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 — Potenz einer Zahl beliebiger Genauigkeit, vermindert um ein angegebenen Modulo
Beschreibung
string bcpowmod
( string $left_operand
, string $right_operand
, string $modulus
[, int $scale
] )
Benutzen Sie diese schnelle Exponentialmethode, um left_operand mit right_operand unter Berücksichtigung des Modulo modulus zu potenzieren.
Parameter-Liste
- left_operand
-
Der linke Operand in Stringform.
- right_operand
-
Der rechte Operand in Stringform.
- modulus
-
Modulo in Stringform.
- scale
-
Dieser optionale Parameter kann benutzt werden um die Anzahl der Nachkommastellen im Ergebnis festzulegen. Sie können mit Hilfe der bcscale() Funktion auch einen globalen Vorgabewert für alle Funktionen festlegen.
Rückgabewerte
Gibt das Ergebnis als String zurück. Ist modulus 0, wird NULL zurückgegeben.
Anmerkungen
Hinweis: Da diese Methode mit der Modulo-Operation arbeitet, können nicht-natürliche Zahlen zu unerwarteten Ergebnissen führen. Eine natürliche Zahl ist jede positive Zahl, die nicht null ist.
Beispiele
Die folgenden zwei Anweisungen sind funktional identisch. Die bcpowmod()-Version ist allerdings schneller und akzeptiert größere Parameter.
<?php
$a = bcpowmod($x, $y, $mod);
$b = bcmod(bcpow($x, $y), $mod);
// $a und $b entsprechen einander.
?>
add a note
User Contributed Notesbcpowmod
william at example dot com
21-Feb-2007 02:04
21-Feb-2007 02:04
laysoft at gmail dot com
30-Jan-2007 05:34
30-Jan-2007 05:34
I found a better way to emulate bcpowmod on PHP 4, which works with very big numbers too:
function powmod($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);
}