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<?php
/**
* Ed448
*
* PHP version 5 and 7
*
* @category Crypt
* @package EC
* @author Jim Wigginton <terrafrost@php.net>
* @copyright 2017 Jim Wigginton
* @license http://www.opensource.org/licenses/mit-license.html MIT License
*/
namespace phpseclib3\Crypt\EC\Curves;
use phpseclib3\Crypt\EC\BaseCurves\TwistedEdwards;
use phpseclib3\Math\BigInteger;
use phpseclib3\Crypt\Hash;
use phpseclib3\Crypt\Random;
class Ed448 extends TwistedEdwards
{
const HASH = 'shake256-912';
const SIZE = 57;
public function __construct()
{
// 2^448 - 2^224 - 1
$this->setModulo(new BigInteger(
'FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE' .
'FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF', 16));
$this->setCoefficients(
new BigInteger(1),
// -39081
new BigInteger('FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE' .
'FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6756', 16)
);
$this->setBasePoint(
new BigInteger('4F1970C66BED0DED221D15A622BF36DA9E146570470F1767EA6DE324' .
'A3D3A46412AE1AF72AB66511433B80E18B00938E2626A82BC70CC05E', 16),
new BigInteger('693F46716EB6BC248876203756C9C7624BEA73736CA3984087789C1E' .
'05A0C2D73AD3FF1CE67C39C4FDBD132C4ED7C8AD9808795BF230FA14', 16)
);
$this->setOrder(new BigInteger(
'3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF' .
'7CCA23E9C44EDB49AED63690216CC2728DC58F552378C292AB5844F3', 16));
}
/**
* Recover X from Y
*
* Implements steps 2-4 at https://tools.ietf.org/html/rfc8032#section-5.2.3
*
* Used by EC\Keys\Common.php
*
* @param BigInteger $y
* @param boolean $sign
* @return object[]
*/
public function recoverX(BigInteger $y, $sign)
{
$y = $this->factory->newInteger($y);
$y2 = $y->multiply($y);
$u = $y2->subtract($this->one);
$v = $this->d->multiply($y2)->subtract($this->one);
$x2 = $u->divide($v);
if ($x2->equals($this->zero)) {
if ($sign) {
throw new \RuntimeException('Unable to recover X coordinate (x2 = 0)');
}
return clone $this->zero;
}
// find the square root
$exp = $this->getModulo()->add(new BigInteger(1));
$exp = $exp->bitwise_rightShift(2);
$x = $x2->pow($exp);
if (!$x->multiply($x)->subtract($x2)->equals($this->zero)) {
throw new \RuntimeException('Unable to recover X coordinate');
}
if ($x->isOdd() != $sign) {
$x = $x->negate();
}
return [$x, $y];
}
/**
* Extract Secret Scalar
*
* Implements steps 1-3 at https://tools.ietf.org/html/rfc8032#section-5.2.5
*
* Used by the various key handlers
*
* @param string $str
* @return \phpseclib3\Math\PrimeField\Integer
*/
public function extractSecret($str)
{
if (strlen($str) != 57) {
throw new \LengthException('Private Key should be 57-bytes long');
}
// 1. Hash the 57-byte private key using SHAKE256(x, 114), storing the
// digest in a 114-octet large buffer, denoted h. Only the lower 57
// bytes are used for generating the public key.
$hash = new Hash('shake256-912');
$h = $hash->hash($str);
$h = substr($h, 0, 57);
// 2. Prune the buffer: The two least significant bits of the first
// octet are cleared, all eight bits the last octet are cleared, and
// the highest bit of the second to last octet is set.
$h[0] = $h[0] & chr(0xFC);
$h = strrev($h);
$h[0] = "\0";
$h[1] = $h[1] | chr(0x80);
// 3. Interpret the buffer as the little-endian integer, forming a
// secret scalar s.
$dA = new BigInteger($h, 256);
$dA = $this->factory->newInteger($dA);
$dA->secret = $str;
return $dA;
}
/**
* Encode a point as a string
*
* @param array $point
* @return string
*/
public function encodePoint($point)
{
list($x, $y) = $point;
$y = "\0" . $y->toBytes();
if ($x->isOdd()) {
$y[0] = $y[0] | chr(0x80);
}
$y = strrev($y);
return $y;
}
/**
* Creates a random scalar multiplier
*
* @return \phpseclib3\Math\PrimeField\Integer
*/
public function createRandomMultiplier()
{
return $this->extractSecret(Random::string(57));
}
/**
* Converts an affine point to an extended homogeneous coordinate
*
* From https://tools.ietf.org/html/rfc8032#section-5.2.4 :
*
* A point (x,y) is represented in extended homogeneous coordinates (X, Y, Z, T),
* with x = X/Z, y = Y/Z, x * y = T/Z.
*
* @return \phpseclib3\Math\PrimeField\Integer[]
*/
public function convertToInternal(array $p)
{
if (empty($p)) {
return [clone $this->zero, clone $this->one, clone $this->one];
}
if (isset($p[2])) {
return $p;
}
$p[2] = clone $this->one;
return $p;
}
/**
* 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($p)) {
return [];
}
if (!isset($p[2])) {
throw new \RuntimeException('Affine coordinates need to be manually converted to "Jacobi" coordinates or vice versa');
}
// from https://tools.ietf.org/html/rfc8032#page-18
list($x1, $y1, $z1) = $p;
$b = $x1->add($y1);
$b = $b->multiply($b);
$c = $x1->multiply($x1);
$d = $y1->multiply($y1);
$e = $c->add($d);
$h = $z1->multiply($z1);
$j = $e->subtract($this->two->multiply($h));
$x3 = $b->subtract($e)->multiply($j);
$y3 = $c->subtract($d)->multiply($e);
$z3 = $e->multiply($j);
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 $p;
}
return [];
}
if (!isset($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($q[0])) {
return !$p[1]->equals($q[1]) ? [] : $this->doublePoint($p);
}
// from https://tools.ietf.org/html/rfc8032#page-17
list($x1, $y1, $z1) = $p;
list($x2, $y2, $z2) = $q;
$a = $z1->multiply($z2);
$b = $a->multiply($a);
$c = $x1->multiply($x2);
$d = $y1->multiply($y2);
$e = $this->d->multiply($c)->multiply($d);
$f = $b->subtract($e);
$g = $b->add($e);
$h = $x1->add($y1)->multiply($x2->add($y2));
$x3 = $a->multiply($f)->multiply($h->subtract($c)->subtract($d));
$y3 = $a->multiply($g)->multiply($d->subtract($c));
$z3 = $f->multiply($g);
return [$x3, $y3, $z3];
}
}
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