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<?php |
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/* |
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* The MIT License (MIT) |
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* |
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* Copyright (c) 2014-2016 Spomky-Labs |
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* |
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* This software may be modified and distributed under the terms |
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* of the MIT license. See the LICENSE file for details. |
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*/ |
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namespace Jose\Util; |
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final class BigInteger |
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{ |
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/**#@+ |
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* Array constants |
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* |
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* Rather than create a thousands and thousands of new BigInteger objects in repeated function calls to add() and |
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* multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them. |
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* |
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*/ |
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/** |
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* $result[self::VALUE] contains the value. |
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*/ |
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const VALUE = 0; |
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/** |
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* $result[self::SIGN] contains the sign. |
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*/ |
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const SIGN = 1; |
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/**#@-*/ |
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/**#@+ |
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* Static properties used by the pure-PHP implementation. |
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* |
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* @see __construct() |
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*/ |
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private static $base; |
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private static $baseFull; |
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private static $maxDigit; |
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private static $msb; |
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/** |
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* $max10 in greatest $max10Len satisfying |
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* $max10 = 10**$max10Len <= 2**$base. |
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*/ |
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private static $max10; |
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/** |
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* $max10Len in greatest $max10Len satisfying |
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* $max10 = 10**$max10Len <= 2**$base. |
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*/ |
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private static $max10Len; |
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private static $maxDigit2; |
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/**#@-*/ |
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/** |
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* Holds the BigInteger's value. |
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* |
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* @var resource |
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*/ |
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private $value; |
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/** |
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* Holds the BigInteger's magnitude. |
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* |
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* @var bool |
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*/ |
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private $is_negative = false; |
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/** |
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* Precision. |
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*/ |
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private $precision = -1; |
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/** |
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* Precision Bitmask. |
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*/ |
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private $bitmask = false; |
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/** |
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* Converts base-2, base-10, base-16, and binary strings (base-256) to BigIntegers. |
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* |
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* If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using |
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* two's compliment. The sole exception to this is -10, which is treated the same as 10 is. |
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* |
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* Here's an example: |
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* <code> |
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* <?php |
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* $a = new \Jose\Util\in base-16 |
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* |
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* echo $a->toString(); // outputs 50 |
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* ?> |
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* </code> |
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* |
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* @param $x base-10 number or base-$base number if $base set. |
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* @param int $base |
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*/ |
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public function __construct($x = 0, $base = 10) |
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{ |
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switch (true) { |
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case is_resource($x) && get_resource_type($x) == 'GMP integer': |
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// PHP 5.6 switched GMP from using resources to objects |
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case $x instanceof \GMP: |
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$this->value = $x; |
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return; |
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} |
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$this->value = gmp_init(0); |
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// '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48 |
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// '0' is the only value like this per http://php.net/empty |
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if (empty($x) && (abs($base) != 256 || $x !== '0')) { |
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return; |
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} |
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switch ($base) { |
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case -256: |
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if (ord($x[0]) & 0x80) { |
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$x = ~$x; |
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$this->is_negative = true; |
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} |
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case 256: |
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$sign = $this->is_negative ? '-' : ''; |
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$this->value = gmp_init($sign.'0x'.bin2hex($x)); |
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if ($this->is_negative) { |
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$this->is_negative = false; |
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$temp = $this->add(new static('-1')); |
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$this->value = $temp->value; |
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} |
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break; |
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case 16: |
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case -16: |
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if ($base > 0 && $x[0] == '-') { |
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$this->is_negative = true; |
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$x = substr($x, 1); |
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} |
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$x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x); |
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$is_negative = false; |
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if ($base < 0 && hexdec($x[0]) >= 8) { |
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$this->is_negative = $is_negative = true; |
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$x = bin2hex(~hex2bin($x)); |
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} |
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$temp = $this->is_negative ? '-0x'.$x : '0x'.$x; |
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$this->value = gmp_init($temp); |
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$this->is_negative = false; |
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if ($is_negative) { |
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$temp = $this->add(new static('-1')); |
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$this->value = $temp->value; |
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} |
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break; |
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case 10: |
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case -10: |
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// (?<!^)(?:-).*: find any -'s that aren't at the beginning and then any characters that follow that |
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// (?<=^|-)0*: find any 0's that are preceded by the start of the string or by a - (ie. octals) |
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// [^-0-9].*: find any non-numeric characters and then any characters that follow that |
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$x = preg_replace('#(?<!^)(?:-).*|(?<=^|-)0*|[^-0-9].*#', '', $x); |
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$this->value = gmp_init($x); |
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break; |
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case 2: // base-2 support originally implemented by Lluis Pamies - thanks! |
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case -2: |
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if ($base > 0 && $x[0] == '-') { |
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$this->is_negative = true; |
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$x = substr($x, 1); |
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} |
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$x = preg_replace('#^([01]*).*#', '$1', $x); |
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$x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT); |
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$str = '0x'; |
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while (strlen($x)) { |
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$part = substr($x, 0, 4); |
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$str .= dechex(bindec($part)); |
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$x = substr($x, 4); |
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} |
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if ($this->is_negative) { |
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$str = '-'.$str; |
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} |
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$temp = new static($str, 8 * $base); // ie. either -16 or +16 |
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$this->value = $temp->value; |
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$this->is_negative = $temp->is_negative; |
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break; |
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default: |
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// base not supported, so we'll let $this == 0 |
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} |
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} |
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/** |
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* Converts a BigInteger to a byte string (eg. base-256). |
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* |
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* Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're |
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* saved as two's compliment. |
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* |
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* Here's an example: |
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* <code> |
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* <?php |
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* $a = new \Jose\Util\ger('65'); |
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* |
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* echo $a->toBytes(); // outputs chr(65) |
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* ?> |
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* </code> |
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* |
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* @param bool $twos_compliment |
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* |
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* @return string |
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* |
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*/ |
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public function toBytes($twos_compliment = false) |
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{ |
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if ($twos_compliment) { |
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$comparison = $this->compare(new static()); |
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if ($comparison == 0) { |
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return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; |
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} |
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$temp = $comparison < 0 ? $this->add(new static(1)) : $this; |
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$bytes = $temp->toBytes(); |
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if (empty($bytes)) { // eg. if the number we're trying to convert is -1 |
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$bytes = chr(0); |
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} |
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if (ord($bytes[0]) & 0x80) { |
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$bytes = chr(0).$bytes; |
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} |
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return $comparison < 0 ? ~$bytes : $bytes; |
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} |
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if (gmp_cmp($this->value, gmp_init(0)) == 0) { |
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return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; |
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} |
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$temp = gmp_strval(gmp_abs($this->value), 16); |
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$temp = (strlen($temp) & 1) ? '0'.$temp : $temp; |
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$temp = hex2bin($temp); |
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return $this->precision > 0 ? |
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substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) : |
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ltrim($temp, chr(0)); |
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} |
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/** |
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* Adds two BigIntegers. |
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* |
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* Here's an example: |
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* <code> |
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* <?php |
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* $a = new \Jose\Util\ger('10'); |
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* $b = new \Jose\Util\ger('20'); |
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* |
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* $c = $a->add($b); |
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* |
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* echo $c->toString(); // outputs 30 |
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* ?> |
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* </code> |
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* |
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* @param \Jose\Util\BigInteger $y |
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* |
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* @return \Jose\Util\BigInteger |
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* |
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*/ |
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public function add(BigInteger $y) |
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{ |
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$temp = new static(); |
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$temp->value = gmp_add($this->value, $y->value); |
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return $this->_normalize($temp); |
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} |
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/** |
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* Subtracts two BigIntegers. |
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* |
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* Here's an example: |
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* <code> |
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* <?php |
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* $a = new \Jose\Util\ger('10'); |
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* $b = new \Jose\Util\ger('20'); |
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* |
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* $c = $a->subtract($b); |
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* |
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* echo $c->toString(); // outputs -10 |
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* ?> |
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* </code> |
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* |
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* @param \Jose\Util\BigInteger $y |
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* |
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* @return \Jose\Util\BigInteger |
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* |
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*/ |
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public function subtract(BigInteger $y) |
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{ |
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$temp = new static(); |
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$temp->value = gmp_sub($this->value, $y->value); |
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return $this->_normalize($temp); |
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} |
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308
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/** |
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* Multiplies two BigIntegers. |
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* |
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* Here's an example: |
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* <code> |
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* <?php |
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* $a = new \Jose\Util\ger('10'); |
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* $b = new \Jose\Util\ger('20'); |
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* |
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* $c = $a->multiply($b); |
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* |
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* echo $c->toString(); // outputs 200 |
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* ?> |
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* </code> |
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* |
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* @param \Jose\Util\BigInteger $x |
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* |
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* @return \Jose\Util\BigInteger |
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*/ |
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public function multiply(BigInteger $x) |
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{ |
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$temp = new static(); |
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$temp->value = gmp_mul($this->value, $x->value); |
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332
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return $this->_normalize($temp); |
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} |
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335
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/** |
336
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* Divides two BigIntegers. |
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* |
338
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* Returns an array whose first element contains the quotient and whose second element contains the |
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* "common residue". If the remainder would be positive, the "common residue" and the remainder are the |
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* same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder |
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* and the divisor (basically, the "common residue" is the first positive modulo). |
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* |
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* Here's an example: |
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* <code> |
345
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* <?php |
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* $a = new \Jose\Util\ger('10'); |
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* $b = new \Jose\Util\ger('20'); |
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* |
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* list($quotient, $remainder) = $a->divide($b); |
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* |
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* echo $quotient->toString(); // outputs 0 |
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* echo "\r\n"; |
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* echo $remainder->toString(); // outputs 10 |
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* ?> |
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* </code> |
356
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* |
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* @param \Jose\Util\BigInteger $y |
358
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* |
359
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* @return @return \Jose\Util\BigInteger[] |
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360
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* |
361
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*/ |
362
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public function divide(BigInteger $y) |
363
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{ |
364
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$quotient = new static(); |
365
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$remainder = new static(); |
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367
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list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value); |
368
|
|
|
|
369
|
|
|
if (gmp_sign($remainder->value) < 0) { |
370
|
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|
$remainder->value = gmp_add($remainder->value, gmp_abs($y->value)); |
371
|
|
|
} |
372
|
|
|
|
373
|
|
|
return [$this->_normalize($quotient), $this->_normalize($remainder)]; |
374
|
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|
} |
375
|
|
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|
376
|
|
|
/** |
377
|
|
|
* Performs modular exponentiation. |
378
|
|
|
* |
379
|
|
|
* Here's an example: |
380
|
|
|
* <code> |
381
|
|
|
* <?php |
382
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|
|
* $a = new \Jose\Util\ger('10'); |
383
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|
|
* $b = new \Jose\Util\ger('20'); |
384
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|
|
* $c = new \Jose\Util\ger('30'); |
385
|
|
|
* |
386
|
|
|
* $c = $a->modPow($b, $c); |
387
|
|
|
* |
388
|
|
|
* echo $c->toString(); // outputs 10 |
389
|
|
|
* ?> |
390
|
|
|
* </code> |
391
|
|
|
* |
392
|
|
|
* @param \Jose\Util\BigInteger $e |
393
|
|
|
* @param \Jose\Util\BigInteger $n |
394
|
|
|
* |
395
|
|
|
* @return \Jose\Util\BigInteger |
396
|
|
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* |
397
|
|
|
* and although the approach involving repeated squaring does vastly better, it, too, is impractical |
398
|
|
|
* for our purposes. The reason being that division - by far the most complicated and time-consuming |
399
|
|
|
* of the basic operations (eg. +,-,*,/) - occurs multiple times within it. |
400
|
|
|
* |
401
|
|
|
* Modular reductions resolve this issue. Although an individual modular reduction takes more time |
402
|
|
|
* then an individual division, when performed in succession (with the same modulo), they're a lot faster. |
403
|
|
|
* |
404
|
|
|
* The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction, |
405
|
|
|
* although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the |
406
|
|
|
* base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because |
407
|
|
|
* the product of two odd numbers is odd), but what about when RSA isn't used? |
408
|
|
|
* |
409
|
|
|
* In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a |
410
|
|
|
* Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the |
411
|
|
|
* modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however, |
412
|
|
|
* uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and |
413
|
|
|
* the other, a power of two - and recombine them, later. This is the method that this modPow function uses. |
414
|
|
|
* {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates. |
415
|
|
|
*/ |
416
|
|
|
public function modPow(BigInteger $e, BigInteger $n) |
417
|
|
|
{ |
418
|
|
|
$n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs(); |
|
|
|
|
419
|
|
|
|
420
|
|
|
if ($e->compare(new static()) < 0) { |
421
|
|
|
$e = $e->abs(); |
422
|
|
|
|
423
|
|
|
$temp = $this->modInverse($n); |
|
|
|
|
424
|
|
|
if ($temp === false) { |
425
|
|
|
return false; |
|
|
|
|
426
|
|
|
} |
427
|
|
|
|
428
|
|
|
return $this->_normalize($temp->modPow($e, $n)); |
|
|
|
|
429
|
|
|
} |
430
|
|
|
|
431
|
|
|
$temp = new static(); |
432
|
|
|
$temp->value = gmp_powm($this->value, $e->value, $n->value); |
433
|
|
|
|
434
|
|
|
return $this->_normalize($temp); |
435
|
|
|
} |
436
|
|
|
|
437
|
|
|
/** |
438
|
|
|
* Calculates modular inverses. |
439
|
|
|
* |
440
|
|
|
* Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses. |
441
|
|
|
* |
442
|
|
|
* Here's an example: |
443
|
|
|
* <code> |
444
|
|
|
* <?php |
445
|
|
|
* $a = new \Jose\Util\teger(30); |
446
|
|
|
* $b = new \Jose\Util\teger(17); |
447
|
|
|
* |
448
|
|
|
* $c = $a->modInverse($b); |
449
|
|
|
* echo $c->toString(); // outputs 4 |
450
|
|
|
* |
451
|
|
|
* echo "\r\n"; |
452
|
|
|
* |
453
|
|
|
* $d = $a->multiply($c); |
454
|
|
|
* list(, $d) = $d->divide($b); |
455
|
|
|
* echo $d; // outputs 1 (as per the definition of modular inverse) |
456
|
|
|
* ?> |
457
|
|
|
* </code> |
458
|
|
|
* |
459
|
|
|
* @param \Jose\Util\BigInteger $n |
460
|
|
|
* |
461
|
|
|
* @return \Jose\Util\BigInteger|bool |
462
|
|
|
* |
463
|
|
|
*/ |
464
|
|
|
public function modInverse(BigInteger $n) |
465
|
|
|
{ |
466
|
|
|
$temp = new static(); |
467
|
|
|
$temp->value = gmp_invert($this->value, $n->value); |
468
|
|
|
|
469
|
|
|
return ($temp->value === false) ? false : $this->_normalize($temp); |
470
|
|
|
} |
471
|
|
|
|
472
|
|
|
/** |
473
|
|
|
* Absolute value. |
474
|
|
|
* |
475
|
|
|
* @return \Jose\Util\BigInteger |
476
|
|
|
*/ |
477
|
|
|
public function abs() |
478
|
|
|
{ |
479
|
|
|
$temp = new static(); |
480
|
|
|
|
481
|
|
|
$temp->value = gmp_abs($this->value); |
482
|
|
|
|
483
|
|
|
return $temp; |
484
|
|
|
} |
485
|
|
|
|
486
|
|
|
/** |
487
|
|
|
* Compares two numbers. |
488
|
|
|
* |
489
|
|
|
* Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is |
490
|
|
|
* demonstrated thusly: |
491
|
|
|
* |
492
|
|
|
* $x > $y: $x->compare($y) > 0 |
493
|
|
|
* $x < $y: $x->compare($y) < 0 |
494
|
|
|
* $x == $y: $x->compare($y) == 0 |
495
|
|
|
* |
496
|
|
|
* Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y). |
497
|
|
|
* |
498
|
|
|
* @param \Jose\Util\BigInteger $y |
499
|
|
|
* |
500
|
|
|
* @return int < 0 if $this is less than $y; > 0 if $this is greater than $y, and 0 if they are equal. |
501
|
|
|
* |
502
|
|
|
*/ |
503
|
|
|
public function compare(BigInteger $y) |
504
|
|
|
{ |
505
|
|
|
return gmp_cmp($this->value, $y->value); |
506
|
|
|
} |
507
|
|
|
|
508
|
|
|
/** |
509
|
|
|
* Logical Left Shift. |
510
|
|
|
* |
511
|
|
|
* Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift. |
512
|
|
|
* |
513
|
|
|
* @param int $shift |
514
|
|
|
* |
515
|
|
|
* @return \Jose\Util\BigInteger |
516
|
|
|
* |
517
|
|
|
*/ |
518
|
|
|
public function bitwise_leftShift($shift) |
519
|
|
|
{ |
520
|
|
|
$temp = new static(); |
521
|
|
|
|
522
|
|
|
static $two; |
523
|
|
|
|
524
|
|
|
if (!isset($two)) { |
525
|
|
|
$two = gmp_init('2'); |
526
|
|
|
} |
527
|
|
|
|
528
|
|
|
$temp->value = gmp_mul($this->value, gmp_pow($two, $shift)); |
529
|
|
|
|
530
|
|
|
return $this->_normalize($temp); |
531
|
|
|
} |
532
|
|
|
|
533
|
|
|
/** |
534
|
|
|
* Generates a random BigInteger. |
535
|
|
|
* |
536
|
|
|
* Byte length is equal to $length. Uses \phpseclib\Crypt\Random if it's loaded and mt_rand if it's not. |
537
|
|
|
* |
538
|
|
|
* @param int $size |
539
|
|
|
* |
540
|
|
|
* @return \Jose\Util\BigInteger |
541
|
|
|
*/ |
542
|
|
|
private static function _random_number_helper($size) |
543
|
|
|
{ |
544
|
|
|
return new static(random_bytes($size), 256); |
545
|
|
|
} |
546
|
|
|
|
547
|
|
|
/** |
548
|
|
|
* Generate a random number. |
549
|
|
|
* |
550
|
|
|
* Returns a random number between $min and $max where $min and $max |
551
|
|
|
* can be defined using one of the two methods: |
552
|
|
|
* |
553
|
|
|
* BigInteger::random($min, $max) |
554
|
|
|
* BigInteger::random($max, $min) |
555
|
|
|
* |
556
|
|
|
* @param \Jose\Util\BigInteger $min |
557
|
|
|
* @param \Jose\Util\BigInteger $max |
558
|
|
|
* |
559
|
|
|
* @return \Jose\Util\BigInteger |
560
|
|
|
*/ |
561
|
|
|
public static function random(BigInteger $min, BigInteger $max) |
562
|
|
|
{ |
563
|
|
|
$compare = $max->compare($min); |
564
|
|
|
|
565
|
|
|
if (!$compare) { |
566
|
|
|
return $this->_normalize($min); |
|
|
|
|
567
|
|
|
} elseif ($compare < 0) { |
568
|
|
|
// if $min is bigger then $max, swap $min and $max |
569
|
|
|
$temp = $max; |
570
|
|
|
$max = $min; |
571
|
|
|
$min = $temp; |
572
|
|
|
} |
573
|
|
|
|
574
|
|
|
static $one; |
575
|
|
|
if (!isset($one)) { |
576
|
|
|
$one = new static(1); |
577
|
|
|
} |
578
|
|
|
|
579
|
|
|
$max = $max->subtract($min->subtract($one)); |
580
|
|
|
$size = strlen(ltrim($max->toBytes(), chr(0))); |
581
|
|
|
|
582
|
|
|
/* |
583
|
|
|
doing $random % $max doesn't work because some numbers will be more likely to occur than others. |
584
|
|
|
eg. if $max is 140 and $random's max is 255 then that'd mean both $random = 5 and $random = 145 |
585
|
|
|
would produce 5 whereas the only value of random that could produce 139 would be 139. ie. |
586
|
|
|
not all numbers would be equally likely. some would be more likely than others. |
587
|
|
|
|
588
|
|
|
creating a whole new random number until you find one that is within the range doesn't work |
589
|
|
|
because, for sufficiently small ranges, the likelihood that you'd get a number within that range |
590
|
|
|
would be pretty small. eg. with $random's max being 255 and if your $max being 1 the probability |
591
|
|
|
would be pretty high that $random would be greater than $max. |
592
|
|
|
|
593
|
|
|
phpseclib works around this using the technique described here: |
594
|
|
|
|
595
|
|
|
http://crypto.stackexchange.com/questions/5708/creating-a-small-number-from-a-cryptographically-secure-random-string |
596
|
|
|
*/ |
597
|
|
|
$random_max = new static(chr(1).str_repeat("\0", $size), 256); |
598
|
|
|
$random = self::_random_number_helper($size); |
599
|
|
|
|
600
|
|
|
list($max_multiple) = $random_max->divide($max); |
601
|
|
|
$max_multiple = $max_multiple->multiply($max); |
602
|
|
|
|
603
|
|
|
while ($random->compare($max_multiple) >= 0) { |
604
|
|
|
$random = $random->subtract($max_multiple); |
605
|
|
|
$random_max = $random_max->subtract($max_multiple); |
606
|
|
|
$random = $random->bitwise_leftShift(8); |
607
|
|
|
$random = $random->add(self::_random_number_helper(1)); |
608
|
|
|
$random_max = $random_max->bitwise_leftShift(8); |
609
|
|
|
list($max_multiple) = $random_max->divide($max); |
610
|
|
|
$max_multiple = $max_multiple->multiply($max); |
611
|
|
|
} |
612
|
|
|
list(, $random) = $random->divide($max); |
613
|
|
|
|
614
|
|
|
return $random->add($min); |
615
|
|
|
} |
616
|
|
|
|
617
|
|
|
/** |
618
|
|
|
* Normalize. |
619
|
|
|
* |
620
|
|
|
* Removes leading zeros and truncates (if necessary) to maintain the appropriate precision |
621
|
|
|
* |
622
|
|
|
* @param \Jose\Util\BigInteger $result |
623
|
|
|
* |
624
|
|
|
* @return \Jose\Util\BigInteger |
625
|
|
|
*/ |
626
|
|
|
private function _normalize($result) |
627
|
|
|
{ |
628
|
|
|
$result->precision = $this->precision; |
629
|
|
|
$result->bitmask = $this->bitmask; |
630
|
|
|
|
631
|
|
|
if ($this->bitmask !== false) { |
632
|
|
|
$result->value = gmp_and($result->value, $result->bitmask->value); |
633
|
|
|
} |
634
|
|
|
|
635
|
|
|
return $result; |
636
|
|
|
} |
637
|
|
|
} |
638
|
|
|
|
This check marks private properties in classes that are never used. Those properties can be removed.