bcpowmod()

(PHP 5, PHP 7)

Raise an arbitrary precision number to another, reduced by a specified modulus

说明

bcpowmod(string $base,string $exponent,string $modulus[,int $scale= 0]): string

Use the fast-exponentiation method to raise$baseto the power$exponentwith respect to the modulus$modulus.

参数

$base: The base, as an integral string(i.e. the scale has to be zero).
$exponent: The exponent, as an non-negative, integral string(i.e. the scale has to be zero).
$modulus: The modulus, as an integral string(i.e. the scale has to be zero).
$scale: 此可选参数用于设置结果中小数点后的小数位数。也可通过使用bcscale()来设置全局默认的小数位数，用于所有函数。

返回值

Returns the result as a string, or NULL if$modulusis0or$exponentis negative.

注释

Note:
Because this method uses the modulus operation, numbers which are not positive integers may give unexpected results.

范例

The following two statements are functionally identical. The bcpowmod() version however, executes in less time and can accept larger parameters.

<?php
$a = bcpowmod($x, $y, $mod);
$b = bcmod(bcpow($x, $y), $mod);
// $a and $b are equal to each other.
?>

参见

bcpow()任意精度数字的乘方
bcmod()对一个任意精度数字取模

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);
  }
}

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);
 }

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."