "2988348162058574136915891421498819466320163312926952423791023078876139")
     PROC_hiputdec(2, "2351399303373464486466122544523690094744975233415544072992656881240319")
     PROC_hiputdec(3, "10000000000000000000000000000000000000000")
     h1% = 1 : h2% = 2 : h3% = 3 : h4% = 4
     SYS `hi_PowMod`, ^h1%, ^h2%, ^h3%, ^h4%
     PRINT FN_higetdec(4)</lang>

Output:

1527229998585248450016808958343740453059

C

Given numbers are too big for even 64 bit integers, so might as well take the lazy route and use GMP:

Library: GMP

<lang c>#include <gmp.h>

int main() { mpz_t a, b, m, r;

mpz_init_set_str(a, "2988348162058574136915891421498819466320" "163312926952423791023078876139", 0); mpz_init_set_str(b, "2351399303373464486466122544523690094744" "975233415544072992656881240319", 0); mpz_init(m); mpz_ui_pow_ui(m, 10, 40);

mpz_init(r); mpz_powm(r, a, b, m);

gmp_printf("%Zd\n", r); /* ...16808958343740453059 */

mpz_clear(a); mpz_clear(b); mpz_clear(m); mpz_clear(r);

return 0; }</lang>

Common Lisp

<lang lisp>(defun rosetta-mod-expt (base power divisor)

 "Return BASE raised to the POWER, modulo DIVISOR.
 This function is faster than (MOD (EXPT BASE POWER) DIVISOR), but
 only works when POWER is a non-negative integer."
 (setq base (mod base divisor))
 ;; Multiply product with base until power is zero.
 (do ((product 1))
     ((zerop power) product)
   ;; Square base, and divide power by 2, until power becomes odd.
   (do () ((oddp power))
     (setq base (mod (* base base) divisor)

power (ash power -1)))

   (setq product (mod (* product base) divisor)

power (1- power))))

(let ((a 2988348162058574136915891421498819466320163312926952423791023078876139)

     (b 2351399303373464486466122544523690094744975233415544072992656881240319))
 (format t "~A~%" (rosetta-mod-expt a b (expt 10 40))))</lang>
Works with: CLISP

<lang lisp>;; CLISP provides EXT:MOD-EXPT (let ((a 2988348162058574136915891421498819466320163312926952423791023078876139)

     (b 2351399303373464486466122544523690094744975233415544072992656881240319))
 (format t "~A~%" (mod-expt a b (expt 10 40))))</lang>

Implementation with LOOP

<lang lisp>(defun mod-expt (a n m)

  (loop with c = 1 while (plusp n) do
     (if (oddp n) (setf c (mod (* a c) m)))
     (setf n (ash n -1))
     (setf a (mod (* a a) m))
     finally (return c)))</lang>

D

Translation of: Icon_and_Unicon

<lang d>import std.stdio, std.bigint;

BigInt powMod(BigInt base, BigInt exponent, BigInt modulus) in {

  assert(exponent >= 0);

} body {

   BigInt result = 1;
   while (exponent > 0) {
       if (exponent % 2 == 1)
           result = (result * base) % modulus;
       exponent /= 2;
       base = base ^^ 2 % modulus;
   }
   return result;

}

void main() {

   powMod(BigInt("29883481620585741369158914214988194" ~
                 "66320163312926952423791023078876139"),
          BigInt("235139930337346448646612254452369009" ~
                 "4744975233415544072992656881240319"),
          BigInt(10) ^^ 40).writeln();

}</lang>

Output:
1527229998585248450016808958343740453059

Dc

<lang Dc>2988348162058574136915891421498819466320163312926952423791023078876139 2351399303373464486466122544523690094744975233415544072992656881240319 10 40^|p</lang>

Emacs Lisp

Library: Calc

<lang lisp>(let ((a "2988348162058574136915891421498819466320163312926952423791023078876139")

     (b "2351399303373464486466122544523690094744975233415544072992656881240319"))
 ;; "$ ^ $$ mod (10 ^ 40)" performs modular exponentiation.
 ;; "unpack(-5, x)_1" unpacks the integer from the modulo form.
 (message "%s" (calc-eval "unpack(-5, $ ^ $$ mod (10 ^ 40))_1" nil a b)))</lang>

Go

<lang go>package main

import (

   "fmt"
   "math/big"

)

func main() {

   a, _ := new(big.Int).SetString(
       "2988348162058574136915891421498819466320163312926952423791023078876139", 10)
   b, _ := new(big.Int).SetString(
       "2351399303373464486466122544523690094744975233415544072992656881240319", 10)
   m := big.NewInt(10)
   r := big.NewInt(40)
   m.Exp(m, r, nil)
   r.Exp(a, b, m)
   fmt.Println(r)

}</lang> Output:

1527229998585248450016808958343740453059

Haskell

Kind of a hack. We partially implement a "modular arithmetic" instance of Num, so that we can take advantage of the efficient built-in exponentiation-by-squaring operation without implementing it ourselves. Since there are no "local" instances, we must keep the modulo base around with us in the type, which makes the code inelegant. <lang haskell>-- Private type. Do not use outside of the modPow function newtype ModN = ModN (Integer, Integer) deriving (Eq, Show) instance Num ModN where

 -- actually only multiplication needs to be implemented
 -- but we do some of the other ones too for good measure
 ModN (x, m) + ModN (y, m') | m == m' = ModN ((x + y) `mod` m, m)
                            | otherwise = undefined
 ModN (x, m) * ModN (y, m') | m == m' = ModN ((x * y) `mod` m, m)
                            | otherwise = undefined
 negate (ModN (x, m)) = ModN ((- x) `mod` m, m)
 abs _ = undefined
 signum _ = undefined
 fromInteger _ = undefined

modPow :: Integer -> Integer -> Integer -> Integer modPow _ 0 m = 1 `mod` m modPow a b m = c

 where a' = ModN (a, m)
       ModN (c, _) = a' ^ b

main :: IO () main = print $ modPow a b m

 where a = 2988348162058574136915891421498819466320163312926952423791023078876139
       b = 2351399303373464486466122544523690094744975233415544072992656881240319
       m = 10 ^ 40</lang>

Output:

1527229998585248450016808958343740453059

Icon and Unicon

This uses the exponentiation procedure from RSA Code an example of the right to left binary method. <lang Icon>procedure main()

   a := 2988348162058574136915891421498819466320163312926952423791023078876139
   b := 2351399303373464486466122544523690094744975233415544072992656881240319 
   write("last 40 digits = ",mod_power(a,b,(10^40))   

end

procedure mod_power(base, exponent, modulus) # fast modular exponentation

  if exponent < 0 then runerr(205,m)          # added for this task
  result := 1
  while exponent > 0 do {
     if exponent % 2 = 1 then 
        result := (result * base) % modulus
     exponent /:= 2   
     base := base ^ 2 % modulus
     }  
  return result

end</lang>

Output:
last 40 digits = 1527229998585248450016808958343740453059

J

Solution:<lang j> m&|@^</lang> Example:<lang j> a =: 2988348162058574136915891421498819466320163312926952423791023078876139x

  b =: 2351399303373464486466122544523690094744975233415544072992656881240319x
  m =: 10^40x
  a m&|@^ b

1527229998585248450016808958343740453059</lang> Discussion: The phrase m&|@^ is the natural expression of a^b mod m in J, and is recognized by the interpreter as an opportunity for optimization, by avoiding the exponentiation.

Java

java.math.BigInteger.modPow solves this task. Inside OpenJDK, BigInteger.java implements BigInteger.modPow with a fast algorithm from Colin Plumb's bnlib. This "window algorithm" caches odd powers of the base, to decrease the number of squares and multiplications. It also exploits both the Chinese remainder theorem and the Montgomery reduction.

<lang java>import java.math.BigInteger;

public class PowMod {

   public static void main(String[] args){
       BigInteger a = new BigInteger(
     "2988348162058574136915891421498819466320163312926952423791023078876139");
       BigInteger b = new BigInteger(
     "2351399303373464486466122544523690094744975233415544072992656881240319");
       BigInteger m = new BigInteger("10000000000000000000000000000000000000000");
       
       System.out.println(a.modPow(b, m));
   }

}</lang> Output:

1527229998585248450016808958343740453059

Maple

<lang Maple>a := 2988348162058574136915891421498819466320163312926952423791023078876139: b := 2351399303373464486466122544523690094744975233415544072992656881240319: a &^ b mod 10^40;</lang> Output:

1527229998585248450016808958343740453059

Mathematica

<lang Mathematica>a = 2988348162058574136915891421498819466320163312926952423791023078876139; b = 2351399303373464486466122544523690094744975233415544072992656881240319; m = 10^40; PowerMod[a, b, m] -> 1527229998585248450016808958343740453059</lang>

Maxima

<lang maxima>a: 2988348162058574136915891421498819466320163312926952423791023078876139$ b: 2351399303373464486466122544523690094744975233415544072992656881240319$ power_mod(a, b, 10^40); /* 1527229998585248450016808958343740453059 */</lang>

PARI/GP

<lang parigp>a=2988348162058574136915891421498819466320163312926952423791023078876139; b=2351399303373464486466122544523690094744975233415544072992656881240319; lift(Mod(a,10^40)^b)</lang>

Pascal

Works with: Free_Pascal
Library: GMP

A port of the C example using gmp. <lang pascal>Program ModularExponentiation(output);

uses

 gmp;
 

var

 a, b, m, r: mpz_t;
 fmt: pchar;

begin

 mpz_init_set_str(a, '2988348162058574136915891421498819466320163312926952423791023078876139', 10);
 mpz_init_set_str(b, '2351399303373464486466122544523690094744975233415544072992656881240319', 10);
 mpz_init(m);
 mpz_ui_pow_ui(m, 10, 40);
 mpz_init(r);
 mpz_powm(r, a, b, m);
 fmt := '%Zd' + chr(13) + chr(10);
 mp_printf(fmt, @r); (* ...16808958343740453059 *)
 
 mpz_clear(a);
 mpz_clear(b);
 mpz_clear(m);
 mpz_clear(r);

end.</lang> Output:

% ./ModularExponentiation
1527229998585248450016808958343740453059

Perl

<lang perl>use bigint;

my $a = 2988348162058574136915891421498819466320163312926952423791023078876139; my $b = 2351399303373464486466122544523690094744975233415544072992656881240319; my $m = 10 ** 40; print $a->bmodpow($b, $m), "\n";</lang> Output:

1527229998585248450016808958343740453059

Perl 6

This is specced as a built-in, but here's an explicit version: <lang perl6>sub expmod(Int $a is copy, Int $b is copy, $n) {

   my $c = 1;
   repeat while $b div= 2 {
       ($c *= $a) %= $n if $b % 2;
       ($a *= $a) %= $n;
   }
   $c;

}

say expmod

   2988348162058574136915891421498819466320163312926952423791023078876139,
   2351399303373464486466122544523690094744975233415544072992656881240319,
   10**40;</lang>

Output:

1527229998585248450016808958343740453059

PHP

<lang php><?php $a = '2988348162058574136915891421498819466320163312926952423791023078876139'; $b = '2351399303373464486466122544523690094744975233415544072992656881240319'; $m = '1' . str_repeat('0', 40); echo bcpowmod($a, $b, $m), "\n"; ?></lang> Output:

1527229998585248450016808958343740453059

PicoLisp

The following function is taken from "lib/rsa.l": <lang PicoLisp>(de **Mod (X Y N)

  (let M 1
     (loop
        (when (bit? 1 Y)
           (setq M (% (* M X) N)) )
        (T (=0 (setq Y (>> 1 Y)))
           M )
        (setq X (% (* X X) N)) ) ) )</lang>

Test: <lang PicoLisp>: (**Mod

  2988348162058574136915891421498819466320163312926952423791023078876139
  2351399303373464486466122544523690094744975233415544072992656881240319
  10000000000000000000000000000000000000000 )

-> 1527229998585248450016808958343740453059</lang>

Python

<lang python>a = 2988348162058574136915891421498819466320163312926952423791023078876139 b = 2351399303373464486466122544523690094744975233415544072992656881240319 m = 10 ** 40 print(pow(a, b, m))</lang> Output:

1527229998585248450016808958343740453059

REXX

This REXX program attempts to handle any a,b, or m, but there are limits for any computer language.
For REXX, it's around eight million digits, unless or exceeds that. <lang rexx>/*REXX program to show modular exponentation: a**b mod M */ parse arg a b mm /*get the arguments (maybe).*/ if a== | a==',' then a=,

   2988348162058574136915891421498819466320163312926952423791023078876139

if b== | b==',' then b=,

   2351399303373464486466122544523690094744975233415544072992656881240319

if mm== then mm=40 say 'a=' a; say ' ('length(a) "digits)" say 'b=' b; say ' ('length(b) "digits)"

     do j=1 for words(mm);   m=word(mm,j);   say copies('─',linesize()-1)
     say 'a**b (mod 10**'m")=" powerModulated(a,b,10**m)
     end   /*j*/

exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────────POWERMODULATED subroutine───────*/ powerModulated: procedure; parse arg x,p,n /*fast modular exponentation*/ if p==0 then return 1 /*special case. */ if p==1 then return x /*special case. */ if p<0 then do; say '***error!*** power is negative:' p; exit 13; end parse value max(x**2,p,n)'E0' with "E" e /*pick biggest of the three.*/ numeric digits max(20,e*2) /*big enough to handle A² */ _=1

         do while p\==0;   if p//2==1 then _=_*x//n
         p=p%2;     x=x*x // n
         end    /*while*/

return _</lang> output when using the input of: 40 80 180 888
Note the REXX program was executing within a window of 600 bytes wide.

a= 2988348162058574136915891421498819466320163312926952423791023078876139
        (70 digits)
b= 2351399303373464486466122544523690094744975233415544072992656881240319
        (70 digits)
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
a**b (mod 10**40)= 1527229998585248450016808958343740453059
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
a**b (mod 10**80)= 53259517041910225328867076245698908287781527229998585248450016808958343740453059
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
a**b (mod 10**180)= 31857295076204937005344367438778481743660325586328069392203762862423884839076695547212682454523811053259517041910225328867076245698908287781527229998585248450016808958343740453059
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
a**b (mod 10**888)= 2612849643808365153970307063634422265713972370574889513136845452410856423299436762487557161242604471887885300171829510516527484255607339748359444160694661767131561827274483018385170003434853270016569482853811730383390737793312301323406698998964489388587853627711904603124125798753498716559994462054260496622614506334484689315735068762556447491553489235236807309998697854727791160093566968169527719659307289405305177993299425901141782840092602984267350865792542825912897568403588118221513074793528568569833937153488707152390200379629380198479929609788498528506130631774711751914442
62586321233906926671000476591123695550566585083205841790404069511972417770392822283604206143472509425391114072344402850867571806031857295076204937005344367438778481743660325586328069392203762862423884839076695547212682454523811053259517041910225328867076245698908287781527229998585248450016808958343740453059

Ruby

Ruby's core library has no modular exponentiation. OpenSSL, in Ruby's standard library, provides OpenSSL::BN#mod_exp. To reach this method, we call Integer#to_bn to convert a from Integer to OpenSSL::BN. The method implicitly converts b and m.

Library: OpenSSL

<lang ruby>require 'openssl'

a = 2988348162058574136915891421498819466320163312926952423791023078876139 b = 2351399303373464486466122544523690094744975233415544072992656881240319 m = 10 ** 40 puts a.to_bn.mod_exp(b, m)</lang>

Or we can implement a custom method, Integer#rosetta_mod_exp, to calculate the same result. This method does exponentiation by successive squaring, but replaces each intermediate product with a congruent value. (Program needs Ruby 1.8.7 for Integer#odd?.)

Works with: Ruby version 1.8.7

<lang ruby>class Integer

 def rosetta_mod_exp(exp, mod)
   exp < 0 and raise ArgumentError, "negative exponent"
   prod = 1
   base = self % mod
   until exp.zero?
     exp.odd? and prod = (prod * base) % mod
     exp >>= 1
     base = (base * base) % mod
   end
   prod
 end

end

a = 2988348162058574136915891421498819466320163312926952423791023078876139 b = 2351399303373464486466122544523690094744975233415544072992656881240319 m = 10 ** 40 puts a.rosetta_mod_exp(b, m)</lang>

Scala

<lang scala>import scala.math.BigInt

val a = BigInt(

 "2988348162058574136915891421498819466320163312926952423791023078876139")

val b = BigInt(

 "2351399303373464486466122544523690094744975233415544072992656881240319")

println(a.modPow(b, BigInt(10).pow(40)))</lang>

Tcl

While Tcl does have arbitrary-precision arithmetic (from 8.5 onwards), it doesn't expose a modular exponentiation function. Thus we implement one ourselves.

Recursive

<lang tcl>package require Tcl 8.5

  1. Algorithm from http://introcs.cs.princeton.edu/java/78crypto/ModExp.java.html
  2. but Tcl has arbitrary-width integers and an exponentiation operator, which
  3. helps simplify the code.

proc tcl::mathfunc::modexp {a b n} {

   if {$b == 0} {return 1}
   set c [expr {modexp($a, $b / 2, $n)**2 % $n}]
   if {$b & 1} {

set c [expr {($c * $a) % $n}]

   }
   return $c

}</lang> Demonstrating: <lang tcl>set a 2988348162058574136915891421498819466320163312926952423791023078876139 set b 2351399303373464486466122544523690094744975233415544072992656881240319 set n [expr {10**40}] puts [expr {modexp($a,$b,$n)}]</lang> Output:

1527229998585248450016808958343740453059

Iterative

<lang tcl>package require Tcl 8.5 proc modexp {a b n} {

   for {set c 1} {$b} {set a [expr {$a*$a % $n}]} {

if {$b & 1} { set c [expr {$c*$a % $n}] } set b [expr {$b >> 1}]

   }
   return $c 

}</lang> Demonstrating: <lang tcl>set a 2988348162058574136915891421498819466320163312926952423791023078876139 set b 2351399303373464486466122544523690094744975233415544072992656881240319 set n [expr {10**40}] puts [modexp $a $b $n]</lang> Output:

1527229998585248450016808958343740453059

TXR

<lang txr>@(bind result @(exptmod 2988348162058574136915891421498819466320163312926952423791023078876139

                       2351399303373464486466122544523690094744975233415544072992656881240319
                       (expt 10 40)))</lang>
$ ./txr rosetta/modexp.txr
result="1527229998585248450016808958343740453059"