Developer
<?php
/**
* Curves over y^2 = x^3 + a*x + b
*
* These are curves used in SEC 2 over prime fields: http://www.secg.org/SEC2-Ver-1.0.pdf
* The curve is a weierstrass curve with a[1], a[3] and a[2] set to 0.
*
* Uses Jacobian Coordinates for speed if able:
*
* https://en.wikipedia.org/wiki/Jacobian_curve
* https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
*
* PHP version 5 and 7
*
* @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\Crypt\EC\BaseCurves;
use phpseclib3\Common\Functions\Strings;
use phpseclib3\Math\BigInteger;
use phpseclib3\Math\Common\FiniteField\Integer;
use phpseclib3\Math\PrimeField;
use phpseclib3\Math\PrimeField\Integer as PrimeInteger;
/**
* Curves over y^2 = x^3 + a*x + b
*
* @author Jim Wigginton <terrafrost@php.net>
*/
class Prime extends Base
{
/**
* Prime Field Integer factory
*
* @var \phpseclib3\Math\PrimeFields
*/
protected $factory;
/**
* Cofficient for x^1
*
* @var object
*/
protected $a;
/**
* Cofficient for x^0
*
* @var object
*/
protected $b;
/**
* Base Point
*
* @var object
*/
protected $p;
/**
* The number one over the specified finite field
*
* @var object
*/
protected $one;
/**
* The number two over the specified finite field
*
* @var object
*/
protected $two;
/**
* The number three over the specified finite field
*
* @var object
*/
protected $three;
/**
* The number four over the specified finite field
*
* @var object
*/
protected $four;
/**
* The number eight over the specified finite field
*
* @var object
*/
protected $eight;
/**
* The modulo
*
* @var BigInteger
*/
protected $modulo;
/**
* The Order
*
* @var BigInteger
*/
protected $order;
/**
* Sets the modulo
*/
public function setModulo(BigInteger $modulo)
{
$this->modulo = $modulo;
$this->factory = new PrimeField($modulo);
$this->two = $this->factory->newInteger(new BigInteger(2));
$this->three = $this->factory->newInteger(new BigInteger(3));
// used by jacobian coordinates
$this->one = $this->factory->newInteger(new BigInteger(1));
$this->four = $this->factory->newInteger(new BigInteger(4));
$this->eight = $this->factory->newInteger(new BigInteger(8));
}
/**
* Set coefficients a and b
*/
public function setCoefficients(BigInteger $a, BigInteger $b)
{
if (!isset($this->factory)) {
throw new \RuntimeException('setModulo needs to be called before this method');
}
$this->a = $this->factory->newInteger($a);
$this->b = $this->factory->newInteger($b);
}
/**
* Set x and y coordinates for the base point
*
* @param BigInteger|PrimeInteger $x
* @param BigInteger|PrimeInteger $y
* @return PrimeInteger[]
*/
public function setBasePoint($x, $y)
{
switch (true) {
case !$x instanceof BigInteger && !$x instanceof PrimeInteger:
throw new \UnexpectedValueException('Argument 1 passed to Prime::setBasePoint() must be an instance of either BigInteger or PrimeField\Integer');
case !$y instanceof BigInteger && !$y instanceof PrimeInteger:
throw new \UnexpectedValueException('Argument 2 passed to Prime::setBasePoint() must be an instance of either BigInteger or PrimeField\Integer');
}
if (!isset($this->factory)) {
throw new \RuntimeException('setModulo needs to be called before this method');
}
$this->p = [
$x instanceof BigInteger ? $this->factory->newInteger($x) : $x,
$y instanceof BigInteger ? $this->factory->newInteger($y) : $y
];
}
/**
* Retrieve the base point as an array
*
* @return array
*/
public function getBasePoint()
{
if (!isset($this->factory)) {
throw new \RuntimeException('setModulo needs to be called before this method');
}
/*
if (!isset($this->p)) {
throw new \RuntimeException('setBasePoint needs to be called before this method');
}
*/
return $this->p;
}
/**
* Adds two "fresh" jacobian form on the curve
*
* @return FiniteField[]
*/
protected function jacobianAddPointMixedXY(array $p, array $q)
{
list($u1, $s1) = $p;
list($u2, $s2) = $q;
if ($u1->equals($u2)) {
if (!$s1->equals($s2)) {
return [];
} else {
return $this->doublePoint($p);
}
}
$h = $u2->subtract($u1);
$r = $s2->subtract($s1);
$h2 = $h->multiply($h);
$h3 = $h2->multiply($h);
$v = $u1->multiply($h2);
$x3 = $r->multiply($r)->subtract($h3)->subtract($v->multiply($this->two));
$y3 = $r->multiply(
$v->subtract($x3)
)->subtract(
$s1->multiply($h3)
);
return [$x3, $y3, $h];
}
/**
* Adds one "fresh" jacobian form on the curve
*
* The second parameter should be the "fresh" one
*
* @return FiniteField[]
*/
protected function jacobianAddPointMixedX(array $p, array $q)
{
list($u1, $s1, $z1) = $p;
list($x2, $y2) = $q;
$z12 = $z1->multiply($z1);
$u2 = $x2->multiply($z12);
$s2 = $y2->multiply($z12->multiply($z1));
if ($u1->equals($u2)) {
if (!$s1->equals($s2)) {
return [];
} else {
return $this->doublePoint($p);
}
}
$h = $u2->subtract($u1);
$r = $s2->subtract($s1);
$h2 = $h->multiply($h);
$h3 = $h2->multiply($h);
$v = $u1->multiply($h2);
$x3 = $r->multiply($r)->subtract($h3)->subtract($v->multiply($this->two));
$y3 = $r->multiply(
$v->subtract($x3)
)->subtract(
$s1->multiply($h3)
);
$z3 = $h->multiply($z1);
return [$x3, $y3, $z3];
}
/**
* Adds two jacobian coordinates on the curve
*
* @return FiniteField[]
*/
protected function jacobianAddPoint(array $p, array $q)
{
list($x1, $y1, $z1) = $p;
list($x2, $y2, $z2) = $q;
$z12 = $z1->multiply($z1);
$z22 = $z2->multiply($z2);
$u1 = $x1->multiply($z22);
$u2 = $x2->multiply($z12);
$s1 = $y1->multiply($z22->multiply($z2));
$s2 = $y2->multiply($z12->multiply($z1));
if ($u1->equals($u2)) {
if (!$s1->equals($s2)) {
return [];
} else {
return $this->doublePoint($p);
}
}
$h = $u2->subtract($u1);
$r = $s2->subtract($s1);
$h2 = $h->multiply($h);
$h3 = $h2->multiply($h);
$v = $u1->multiply($h2);
$x3 = $r->multiply($r)->subtract($h3)->subtract($v->multiply($this->two));
$y3 = $r->multiply(
$v->subtract($x3)
)->subtract(
$s1->multiply($h3)
);
$z3 = $h->multiply($z1)->multiply($z2);
return [$x3, $y3, $z3];
}
/**
* Adds two points on the curve
*
* @return FiniteField[]
*/
public function addPoint(array $p, array $q)
{
if (!isset($this->factory)) {
throw new \RuntimeException('setModulo needs to be called before this method');
}
if (!count($p) || !count($q)) {
if (count($q)) {
return $q;
}
if (count($p)) {
return $p;
}
return [];
}
// use jacobian coordinates
if (isset($p[2]) && isset($q[2])) {
if (isset($p['fresh']) && isset($q['fresh'])) {
return $this->jacobianAddPointMixedXY($p, $q);
}
if (isset($p['fresh'])) {
return $this->jacobianAddPointMixedX($q, $p);
}
if (isset($q['fresh'])) {
return $this->jacobianAddPointMixedX($p, $q);
}
return $this->jacobianAddPoint($p, $q);
}
if (isset($p[2]) || isset($q[2])) {
throw new \RuntimeException('Affine coordinates need to be manually converted to Jacobi coordinates or vice versa');
}
if ($p[0]->equals($q[0])) {
if (!$p[1]->equals($q[1])) {
return [];
} else { // eg. doublePoint
list($numerator, $denominator) = $this->doublePointHelper($p);
}
} else {
$numerator = $q[1]->subtract($p[1]);
$denominator = $q[0]->subtract($p[0]);
}
$slope = $numerator->divide($denominator);
$x = $slope->multiply($slope)->subtract($p[0])->subtract($q[0]);
$y = $slope->multiply($p[0]->subtract($x))->subtract($p[1]);
return [$x, $y];
}
/**
* Returns the numerator and denominator of the slope
*
* @return FiniteField[]
*/
protected function doublePointHelper(array $p)
{
$numerator = $this->three->multiply($p[0])->multiply($p[0])->add($this->a);
$denominator = $this->two->multiply($p[1]);
return [$numerator, $denominator];
}
/**
* Doubles a jacobian coordinate on the curve
*
* @return FiniteField[]
*/
protected function jacobianDoublePoint(array $p)
{
list($x, $y, $z) = $p;
$x2 = $x->multiply($x);
$y2 = $y->multiply($y);
$z2 = $z->multiply($z);
$s = $this->four->multiply($x)->multiply($y2);
$m1 = $this->three->multiply($x2);
$m2 = $this->a->multiply($z2->multiply($z2));
$m = $m1->add($m2);
$x1 = $m->multiply($m)->subtract($this->two->multiply($s));
$y1 = $m->multiply($s->subtract($x1))->subtract(
$this->eight->multiply($y2->multiply($y2))
);
$z1 = $this->two->multiply($y)->multiply($z);
return [$x1, $y1, $z1];
}
/**
* Doubles a "fresh" jacobian coordinate on the curve
*
* @return FiniteField[]
*/
protected function jacobianDoublePointMixed(array $p)
{
list($x, $y) = $p;
$x2 = $x->multiply($x);
$y2 = $y->multiply($y);
$s = $this->four->multiply($x)->multiply($y2);
$m1 = $this->three->multiply($x2);
$m = $m1->add($this->a);
$x1 = $m->multiply($m)->subtract($this->two->multiply($s));
$y1 = $m->multiply($s->subtract($x1))->subtract(
$this->eight->multiply($y2->multiply($y2))
);
$z1 = $this->two->multiply($y);
return [$x1, $y1, $z1];
}
/**
* Doubles a point on a curve
*
* @return FiniteField[]
*/
public function doublePoint(array $p)
{
if (!isset($this->factory)) {
throw new \RuntimeException('setModulo needs to be called before this method');
}
if (!count($p)) {
return [];
}
// use jacobian coordinates
if (isset($p[2])) {
if (isset($p['fresh'])) {
return $this->jacobianDoublePointMixed($p);
}
return $this->jacobianDoublePoint($p);
}
list($numerator, $denominator) = $this->doublePointHelper($p);
$slope = $numerator->divide($denominator);
$x = $slope->multiply($slope)->subtract($p[0])->subtract($p[0]);
$y = $slope->multiply($p[0]->subtract($x))->subtract($p[1]);
return [$x, $y];
}
/**
* Returns the X coordinate and the derived Y coordinate
*
* @return array
*/
public function derivePoint($m)
{
$y = ord(Strings::shift($m));
$x = new BigInteger($m, 256);
$xp = $this->convertInteger($x);
switch ($y) {
case 2:
$ypn = false;
break;
case 3:
$ypn = true;
break;
default:
throw new \RuntimeException('Coordinate not in recognized format');
}
$temp = $xp->multiply($this->a);
$temp = $xp->multiply($xp)->multiply($xp)->add($temp);
$temp = $temp->add($this->b);
$b = $temp->squareRoot();
if (!$b) {
throw new \RuntimeException('Unable to derive Y coordinate');
}
$bn = $b->isOdd();
$yp = $ypn == $bn ? $b : $b->negate();
return [$xp, $yp];
}
/**
* Tests whether or not the x / y values satisfy the equation
*
* @return boolean
*/
public function verifyPoint(array $p)
{
list($x, $y) = $p;
$lhs = $y->multiply($y);
$temp = $x->multiply($this->a);
$temp = $x->multiply($x)->multiply($x)->add($temp);
$rhs = $temp->add($this->b);
return $lhs->equals($rhs);
}
/**
* Returns the modulo
*
* @return \phpseclib3\Math\BigInteger
*/
public function getModulo()
{
return $this->modulo;
}
/**
* Returns the a coefficient
*
* @return \phpseclib3\Math\PrimeField\Integer
*/
public function getA()
{
return $this->a;
}
/**
* Returns the a coefficient
*
* @return \phpseclib3\Math\PrimeField\Integer
*/
public function getB()
{
return $this->b;
}
/**
* Multiply and Add Points
*
* Adapted from:
* https://github.com/indutny/elliptic/blob/725bd91/lib/elliptic/curve/base.js#L125
*
* @return int[]
*/
public function multiplyAddPoints(array $points, array $scalars)
{
$length = count($points);
foreach ($points as &$point) {
$point = $this->convertToInternal($point);
}
$wnd = [$this->getNAFPoints($points[0], 7)];
$wndWidth = [isset($points[0]['nafwidth']) ? $points[0]['nafwidth'] : 7];
for ($i = 1; $i < $length; $i++) {
$wnd[] = $this->getNAFPoints($points[$i], 1);
$wndWidth[] = isset($points[$i]['nafwidth']) ? $points[$i]['nafwidth'] : 1;
}
$naf = [];
// comb all window NAFs
$max = 0;
for ($i = $length - 1; $i >= 1; $i -= 2) {
$a = $i - 1;
$b = $i;
if ($wndWidth[$a] != 1 || $wndWidth[$b] != 1) {
$naf[$a] = $scalars[$a]->getNAF($wndWidth[$a]);
$naf[$b] = $scalars[$b]->getNAF($wndWidth[$b]);
$max = max(count($naf[$a]), count($naf[$b]), $max);
continue;
}
$comb = [
$points[$a], // 1
null, // 3
null, // 5
$points[$b] // 7
];
$comb[1] = $this->addPoint($points[$a], $points[$b]);
$comb[2] = $this->addPoint($points[$a], $this->negatePoint($points[$b]));
$index = [
-3, /* -1 -1 */
-1, /* -1 0 */
-5, /* -1 1 */
-7, /* 0 -1 */
0, /* 0 -1 */
7, /* 0 1 */
5, /* 1 -1 */
1, /* 1 0 */
3 /* 1 1 */
];
$jsf = self::getJSFPoints($scalars[$a], $scalars[$b]);
$max = max(count($jsf[0]), $max);
if ($max > 0) {
$naf[$a] = array_fill(0, $max, 0);
$naf[$b] = array_fill(0, $max, 0);
} else {
$naf[$a] = [];
$naf[$b] = [];
}
for ($j = 0; $j < $max; $j++) {
$ja = isset($jsf[0][$j]) ? $jsf[0][$j] : 0;
$jb = isset($jsf[1][$j]) ? $jsf[1][$j] : 0;
$naf[$a][$j] = $index[3 * ($ja + 1) + $jb + 1];
$naf[$b][$j] = 0;
$wnd[$a] = $comb;
}
}
$acc = [];
$temp = [0, 0, 0, 0];
for ($i = $max; $i >= 0; $i--) {
$k = 0;
while ($i >= 0) {
$zero = true;
for ($j = 0; $j < $length; $j++) {
$temp[$j] = isset($naf[$j][$i]) ? $naf[$j][$i] : 0;
if ($temp[$j] != 0) {
$zero = false;
}
}
if (!$zero) {
break;
}
$k++;
$i--;
}
if ($i >= 0) {
$k++;
}
while ($k--) {
$acc = $this->doublePoint($acc);
}
if ($i < 0) {
break;
}
for ($j = 0; $j < $length; $j++) {
$z = $temp[$j];
$p = null;
if ($z == 0) {
continue;
}
$p = $z > 0 ?
$wnd[$j][($z - 1) >> 1] :
$this->negatePoint($wnd[$j][(-$z - 1) >> 1]);
$acc = $this->addPoint($acc, $p);
}
}
return $this->convertToAffine($acc);
}
/**
* Precomputes NAF points
*
* Adapted from:
* https://github.com/indutny/elliptic/blob/725bd91/lib/elliptic/curve/base.js#L351
*
* @return int[]
*/
private function getNAFPoints(array $point, $wnd)
{
if (isset($point['naf'])) {
return $point['naf'];
}
$res = [$point];
$max = (1 << $wnd) - 1;
$dbl = $max == 1 ? null : $this->doublePoint($point);
for ($i = 1; $i < $max; $i++) {
$res[] = $this->addPoint($res[$i - 1], $dbl);
}
$point['naf'] = $res;
/*
$str = '';
foreach ($res as $re) {
$re[0] = bin2hex($re[0]->toBytes());
$re[1] = bin2hex($re[1]->toBytes());
$str.= " ['$re[0]', '$re[1]'],\r\n";
}
file_put_contents('temp.txt', $str);
exit;
*/
return $res;
}
/**
* Precomputes points in Joint Sparse Form
*
* Adapted from:
* https://github.com/indutny/elliptic/blob/725bd91/lib/elliptic/utils.js#L96
*
* @return int[]
*/
private static function getJSFPoints(Integer $k1, Integer $k2)
{
static $three;
if (!isset($three)) {
$three = new BigInteger(3);
}
$jsf = [[], []];
$k1 = $k1->toBigInteger();
$k2 = $k2->toBigInteger();
$d1 = 0;
$d2 = 0;
while ($k1->compare(new BigInteger(-$d1)) > 0 || $k2->compare(new BigInteger(-$d2)) > 0) {
// first phase
$m14 = $k1->testBit(0) + 2 * $k1->testBit(1);
$m14 += $d1;
$m14 &= 3;
$m24 = $k2->testBit(0) + 2 * $k2->testBit(1);
$m24 += $d2;
$m24 &= 3;
if ($m14 == 3) {
$m14 = -1;
}
if ($m24 == 3) {
$m24 = -1;
}
$u1 = 0;
if ($m14 & 1) { // if $m14 is odd
$m8 = $k1->testBit(0) + 2 * $k1->testBit(1) + 4 * $k1->testBit(2);
$m8 += $d1;
$m8 &= 7;
$u1 = ($m8 == 3 || $m8 == 5) && $m24 == 2 ? -$m14 : $m14;
}
$jsf[0][] = $u1;
$u2 = 0;
if ($m24 & 1) { // if $m24 is odd
$m8 = $k2->testBit(0) + 2 * $k2->testBit(1) + 4 * $k2->testBit(2);
$m8 += $d2;
$m8 &= 7;
$u2 = ($m8 == 3 || $m8 == 5) && $m14 == 2 ? -$m24 : $m24;
}
$jsf[1][] = $u2;
// second phase
if (2 * $d1 == $u1 + 1) {
$d1 = 1 - $d1;
}
if (2 * $d2 == $u2 + 1) {
$d2 = 1 - $d2;
}
$k1 = $k1->bitwise_rightShift(1);
$k2 = $k2->bitwise_rightShift(1);
}
return $jsf;
}
/**
* Returns the affine point
*
* A Jacobian Coordinate is of the form (x, y, z).
* To convert a Jacobian Coordinate to an Affine Point
* you do (x / z^2, y / z^3)
*
* @return \phpseclib3\Math\PrimeField\Integer[]
*/
public function convertToAffine(array $p)
{
if (!isset($p[2])) {
return $p;
}
list($x, $y, $z) = $p;
$z = $this->one->divide($z);
$z2 = $z->multiply($z);
return [
$x->multiply($z2),
$y->multiply($z2)->multiply($z)
];
}
/**
* Converts an affine point to a jacobian coordinate
*
* @return \phpseclib3\Math\PrimeField\Integer[]
*/
public function convertToInternal(array $p)
{
if (isset($p[2])) {
return $p;
}
$p[2] = clone $this->one;
$p['fresh'] = true;
return $p;
}
}
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