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 — Aumenta um número de precisão arbitrária para outro, reduzido por um módulo especificado
Descrição
string bcpowmod
( string $left_operand
, string $right_operand
, string $modulus
[, int $scale
] )
Use o método de exponenciação rápida para elevar left_operand para a potência right_operand levando em conta o módulo modulus .
Parâmetros
- left_operand
-
O operando da esquerda, do tipo string.
- right_operand
-
O operando da direita, do tipo string.
- modulus
-
O módulo, do tipo string.
- scale
-
Este parâmetro opcional é usado para definir o número de digitos depois do ponto decimal no resultado. Você pode definir a escala padrão global para todas as funções usando bcscale().
Valor Retornado
Retorna um string com resultado ou NULL se modulus for 0.
Notas
Nota: Por que esse método usa a operação de módulo, números não-naturais pode dar resultados inexperados. Um número natural é qualquer inteiro positivo diferente de zero.
Exemplos
As seguintes instruções têm funcionalidades idênticas. A versão bcpowmod(), no entanto, executa em menos tempo e pode aceitar parâmetros maiores.
<?php
$a = bcpowmod($x, $y, $mod);
$b = bcmod(bcpow($x, $y), $mod);
// $a and $b are equal to each other.
?>
add a note
User Contributed Notesbcpowmod
william at example dot com
21-Feb-2007 11:04
21-Feb-2007 11:04
laysoft at gmail dot com
30-Jan-2007 02:34
30-Jan-2007 02:34
I found a better way to emulate bcpowmod on PHP 4, which works with very big numbers too:
function powmod($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);
}