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 — Eleva un número de precisión arbitraria a otro, reducido por un módulo especificado
Descripción
string bcpowmod
( string $operando_izquierda
, string $operando_derecha
, string $modulo
[, int $escala
] )
Utiliza el método de exponenciación-rápida para elevar operando_izquierda a la potencia operando_derecha con respecto al módulo modulo .
Lista de parámetros
- operando_izquierda
-
El operando de la izquierda, como una cadena.
- operando_derecha
-
El operando de la derecha, como una cadena.
- modulo
-
El módulo, como una cadena.
- scale
-
This optional parameter is used to set the number of digits after the decimal place in the result. You can also set the global default scale for all functions by using bcscale().
Valores retornados
Devuelve el resultado como una cadena, o NULL si modulo es 0.
Notes
Note: Dado que éste método usa la operación de módulo, los números no-naturales pueden dar resultados inesperados. Un número natural es cualquier entero positivo diferente de cero.
Ejemplos
Las siguientes dos declaraciones son funcionalmente idénticas. Sin embargo, la versión que usa bcpowmod() se ejecuta en menos tiempo y puede aceptar parámetros más grandes.
<?php
$a = bcpowmod($x, $y, $mod);
$b = bcmod(bcpow($x, $y),$mod);
// $a y $b poseen valores iguales.
?>
add a note
User Contributed Notesbcpowmod
william at example dot com
21-Feb-2007 11:04
21-Feb-2007 11:04
laysoft at gmail dot com
30-Jan-2007 02:34
30-Jan-2007 02:34
I found a better way to emulate bcpowmod on PHP 4, which works with very big numbers too:
function powmod($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 11:46
15-May-2006 11:46
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
28-Sep-2005 06:46
28-Sep-2005 06:46
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);
}