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
<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:
<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>
<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
<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
<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
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.
<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?.)
<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
- Algorithm from http://introcs.cs.princeton.edu/java/78crypto/ModExp.java.html
- but Tcl has arbitrary-width integers and an exponentiation operator, which
- 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"