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
说明
string bcpowmod ( string left_operand, string right_operand, string modulus [, int scale] )
Use the fast-exponentiation method to raise
left_operand to the power
right_operand with respect to the modulus
modulus.
参数
left_operandThe left operand, as a string.
right_operandThe right operand, as a string.
modulusThe modulus, as a string.
scaleThis 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().
注释
注意: Because this method uses the modulus operation, non-natural numbers may give unexpected results. A natural number is any positive non-zero integer.
范例
The following two statements are functionally identical. The bcpowmod() version however, executes in less time and can accept larger parameters.
add a note
User Contributed Notes
william at example dot com
21-Feb-2007 10:04
21-Feb-2007 10:04
laysoft at gmail dot com
30-Jan-2007 01:34
30-Jan-2007 01:34
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 09:46
15-May-2006 09:46
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 04:46
28-Sep-2005 04:46
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);
}