bcpow
The author's statement:
"A natural number is any positive non-zero integer."
should be, imo, something like:
"In this context only positive non-zero integers are considered to be natural numbers."
http://mathworld.wolfram.com/NaturalNumber.html
http://planetmath.org/encyclopedia/NaturalNumber.html
http://en.wikipedia.org/wiki/Natural_number
bcpowmod
(PHP 5)
bcpowmod — Umocniť číslo s ľubovoľnou presnosťou na iné, redukované o určený modulus.
Popis
string bcpowmod
( string $x
, string $y
, string $modulus
[, int $scale
] )
Využíva metódu rýchleho umocňovania x na y s ohľadom na modulus modulus . Voliteľný scale sa može použiť na nastavenie počtu číslic za desatinou čiarkou vo výsledku.
Následujúce dva príkazy sú funkčne identické. Avšak verzia bcpowmod() je rýchlejšia a môže pracovať s väčšími parametrami.
<?php
$a = bcpowmod($x, $y, $mod);
$b = bcmod(bcpow($x, $y), $mod);
?>
Note: Pretože táto metóda využíva operáciu modulus, neprirodzené čísla môžu mať za dôsledok nečakané výsledky. Prirodzené číslo je kladný nenulový integer.
add a note
User Contributed Notesbcpowmod
william at example dot com
21-Feb-2007 02:04
21-Feb-2007 02:04
laysoft at gmail dot com
30-Jan-2007 05:34
30-Jan-2007 05:34
I found a better way to emulate bcpowmod on PHP 4, which works with very big numbers too:
function powmod($m,$e,$n) {
if (intval(PHP_VERSION)>4) {
return(bcpowmod($m,$e,$n));
} else {
$r="";
while ($e!="0") {
$t=bcmod($e,"4096");
$r=substr("000000000000".decbin(intval($t)),-12).$r;
$e=bcdiv($e,"4096");
}
$r=preg_replace("!^0+!","",$r);
if ($r=="") $r="0";
$m=bcmod($m,$n);
$erb=strrev($r);
$q="1";
$a[0]=$m;
for ($i=1;$i<strlen($erb);$i++) {
$a[$i]=bcmod(bcmul($a[$i-1],$a[$i-1]),$n);
}
for ($i=0;$i<strlen($erb);$i++) {
if ($erb[$i]=="1") {
$q=bcmod(bcmul($q,$a[$i]),$n);
}
}
return($q);
}
}
rrasss at gmail dot com
15-May-2006 02:46
15-May-2006 02:46
However, if you read his full note, you see this paragraph:
"The function bcpowmod(v, e, m) is supposedly equivalent to bcmod(bcpow(v, e), m). However, for the large numbers used as keys in the RSA algorithm, the bcpow function generates a number so big as to overflow it. For any exponent greater than a few tens of thousands, bcpow overflows and returns 1."
So you still can, and should (over bcmod(bcpow(v, e), m) ), use his function if you are using larger exponents, "any exponent greater than a few tens of thousand."
ewilde aht bsmdevelopment dawt com
27-Sep-2005 09:46
27-Sep-2005 09:46
Versions of PHP prior to 5 do not have bcpowmod in their repertoire. This routine simulates this function using bcdiv, bcmod and bcmul. It is useful to have bcpowmod available because it is commonly used to implement the RSA algorithm.
The function bcpowmod(v, e, m) is supposedly equivalent to bcmod(bcpow(v, e), m). However, for the large numbers used as keys in the RSA algorithm, the bcpow function generates a number so big as to overflow it. For any exponent greater than a few tens of thousands, bcpow overflows and returns 1.
This routine will iterate through a loop squaring the result, modulo the modulus, for every one-bit in the exponent. The exponent is shifted right by one bit for each iteration. When it has been reduced to zero, the calculation ends.
This method may be slower than bcpowmod but at least it works.
function PowModSim($Value, $Exponent, $Modulus)
{
// Check if simulation is even necessary.
if (function_exists("bcpowmod"))
return (bcpowmod($Value, $Exponent, $Modulus));
// Loop until the exponent is reduced to zero.
$Result = "1";
while (TRUE)
{
if (bcmod($Exponent, 2) == "1")
$Result = bcmod(bcmul($Result, $Value), $Modulus);
if (($Exponent = bcdiv($Exponent, 2)) == "0") break;
$Value = bcmod(bcmul($Value, $Value), $Modulus);
}
return ($Result);
}