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<?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 phpseclib3\Math\BigInteger\Engines;
use ParagonIE\ConstantTime\Hex;
use phpseclib3\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 $base
* @see parent::__construct()
* @return \phpseclib3\Math\BigInteger\Engines\PHP
*/
public function __construct($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 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->bitmask = false;
$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 = $this->bitwise_small_split(8);
$result = implode('', array_map('chr', $result));
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 == $y_value ) {
// return [
// self::VALUE => self::square($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 \phpseclib3\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;
if (count($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 \phpseclib3\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 $result)
{
unset($result->reduce);
$result->precision = $this->precision;
$result->bitmask = $this->bitmask;
$value = &$result->value;
if (!count($value)) {
$result->is_negative = false;
return $result;
}
$value = static::trim($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;
}
/**
* Absolute value.
*
* @return \phpseclib3\Math\BigInteger\Engines\PHP
*/
public function abs()
{
$temp = new static();
$temp->value = $this->value;
return $temp;
}
/**
* 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 BigInteger's by $shift bits, effectively dividing by 2**$shift.
*
* @param int $shift
* @return \phpseclib3\Math\BigInteger\Engines\PHP
*/
public function bitwise_rightShift($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 $shift
* @return \phpseclib3\Math\BigInteger\Engines\PHP
*/
public function bitwise_leftShift($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));
}
/**
* 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;
}
$this->value = static::trim($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
*/
public 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 $temp;
}
/**
* Is Odd?
*
* @return boolean
*/
public function isOdd()
{
return (bool) ($this->value[0] & 1);
}
/**
* Tests if a bit is set
*
* @return boolean
*/
public function testBit($x)
{
$digit = floor($x / static::BASE);
$bit = $x % static::BASE;
if (!isset($this->value[$digit])) {
return false;
}
return (bool) ($this->value[$digit] & (1 << $bit));
}
/**
* Is Negative?
*
* @return boolean
*/
public function isNegative()
{
return $this->is_negative;
}
/**
* Negate
*
* Given $k, returns -$k
*
* @return BigInteger
*/
public function negate()
{
$temp = clone $this;
$temp->is_negative = !$temp->is_negative;
return $temp;
}
/**
* Bitwise Split
*
* Splits BigInteger's into chunks of $split bits
*
* @param int $split
* @return \phpseclib3\Math\BigInteger\Engines\PHP[]
*/
public function bitwise_split($split)
{
if ($split < 1) {
throw new \RuntimeException('Offset must be greater than 1');
}
$width = (int) ($split / static::BASE);
if (!$width) {
$arr = $this->bitwise_small_split($split);
return array_map(function ($digit) {
$temp = new static();
$temp->value = $digit != 0 ? [$digit] : [];
return $temp;
}, $arr);
}
$vals = [];
$val = $this->value;
$i = $overflow = 0;
$len = count($val);
while ($i < $len) {
$digit = [];
if (!$overflow) {
$digit = array_slice($val, $i, $width);
$i+= $width;
$overflow = $split % static::BASE;
if ($overflow) {
$mask = (1 << $overflow) - 1;
$temp = isset($val[$i]) ? $val[$i] : 0;
$digit[] = $temp & $mask;
}
} else {
$remaining = static::BASE - $overflow;
$tempsplit = $split - $remaining;
$tempwidth = (int) ($tempsplit / static::BASE + 1);
$digit = array_slice($val, $i, $tempwidth);
$i+= $tempwidth;
$tempoverflow = $tempsplit % static::BASE;
if ($tempoverflow) {
$tempmask = (1 << $tempoverflow) - 1;
$temp = isset($val[$i]) ? $val[$i] : 0;
$digit[] = $temp & $tempmask;
}
$newbits = 0;
for ($j = count($digit) - 1; $j >= 0; $j--) {
$temp = $digit[$j] & $mask;
$digit[$j] = ($digit[$j] >> $overflow) | ($newbits << $remaining);
$newbits = $temp;
}
$overflow = $tempoverflow;
$mask = $tempmask;
}
$temp = new static();
$temp->value = static::trim($digit);
$vals[] = $temp;
}
return array_reverse($vals);
}
/**
* Bitwise Split where $split < static::BASE
*
* @param int $split
* @return \phpseclib3\Math\BigInteger\Engines\PHP[]
*/
private function bitwise_small_split($split)
{
$vals = [];
$val = $this->value;
$mask = (1 << $split) - 1;
$i = $overflow = 0;
$len = count($val);
$val[] = 0;
$remaining = static::BASE;
while ($i != $len) {
$digit = $val[$i] & $mask;
$val[$i]>>= $split;
if (!$overflow) {
$remaining-= $split;
$overflow = $split <= $remaining ? 0 : $split - $remaining;
if (!$remaining) {
$i++;
$remaining = static::BASE;
$overflow = 0;
}
} else if (++$i != $len) {
$tempmask = (1 << $overflow) - 1;
$digit|= ($val[$i] & $tempmask) << $remaining;
$val[$i]>>= $overflow;
$remaining = static::BASE - $overflow;
$overflow = $split <= $remaining ? 0 : $split - $remaining;
}
$vals[] = $digit;
}
while ($vals[count($vals) - 1] == 0) {
unset($vals[count($vals) - 1]);
}
return array_reverse($vals);
}
}
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