* @copyright 2017 Jim Wigginton * @license http://www.opensource.org/licenses/mit-license.html MIT License */ namespace phpseclib\Crypt\EC\Curves; use phpseclib\Crypt\EC\BaseCurves\TwistedEdwards; use phpseclib\Math\BigInteger; use phpseclib\Crypt\Hash; use phpseclib\Crypt\Random; class Ed25519 extends TwistedEdwards { const HASH = 'sha512'; /* Per https://tools.ietf.org/html/rfc8032#page-6 EdDSA has several parameters, one of which is b: 2. An integer b with 2^(b-1) > p. EdDSA public keys have exactly b bits, and EdDSA signatures have exactly 2*b bits. b is recommended to be a multiple of 8, so public key and signature lengths are an integral number of octets. SIZE corresponds to b */ const SIZE = 32; public function __construct() { // 2^255 - 19 $this->setModulo(new BigInteger('7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFED', 16)); $this->setCoefficients( // -1 new BigInteger('7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEC', 16), // a // -121665/121666 new BigInteger('52036CEE2B6FFE738CC740797779E89800700A4D4141D8AB75EB4DCA135978A3', 16) // d ); $this->setBasePoint( new BigInteger('216936D3CD6E53FEC0A4E231FDD6DC5C692CC7609525A7B2C9562D608F25D51A', 16), new BigInteger('6666666666666666666666666666666666666666666666666666666666666658', 16) ); $this->setOrder(new BigInteger('1000000000000000000000000000000014DEF9DEA2F79CD65812631A5CF5D3ED', 16)); // algorithm 14.47 from http://cacr.uwaterloo.ca/hac/about/chap14.pdf#page=16 /* $this->setReduction(function($x) { $parts = $x->bitwise_split(255); $className = $this->className; if (count($parts) > 2) { list(, $r) = $x->divide($className::$modulo); return $r; } $zero = new BigInteger(); $c = new BigInteger(19); switch (count($parts)) { case 2: list($qi, $ri) = $parts; break; case 1: $qi = $zero; list($ri) = $parts; break; case 0: return $zero; } $r = $ri; while ($qi->compare($zero) > 0) { $temp = $qi->multiply($c)->bitwise_split(255); if (count($temp) == 2) { list($qi, $ri) = $temp; } else { $qi = $zero; list($ri) = $temp; } $r = $r->add($ri); } while ($r->compare($className::$modulo) > 0) { $r = $r->subtract($className::$modulo); } return $r; }); */ } /** * Recover X from Y * * Implements steps 2-4 at https://tools.ietf.org/html/rfc8032#section-5.1.3 * * Used by EC\Keys\Common.php * * @param BigInteger $x * @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)->add($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 /* we don't do $x2->squareRoot() because, quoting from https://tools.ietf.org/html/rfc8032#section-5.1.1: "For point decoding or "decompression", square roots modulo p are needed. They can be computed using the Tonelli-Shanks algorithm or the special case for p = 5 (mod 8). To find a square root of a, first compute the candidate root x = a^((p+3)/8) (mod p)." */ $exp = $this->getModulo()->add(new BigInteger(3)); $exp = $exp->bitwise_rightShift(3); $x = $x2->pow($exp); // If v x^2 = -u (mod p), set x <-- x * 2^((p-1)/4), which is a square root. if (!$x->multiply($x)->subtract($x2)->equals($this->zero)) { $temp = $this->getModulo()->subtract(new BigInteger(1)); $temp = $temp->bitwise_rightShift(2); $temp = $this->two->pow($temp); $x = $x->multiply($temp); 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-5.1.5 * * Used by the various key handlers * * @param string $str * @return \phpseclib\Math\PrimeField\Integer */ public function extractSecret($str) { if (strlen($str) != 32) { throw new \LengthException('Private Key should be 32-bytes long'); } // 1. Hash the 32-byte private key using SHA-512, storing the digest in // a 64-octet large buffer, denoted h. Only the lower 32 bytes are // used for generating the public key. $hash = new Hash('sha512'); $h = $hash->hash($str); $h = substr($h, 0, 32); // 2. Prune the buffer: The lowest three bits of the first octet are // cleared, the highest bit of the last octet is cleared, and the // second highest bit of the last octet is set. $h[0] = $h[0] & chr(0xF8); $h = strrev($h); $h[0] = ($h[0] & chr(0x3F)) | chr(0x40); // 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 string $str * @return string */ public function encodePoint($point) { list($x, $y) = $point; $y = $y->toBytes(); $y[0] = $y[0] & chr(0x7F); if ($x->isOdd()) { $y[0] = $y[0] | chr(0x80); } $y = strrev($y); return $y; } /** * Creates a random scalar multiplier * * @return \phpseclib\Math\PrimeField\Integer */ public function createRandomMultiplier() { return $this->extractSecret(Random::string(32)); } /** * Converts an affine point to an extended homogeneous coordinate * * From https://tools.ietf.org/html/rfc8032#section-5.1.4 : * * 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 \phpseclib\Math\PrimeField\Integer[] */ public function convertToInternal(array $p) { if (empty($p)) { return [clone $this->zero, clone $this->one, clone $this->one, clone $this->zero]; } if (isset($p[2])) { return $p; } $p[2] = clone $this->one; $p[3] = $p[0]->multiply($p[1]); 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-12 list($x1, $y1, $z1, $t1) = $p; $a = $x1->multiply($x1); $b = $y1->multiply($y1); $c = $this->two->multiply($z1)->multiply($z1); $h = $a->add($b); $temp = $x1->add($y1); $e = $h->subtract($temp->multiply($temp)); $g = $a->subtract($b); $f = $c->add($g); $x3 = $e->multiply($f); $y3 = $g->multiply($h); $t3 = $e->multiply($h); $z3 = $f->multiply($g); return [$x3, $y3, $z3, $t3]; } /** * 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-12 list($x1, $y1, $z1, $t1) = $p; list($x2, $y2, $z2, $t2) = $q; $a = $y1->subtract($x1)->multiply($y2->subtract($x2)); $b = $y1->add($x1)->multiply($y2->add($x2)); $c = $t1->multiply($this->two)->multiply($this->d)->multiply($t2); $d = $z1->multiply($this->two)->multiply($z2); $e = $b->subtract($a); $f = $d->subtract($c); $g = $d->add($c); $h = $b->add($a); $x3 = $e->multiply($f); $y3 = $g->multiply($h); $t3 = $e->multiply($h); $z3 = $f->multiply($g); return [$x3, $y3, $z3, $t3]; } }