tgseclib/phpseclib/Math/BigInteger/Engines/PHP.php

<?php

/**
 * Pure-PHP BigInteger Engine
 *
 * PHP version 5 and 7
 *
 * @category  Math
 * @package   BigInteger
 * @author    Jim Wigginton <terrafrost@php.net>
 * @copyright 2017 Jim Wigginton
 * @license   http://www.opensource.org/licenses/mit-license.html  MIT License
 * @link      http://pear.php.net/package/Math_BigInteger
 */

namespace phpseclib\Math\BigInteger\Engines;

use ParagonIE\ConstantTime\Hex;
use phpseclib\Exception\BadConfigurationException;

/**
 * pure-PHP Engine.
 *
 * @package PHP
 * @author  Jim Wigginton <terrafrost@php.net>
 * @access  public
 */
abstract class PHP extends Engine
{
    /**#@+
     * Array constants
     *
     * Rather than create a thousands and thousands of new BigInteger objects in repeated function calls to add() and
     * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.
     *
     * @access protected
    */
    /**
     * $result[self::VALUE] contains the value.
     */
    const VALUE = 0;
    /**
     * $result[self::SIGN] contains the sign.
     */
    const SIGN = 1;
    /**#@-*/

    /**
     * Karatsuba Cutoff
     *
     * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
     *
     * @access private
     */
    const KARATSUBA_CUTOFF = 25;

    /**
     * Can Bitwise operations be done fast?
     *
     * @see parent::bitwise_leftRotate()
     * @see parent::bitwise_rightRotate()
     * @access protected
     */
    const FAST_BITWISE = true;

    /**
     * Engine Directory
     *
     * @see parent::setModExpEngine
     * @access protected
     */
    const ENGINE_DIR = 'PHP';

    /**
     * Default constructor
     *
     * @param mixed $x integer Base-10 number or base-$base number if $base set.
     * @param int $base
     * @see parent::__construct()
     * @return \phpseclib\Math\BigInteger\Engines\PHP
     */
    public function __construct($x = 0, $base = 10)
    {
        if (!isset(static::$isValidEngine)) {
            static::$isValidEngine = static::isValidEngine();
        }
        if (!static::$isValidEngine) {
            throw new BadConfigurationException(static::class . ' is not setup correctly on this system');
        }

        $this->value = [];
        parent::__construct($x, $base);
    }

    /**
     * Initialize a PHP BigInteger Engine instance
     *
     * @param int $base
     * @see parent::__construct()
     */
    protected function initialize($base)
    {
        switch (abs($base)) {
            case 256:
                $x = $this->value;
                $this->value = [];

                while (strlen($x)) {
                    $this->value[] = self::bytes2int(self::base256_rshift($x, static::BASE));
                }

                if ($this->value == [0]) {
                    $this->value = [];
                }
                break;
            case 16:
                $x = (strlen($this->value) & 1) ? '0' . $this->value : $this->value;
                $temp = new static(Hex::decode($x), 256);
                $this->value = $temp->value;
                break;
            case 10:
                $temp = new static();

                $multiplier = new static();
                $multiplier->value = [static::MAX10];

                $x = $this->value;

                if ($x[0] == '-') {
                    $this->is_negative = true;
                    $x = substr($x, 1);
                }

                $x = str_pad($x, strlen($x) + ((static::MAX10LEN - 1) * strlen($x)) % static::MAX10LEN, 0, STR_PAD_LEFT);
                while (strlen($x)) {
                    $temp = $temp->multiply($multiplier);
                    $temp = $temp->add(new static($this->int2bytes(substr($x, 0, static::MAX10LEN)), 256));
                    $x = substr($x, static::MAX10LEN);
                }

                $this->value = $temp->value;
        }
    }

    /**
     * Pads strings so that unpack may be used on them
     *
     * @param string $str
     * @return string
     */
    protected function pad($str)
    {
        $length = strlen($str);

        $pad = 4 - (strlen($str) % 4);

        return str_pad($str, $length + $pad, "\0", STR_PAD_LEFT);
    }

    /**
     * Converts a BigInteger to a base-10 number.
     *
     * @return string
     */
    public function toString()
    {
        if (!count($this->value)) {
            return '0';
        }

        $temp = clone $this;
        $temp->is_negative = false;

        $divisor = new static();
        $divisor->value = [static::MAX10];
        $result = '';
        while (count($temp->value)) {
            list($temp, $mod) = $temp->divide($divisor);
            $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', static::MAX10LEN, '0', STR_PAD_LEFT) . $result;
        }
        $result = ltrim($result, '0');
        if (empty($result)) {
            $result = '0';
        }

        if ($this->is_negative) {
            $result = '-' . $result;
        }

        return $result;
    }

    /**
     * Converts a BigInteger to a byte string (eg. base-256).
     *
     * @param bool $twos_compliment
     * @return string
     */
    public function toBytes($twos_compliment = false)
    {
        if ($twos_compliment) {
            return $this->toBytesHelper();
        }

        if (!count($this->value)) {
            return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
        }
        $result = self::int2bytes($this->value[count($this->value) - 1]);

        for ($i = count($this->value) - 2; $i >= 0; --$i) {
            self::base256_lshift($result, static::BASE);
            $result = $result | str_pad(self::int2bytes($this->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);
        }

        return $this->precision > 0 ?
            str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 3, chr(0), STR_PAD_LEFT) :
            $result;
    }

    /**
     * Performs addition.
     *
     * @param array $x_value
     * @param bool $x_negative
     * @param array $y_value
     * @param bool $y_negative
     * @return array
     */
    protected static function addHelper(array $x_value, $x_negative, array $y_value, $y_negative)
    {
        $x_size = count($x_value);
        $y_size = count($y_value);

        if ($x_size == 0) {
            return [
                self::VALUE => $y_value,
                self::SIGN => $y_negative
            ];
        } elseif ($y_size == 0) {
            return [
                self::VALUE => $x_value,
                self::SIGN => $x_negative
            ];
        }

        // subtract, if appropriate
        if ($x_negative != $y_negative) {
            if ($x_value == $y_value) {
                return [
                    self::VALUE => [],
                    self::SIGN => false
                ];
            }

            $temp = self::subtractHelper($x_value, false, $y_value, false);
            $temp[self::SIGN] = self::compareHelper($x_value, false, $y_value, false) > 0 ?
                                          $x_negative : $y_negative;

            return $temp;
        }

        if ($x_size < $y_size) {
            $size = $x_size;
            $value = $y_value;
        } else {
            $size = $y_size;
            $value = $x_value;
        }

        $value[count($value)] = 0; // just in case the carry adds an extra digit

        $carry = 0;
        for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {
            //$sum = $x_value[$j] * static::BASE_FULL + $x_value[$i] + $y_value[$j] * static::BASE_FULL + $y_value[$i] + $carry;
            $sum = ($x_value[$j] + $y_value[$j]) * static::BASE_FULL + $x_value[$i] + $y_value[$i] + $carry;
            $carry = $sum >= static::MAX_DIGIT2; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum - static::MAX_DIGIT2 : $sum;

            $temp = static::BASE === 26 ? intval($sum / 0x4000000) : ($sum >> 31);

            $value[$i] = (int) ($sum - static::BASE_FULL * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)
            $value[$j] = $temp;
        }

        if ($j == $size) { // ie. if $y_size is odd
            $sum = $x_value[$i] + $y_value[$i] + $carry;
            $carry = $sum >= static::BASE_FULL;
            $value[$i] = $carry ? $sum - static::BASE_FULL : $sum;
            ++$i; // ie. let $i = $j since we've just done $value[$i]
        }

        if ($carry) {
            for (; $value[$i] == static::MAX_DIGIT; ++$i) {
                $value[$i] = 0;
            }
            ++$value[$i];
        }

        return [
            self::VALUE => self::trim($value),
            self::SIGN => $x_negative
        ];
    }

    /**
     * Performs subtraction.
     *
     * @param array $x_value
     * @param bool $x_negative
     * @param array $y_value
     * @param bool $y_negative
     * @return array
     */
    static function subtractHelper(array $x_value, $x_negative, array $y_value, $y_negative)
    {
        $x_size = count($x_value);
        $y_size = count($y_value);

        if ($x_size == 0) {
            return [
                self::VALUE => $y_value,
                self::SIGN => !$y_negative
            ];
        } elseif ($y_size == 0) {
            return [
                self::VALUE => $x_value,
                self::SIGN => $x_negative
            ];
        }

        // add, if appropriate (ie. -$x - +$y or +$x - -$y)
        if ($x_negative != $y_negative) {
            $temp = self::addHelper($x_value, false, $y_value, false);
            $temp[self::SIGN] = $x_negative;

            return $temp;
        }

        $diff = self::compareHelper($x_value, $x_negative, $y_value, $y_negative);

        if (!$diff) {
            return [
                self::VALUE => [],
                self::SIGN => false
            ];
        }

        // switch $x and $y around, if appropriate.
        if ((!$x_negative && $diff < 0) || ($x_negative && $diff > 0)) {
            $temp = $x_value;
            $x_value = $y_value;
            $y_value = $temp;

            $x_negative = !$x_negative;

            $x_size = count($x_value);
            $y_size = count($y_value);
        }

        // at this point, $x_value should be at least as big as - if not bigger than - $y_value

        $carry = 0;
        for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) {
            $sum = ($x_value[$j] - $y_value[$j]) * static::BASE_FULL + $x_value[$i] - $y_value[$i] - $carry;

            $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum + static::MAX_DIGIT2 : $sum;

            $temp = static::BASE === 26 ? intval($sum / 0x4000000) : ($sum >> 31);

            $x_value[$i] = (int) ($sum - static::BASE_FULL * $temp);
            $x_value[$j] = $temp;
        }

        if ($j == $y_size) { // ie. if $y_size is odd
            $sum = $x_value[$i] - $y_value[$i] - $carry;
            $carry = $sum < 0;
            $x_value[$i] = $carry ? $sum + static::BASE_FULL : $sum;
            ++$i;
        }

        if ($carry) {
            for (; !$x_value[$i]; ++$i) {
                $x_value[$i] = static::MAX_DIGIT;
            }
            --$x_value[$i];
        }

        return [
            self::VALUE => self::trim($x_value),
            self::SIGN => $x_negative
        ];
    }

    /**
     * Performs multiplication.
     *
     * @param array $x_value
     * @param bool $x_negative
     * @param array $y_value
     * @param bool $y_negative
     * @return array
     */
    protected static function multiplyHelper(array $x_value, $x_negative, array $y_value, $y_negative)
    {
        //if ( $x_value == $y_value ) {
        //    return [
        //        self::VALUE => $this->_square($x_value),
        //        self::SIGN => $x_sign != $y_value
        //    ];
        //}

        $x_length = count($x_value);
        $y_length = count($y_value);

        if (!$x_length || !$y_length) { // a 0 is being multiplied
            return [
                self::VALUE => [],
                self::SIGN => false
            ];
        }

        return [
            self::VALUE => min($x_length, $y_length) < 2 * self::KARATSUBA_CUTOFF ?
                self::trim(self::regularMultiply($x_value, $y_value)) :
                self::trim(self::karatsuba($x_value, $y_value)),
            self::SIGN => $x_negative != $y_negative
        ];
    }

    /**
     * Performs Karatsuba multiplication on two BigIntegers
     *
     * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}.
     *
     * @param array $x_value
     * @param array $y_value
     * @return array
     */
    private static function karatsuba(array $x_value, array $y_value)
    {
        $m = min(count($x_value) >> 1, count($y_value) >> 1);

        if ($m < self::KARATSUBA_CUTOFF) {
            return self::regularMultiply($x_value, $y_value);
        }

        $x1 = array_slice($x_value, $m);
        $x0 = array_slice($x_value, 0, $m);
        $y1 = array_slice($y_value, $m);
        $y0 = array_slice($y_value, 0, $m);

        $z2 = self::karatsuba($x1, $y1);
        $z0 = self::karatsuba($x0, $y0);

        $z1 = self::addHelper($x1, false, $x0, false);
        $temp = self::addHelper($y1, false, $y0, false);
        $z1 = self::karatsuba($z1[self::VALUE], $temp[self::VALUE]);
        $temp = self::addHelper($z2, false, $z0, false);
        $z1 = self::subtractHelper($z1, false, $temp[self::VALUE], false);

        $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
        $z1[self::VALUE] = array_merge(array_fill(0, $m, 0), $z1[self::VALUE]);

        $xy = self::addHelper($z2, false, $z1[self::VALUE], $z1[self::SIGN]);
        $xy = self::addHelper($xy[self::VALUE], $xy[self::SIGN], $z0, false);

        return $xy[self::VALUE];
    }

    /**
     * Performs long multiplication on two BigIntegers
     *
     * Modeled after 'multiply' in MutableBigInteger.java.
     *
     * @param array $x_value
     * @param array $y_value
     * @return array
     */
    protected static function regularMultiply(array $x_value, array $y_value)
    {
        $x_length = count($x_value);
        $y_length = count($y_value);

        if (!$x_length || !$y_length) { // a 0 is being multiplied
            return [];
        }

        $product_value = self::array_repeat(0, $x_length + $y_length);

        // the following for loop could be removed if the for loop following it
        // (the one with nested for loops) initially set $i to 0, but
        // doing so would also make the result in one set of unnecessary adds,
        // since on the outermost loops first pass, $product->value[$k] is going
        // to always be 0

        $carry = 0;
        for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0
            $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
            $carry = static::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
            $product_value[$j] = (int) ($temp - static::BASE_FULL * $carry);
        }

        $product_value[$j] = $carry;

        // the above for loop is what the previous comment was talking about.  the
        // following for loop is the "one with nested for loops"
        for ($i = 1; $i < $y_length; ++$i) {
            $carry = 0;

            for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) {
                $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
                $carry = static::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                $product_value[$k] = (int) ($temp - static::BASE_FULL * $carry);
            }

            $product_value[$k] = $carry;
        }

        return $product_value;
    }

    /**
     * Divides two BigIntegers.
     *
     * Returns an array whose first element contains the quotient and whose second element contains the
     * "common residue".  If the remainder would be positive, the "common residue" and the remainder are the
     * same.  If the remainder would be negative, the "common residue" is equal to the sum of the remainder
     * and the divisor (basically, the "common residue" is the first positive modulo).
     *
     * @param \phpseclib\Math\BigInteger\engines\PHP $y
     * @return array
     * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.
     */
    protected function divideHelper(PHP $y)
    {
        if (count($y->value) == 1) {
            list($q, $r) = $this->divide_digit($this->value, $y->value[0]);
            $quotient = new static();
            $remainder = new static();
            $quotient->value = $q;
            $remainder->value = [$r];
            $quotient->is_negative = $this->is_negative != $y->is_negative;
            return [$this->normalize($quotient), $this->normalize($remainder)];
        }

        $x = clone $this;
        $y = clone $y;

        $x_sign = $x->is_negative;
        $y_sign = $y->is_negative;

        $x->is_negative = $y->is_negative = false;

        $diff = $x->compare($y);

        if (!$diff) {
            $temp = new static();
            $temp->value = [1];
            $temp->is_negative = $x_sign != $y_sign;
            return [$this->normalize($temp), $this->normalize(static::$zero)];
        }

        if ($diff < 0) {
            // if $x is negative, "add" $y.
            if ($x_sign) {
                $x = $y->subtract($x);
            }
            return [$this->normalize(static::$zero), $this->normalize($x)];
        }

        // normalize $x and $y as described in HAC 14.23 / 14.24
        $msb = $y->value[count($y->value) - 1];
        for ($shift = 0; !($msb & static::MSB); ++$shift) {
            $msb <<= 1;
        }
        $x->lshift($shift);
        $y->lshift($shift);
        $y_value = &$y->value;

        $x_max = count($x->value) - 1;
        $y_max = count($y->value) - 1;

        $quotient = new static();
        $quotient_value = &$quotient->value;
        $quotient_value = self::array_repeat(0, $x_max - $y_max + 1);

        static $temp, $lhs, $rhs;
        if (!isset($temp)) {
            $temp = new static();
            $lhs =  new static();
            $rhs =  new static();
        }
        $temp_value = &$temp->value;
        $rhs_value =  &$rhs->value;

        // $temp = $y << ($x_max - $y_max-1) in base 2**26
        $temp_value = array_merge(self::array_repeat(0, $x_max - $y_max), $y_value);

        while ($x->compare($temp) >= 0) {
            // calculate the "common residue"
            ++$quotient_value[$x_max - $y_max];
            $x = $x->subtract($temp);
            $x_max = count($x->value) - 1;
        }

        for ($i = $x_max; $i >= $y_max + 1; --$i) {
            $x_value = &$x->value;
            $x_window = [
                isset($x_value[$i]) ? $x_value[$i] : 0,
                isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0,
                isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0
            ];
            $y_window = [
                $y_value[$y_max],
                ($y_max > 0) ? $y_value[$y_max - 1] : 0
            ];

            $q_index = $i - $y_max - 1;
            if ($x_window[0] == $y_window[0]) {
                $quotient_value[$q_index] = static::MAX_DIGIT;
            } else {
                $quotient_value[$q_index] = self::safe_divide(
                    $x_window[0] * static::BASE_FULL + $x_window[1],
                    $y_window[0]
                );
            }

            $temp_value = [$y_window[1], $y_window[0]];

            $lhs->value = [$quotient_value[$q_index]];
            $lhs = $lhs->multiply($temp);

            $rhs_value = [$x_window[2], $x_window[1], $x_window[0]];

            while ($lhs->compare($rhs) > 0) {
                --$quotient_value[$q_index];

                $lhs->value = [$quotient_value[$q_index]];
                $lhs = $lhs->multiply($temp);
            }

            $adjust = self::array_repeat(0, $q_index);
            $temp_value = [$quotient_value[$q_index]];
            $temp = $temp->multiply($y);
            $temp_value = &$temp->value;
            $temp_value = array_merge($adjust, $temp_value);

            $x = $x->subtract($temp);

            if ($x->compare(static::$zero) < 0) {
                $temp_value = array_merge($adjust, $y_value);
                $x = $x->add($temp);

                --$quotient_value[$q_index];
            }

            $x_max = count($x_value) - 1;
        }

        // unnormalize the remainder
        $x->rshift($shift);

        $quotient->is_negative = $x_sign != $y_sign;

        // calculate the "common residue", if appropriate
        if ($x_sign) {
            $y->rshift($shift);
            $x = $y->subtract($x);
        }

        return [$this->normalize($quotient), $this->normalize($x)];
    }

    /**
     * Divides a BigInteger by a regular integer
     *
     * abc / x = a00 / x + b0 / x + c / x
     *
     * @param array $dividend
     * @param int $divisor
     * @return array
     */
    private static function divide_digit(array $dividend, $divisor)
    {
        $carry = 0;
        $result = [];

        for ($i = count($dividend) - 1; $i >= 0; --$i) {
            $temp = static::BASE_FULL * $carry + $dividend[$i];
            $result[$i] = self::safe_divide($temp, $divisor);
            $carry = (int) ($temp - $divisor * $result[$i]);
        }

        return [$result, $carry];
    }

    /**
     * Single digit division
     *
     * Even if int64 is being used the division operator will return a float64 value
     * if the dividend is not evenly divisible by the divisor. Since a float64 doesn't
     * have the precision of int64 this is a problem so, when int64 is being used,
     * we'll guarantee that the dividend is divisible by first subtracting the remainder.
     *
     * @param int $x
     * @param int $y
     * @return int
     */
    private static function safe_divide($x, $y)
    {
        if (static::BASE === 26) {
            return (int) ($x / $y);
        }

        // static::BASE === 31
        return ($x - ($x % $y)) / $y;
    }

    /*
     * Convert an array / boolean to a PHP BigInteger object
     *
     * @param array $arr
     * @return \phpseclib\Math\BigInteger\Engines\PHP
     */
    protected function convertToObj(array $arr)
    {
        $result = new static();
        $result->value = $arr[self::VALUE];
        $result->is_negative = $arr[self::SIGN];

        return $this->normalize($result);
    }

    /**
     * Normalize
     *
     * Removes leading zeros and truncates (if necessary) to maintain the appropriate precision
     *
     * @param PHP $result
     * @return PHP
     */
    protected function normalize(PHP $result)
    {
        $result->precision = $this->precision;
        $result->bitmask = $this->bitmask;

        $value = &$result->value;

        if (!count($value)) {
            return $result;
        }

        $value = $this->trim($value);

        if (!empty($result->bitmask->value)) {
            $length = min(count($value), count($result->bitmask->value));
            $value = array_slice($value, 0, $length);

            for ($i = 0; $i < $length; ++$i) {
                $value[$i] = $value[$i] & $result->bitmask->value[$i];
            }
        }

        return $result;
    }

    /*
     * Compares two numbers.
     *
     * @param array $x_value
     * @param bool $x_negative
     * @param array $y_value
     * @param bool $y_negative
     * @return int
     * @see static::compare()
     */
    protected static function compareHelper(array $x_value, $x_negative, array $y_value, $y_negative)
    {
        if ($x_negative != $y_negative) {
            return (!$x_negative && $y_negative) ? 1 : -1;
        }

        $result = $x_negative ? -1 : 1;

        if (count($x_value) != count($y_value)) {
            return (count($x_value) > count($y_value)) ? $result : -$result;
        }
        $size = max(count($x_value), count($y_value));

        $x_value = array_pad($x_value, $size, 0);
        $y_value = array_pad($y_value, $size, 0);

        for ($i = count($x_value) - 1; $i >= 0; --$i) {
            if ($x_value[$i] != $y_value[$i]) {
                return ($x_value[$i] > $y_value[$i]) ? $result : -$result;
            }
        }

        return 0;
    }

    /**
     * Calculates the greatest common divisor and Bezout's identity.
     *
     * Say you have 693 and 609.  The GCD is 21.  Bezout's identity states that there exist integers x and y such that
     * 693*x + 609*y == 21.  In point of fact, there are actually an infinite number of x and y combinations and which
     * combination is returned is dependent upon which mode is in use.  See
     * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bezout's identity - Wikipedia} for more information.
     *
     * @param PHP $n
     * @return PHP[]
     * @internal Calculates the GCD using the binary xGCD algorithim described in
     *    {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}.  As the text above 14.61 notes,
     *    the more traditional algorithim requires "relatively costly multiple-precision divisions".
     */
    protected function extendedGCDHelper(PHP $n)
    {
        $y = clone $n;
        $x = clone $this;
        $g = new static();
        $g->value = [1];

        while (!(($x->value[0] & 1)|| ($y->value[0] & 1))) {
            $x->rshift(1);
            $y->rshift(1);
            $g->lshift(1);
        }

        $u = clone $x;
        $v = clone $y;

        $a = new static();
        $b = new static();
        $c = new static();
        $d = new static();

        $a->value = $d->value = $g->value = [1];
        $b->value = $c->value = [];

        while (!empty($u->value)) {
            while (!($u->value[0] & 1)) {
                $u->rshift(1);
                if ((!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1))) {
                    $a = $a->add($y);
                    $b = $b->subtract($x);
                }
                $a->rshift(1);
                $b->rshift(1);
            }

            while (!($v->value[0] & 1)) {
                $v->rshift(1);
                if ((!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1))) {
                    $c = $c->add($y);
                    $d = $d->subtract($x);
                }
                $c->rshift(1);
                $d->rshift(1);
            }

            if ($u->compare($v) >= 0) {
                $u = $u->subtract($v);
                $a = $a->subtract($c);
                $b = $b->subtract($d);
            } else {
                $v = $v->subtract($u);
                $c = $c->subtract($a);
                $d = $d->subtract($b);
            }
        }

        return [
            'gcd' => $this->normalize($g->multiply($v)),
            'x'   => $this->normalize($c),
            'y'   => $this->normalize($d)
        ];
    }

    /**
     * Absolute value.
     *
     * @return \phpseclib\Math\BigInteger\Engines\PHP
     */
    public function abs()
    {
        $temp = new static();
        $temp->value = $this->value;

        return $temp;
    }

    /**
     * Logical And
     *
     * @param PHP $x
     * @return PHP
     */
    protected function bitwiseAndHelper(PHP $x)
    {
        $result = clone $this;

        $length = min(count($x->value), count($this->value));

        $result->value = array_slice($result->value, 0, $length);

        for ($i = 0; $i < $length; ++$i) {
            $result->value[$i]&= $x->value[$i];
        }

        return $this->normalize($result);
    }

    /**
     * Logical Or
     *
     * @param PHP $x
     * @return PHP
     */
    protected function bitwiseOrHelper(PHP $x)
    {
        $length = max(count($this->value), count($x->value));
        $result = clone $this;
        $result->value = array_pad($result->value, $length, 0);
        $x->value = array_pad($x->value, $length, 0);

        for ($i = 0; $i < $length; ++$i) {
            $result->value[$i]|= $x->value[$i];
        }

        return $this->normalize($result);
    }

    /**
     * Logical Exclusive Or
     *
     * @param PHP $x
     * @return PHP
     */
    protected function bitwiseXorHelper(PHP $x)
    {
        $length = max(count($this->value), count($x->value));
        $result = clone $this;
        $result->is_negative = false;
        $result->value = array_pad($result->value, $length, 0);
        $x->value = array_pad($x->value, $length, 0);

        for ($i = 0; $i < $length; ++$i) {
            $result->value[$i]^= $x->value[$i];
        }

        return $this->normalize($result);
    }

    /**
     * Trim
     *
     * Removes leading zeros
     *
     * @param array $value
     * @return PHP
     */
    protected static function trim(array $value)
    {
        for ($i = count($value) - 1; $i >= 0; --$i) {
            if ($value[$i]) {
                break;
            }
            unset($value[$i]);
        }

        return $value;
    }

    /**
     * Logical Right Shift
     *
     * Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
     *
     * @param $x String
     * @param $shift Integer
     * @return string
     */
    private static function base256_rshift(&$x, $shift)
    {
        if ($shift == 0) {
            $x = ltrim($x, chr(0));
            return '';
        }

        $num_bytes = $shift >> 3; // eg. floor($shift/8)
        $shift &= 7; // eg. $shift % 8

        $remainder = '';
        if ($num_bytes) {
            $start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;
            $remainder = substr($x, $start);
            $x = substr($x, 0, -$num_bytes);
        }

        $carry = 0;
        $carry_shift = 8 - $shift;
        for ($i = 0; $i < strlen($x); ++$i) {
            $temp = (ord($x[$i]) >> $shift) | $carry;
            $carry = (ord($x[$i]) << $carry_shift) & 0xFF;
            $x[$i] = chr($temp);
        }
        $x = ltrim($x, chr(0));

        $remainder = chr($carry >> $carry_shift) . $remainder;

        return ltrim($remainder, chr(0));
    }

    /**
     * Logical Right Shift
     *
     * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
     *
     * @param int $shift
     * @return \phpseclib\Math\BigInteger\Engines\PHP
     */
    public function bitwise_rightShift($shift)
    {
        $temp = new static();

        // could just replace lshift with this, but then all lshift() calls would need to be rewritten
        // and I don't want to do that...
        $temp->value = $this->value;
        $temp->rshift($shift);

        return $this->normalize($temp);
    }

    /**
     * Logical Left Shift
     *
     * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
     *
     * @param int $shift
     * @return \phpseclib\Math\BigInteger\Engines\PHP
     */
    public function bitwise_leftShift($shift)
    {
        $temp = new static();
        // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
        // and I don't want to do that...
        $temp->value = $this->value;
        $temp->lshift($shift);

        return $this->normalize($temp);
    }

    /**
     * Converts 32-bit integers to bytes.
     *
     * @param int $x
     * @return string
     */
    private static function int2bytes($x)
    {
        return ltrim(pack('N', $x), chr(0));
    }

    /**
     * Converts bytes to 32-bit integers
     *
     * @param string $x
     * @return int
     */
    private static function bytes2int($x)
    {
        $temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
        return $temp['int'];
    }

    /**
     * Array Repeat
     *
     * @param int $input
     * @param int $multiplier
     * @return array
     */
    protected static function array_repeat($input, $multiplier)
    { 
        return $multiplier ? array_fill(0, $multiplier, $input) : [];
    }

    /**
     * Logical Left Shift
     *
     * Shifts BigInteger's by $shift bits.
     *
     * @param int $shift
     */
    protected function lshift($shift)
    {
        if ($shift == 0) {
            return;
        }

        $num_digits = (int) ($shift / static::BASE);
        $shift %= static::BASE;
        $shift = 1 << $shift;

        $carry = 0;

        for ($i = 0; $i < count($this->value); ++$i) {
            $temp = $this->value[$i] * $shift + $carry;
            $carry = static::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
            $this->value[$i] = (int) ($temp - $carry * static::BASE_FULL);
        }

        if ($carry) {
            $this->value[count($this->value)] = $carry;
        }

        while ($num_digits--) {
            array_unshift($this->value, 0);
        }
    }

    /**
     * Logical Right Shift
     *
     * Shifts BigInteger's by $shift bits.
     *
     * @param int $shift
     */
    protected function rshift($shift)
    {
        if ($shift == 0) {
            return;
        }

        $num_digits = (int) ($shift / static::BASE);
        $shift %= static::BASE;
        $carry_shift = static::BASE - $shift;
        $carry_mask = (1 << $shift) - 1;

        if ($num_digits) {
            $this->value = array_slice($this->value, $num_digits);
        }

        $carry = 0;

        for ($i = count($this->value) - 1; $i >= 0; --$i) {
            $temp = $this->value[$i] >> $shift | $carry;
            $carry = ($this->value[$i] & $carry_mask) << $carry_shift;
            $this->value[$i] = $temp;
        }

        $this->value = $this->trim($this->value);
    }

    /**
     * Performs modular exponentiation.
     *
     * @param PHP $e
     * @param PHP $n
     * @return PHP
     */
    protected function powModInner(PHP $e, PHP $n)
    {
        try {
            $class = static::$modexpEngine;
            return $class::powModHelper($this, $e, $n, static::class);
        } catch (\Exception $err) {
            return PHP\DefaultEngine::powModHelper($this, $e, $n, static::class);
        }
    }

    /**
     * Performs squaring
     *
     * @param array $x
     * @return array
     */
    protected static function square(array $x)
    {
        return count($x) < 2 * self::KARATSUBA_CUTOFF ?
            self::trim(self::baseSquare($x)) :
            self::trim(self::karatsubaSquare($x));
    }

    /**
     * Performs traditional squaring on two BigIntegers
     *
     * Squaring can be done faster than multiplying a number by itself can be.  See
     * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
     *
     * @param array $value
     * @return array
     */
    protected static function baseSquare(array $value)
    {
        if (empty($value)) {
            return [];
        }
        $square_value = self::array_repeat(0, 2 * count($value));

        for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) {
            $i2 = $i << 1;

            $temp = $square_value[$i2] + $value[$i] * $value[$i];
            $carry = static::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
            $square_value[$i2] = (int) ($temp - static::BASE_FULL * $carry);

            // note how we start from $i+1 instead of 0 as we do in multiplication.
            for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) {
                $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;
                $carry = static::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                $square_value[$k] = (int) ($temp - static::BASE_FULL * $carry);
            }

            // the following line can yield values larger 2**15.  at this point, PHP should switch
            // over to floats.
            $square_value[$i + $max_index + 1] = $carry;
        }

        return $square_value;
    }

    /**
     * Performs Karatsuba "squaring" on two BigIntegers
     *
     * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}.
     *
     * @param array $value
     * @return array
     */
    protected static function karatsubaSquare(array $value)
    {
        $m = count($value) >> 1;

        if ($m < self::KARATSUBA_CUTOFF) {
            return self::baseSquare($value);
        }

        $x1 = array_slice($value, $m);
        $x0 = array_slice($value, 0, $m);

        $z2 = self::karatsubaSquare($x1);
        $z0 = self::karatsubaSquare($x0);

        $z1 = self::addHelper($x1, false, $x0, false);
        $z1 = self::karatsubaSquare($z1[self::VALUE]);
        $temp = self::addHelper($z2, false, $z0, false);
        $z1 = self::subtractHelper($z1, false, $temp[self::VALUE], false);

        $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
        $z1[self::VALUE] = array_merge(array_fill(0, $m, 0), $z1[self::VALUE]);

        $xx = self::addHelper($z2, false, $z1[self::VALUE], $z1[self::SIGN]);
        $xx = self::addHelper($xx[self::VALUE], $xx[self::SIGN], $z0, false);

        return $xx[self::VALUE];
    }

    /**
     * Make the current number odd
     *
     * If the current number is odd it'll be unchanged.  If it's even, one will be added to it.
     *
     * @see self::randomPrime()
     */
    protected function make_odd()
    {
        $this->value[0] |= 1;
    }

    /**
     * Test the number against small primes.
     *
     * @see self::isPrime()
     */
    protected function testSmallPrimes()
    {
        if ($this->value == [1]) {
            return false;
        }
        if ($this->value == [2]) {
            return true;
        }
        if (~$this->value[0] & 1) {
            return false;
        }

        $value = $this->value;
        foreach (static::$primes as $prime) {
            list(, $r) = self::divide_digit($value, $prime);
            if (!$r) {
                return count($value) == 1 && $value[0] == $prime;
            }
        }

        return true;
    }

    /**
     * Scan for 1 and right shift by that amount
     *
     * ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));
     *
     * @see self::isPrime()
     * @param PHP $r
     * @return int
     */
    protected static function scan1divide(PHP $r)
    {
        $r_value = &$r->value;
        for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) {
            $temp = ~$r_value[$i] & static::MAX_DIGIT;
            for ($j = 1; ($temp >> $j) & 1; ++$j) {
            }
            if ($j <= static::BASE) {
                break;
            }
        }
        $s = static::BASE * $i + $j;
        $r->rshift($s);
        return $s;
    }

    /**
     * Performs exponentiation.
     *
     * @param PHP $n
     * @return PHP
     */
    protected function powHelper(PHP $n)
    {
        if ($n->compare(static::$zero) == 0) {
            return new static(1);
        } // n^0 = 1


        $temp = clone $this;
        while (!$n->equals(static::$one)) {
            $temp = $temp->multiply($this);
            $n = $n->subtract(static::$one);
        }

        return $temp;
    }
}