<?php
/**
* Generalized Koblitz Curves over y^2 = x^3 + b.
*
* According to http://www.secg.org/SEC2-Ver-1.0.pdf Koblitz curves are over the GF(2**m)
* finite field. Both the $a$ and $b$ coefficients are either 0 or 1. However, SEC2
* generalizes the definition to include curves over GF(P) "which possess an efficiently
* computable endomorphism".
*
* For these generalized Koblitz curves $b$ doesn't have to be 0 or 1. Whether or not $a$
* has any restrictions on it is unclear, however, for all the GF(P) Koblitz curves defined
* in SEC2 v1.0 $a$ is $0$ so all of the methods defined herein will assume that it is.
*
* I suppose we could rename the $b$ coefficient to $a$, however, the documentation refers
* to $b$ so we'll just keep it.
*
* If a later version of SEC2 comes out wherein some $a$ values are non-zero we can create a
* new method for those. eg. KoblitzA1Prime.php or something.
*
* 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
* @link http://pear.php.net/package/Math_BigInteger
*/
namespace phpseclib3\
Crypt\EC\BaseCurves
;
use phpseclib3\Math\BigInteger;
use phpseclib3\Math\PrimeField;
/**
* Curves over y^2 = x^3 + b
*
* @package KoblitzPrime
* @author Jim Wigginton <terrafrost@php.net>
* @access public
*/
class KoblitzPrime extends Prime
{
// don't overwrite setCoefficients() with one that only accepts one parameter so that
// one might be able to switch between KoblitzPrime and Prime more easily (for benchmarking
// purposes).
/**
* Multiply and Add Points
*
* Uses a efficiently computable endomorphism to achieve a slight speedup
*
* Adapted from https://git.io/vxbrP
*
* @return int[]
*/
public function multiplyAddPoints
(array $points, array $scalars) {
static $zero, $one, $two;
$two = new BigInteger(2);
$one = new BigInteger(1);
}
if (!isset($this->beta)) { // get roots
$inv = $this->one->divide($this->two)->negate();
$s = $this->three->negate()->squareRoot()->multiply($inv);
$betas = [
$inv->add($s),
$inv->subtract($s)
];
$this->beta = $betas[0]->compare($betas[1]) < 0 ? $betas[0] : $betas[1];
//echo strtoupper($this->beta->toHex(true)) . "\n"; exit;
}
if (!isset($this->basis)) { $factory = new PrimeField($this->order);
$tempOne = $factory->newInteger($one);
$tempTwo = $factory->newInteger($two);
$tempThree = $factory->newInteger(new BigInteger(3));
$inv = $tempOne->divide($tempTwo)->negate();
$s = $tempThree->negate()->squareRoot()->multiply($inv);
$lambdas = [
$inv->add($s),
$inv->subtract($s)
];
$lhs = $this->multiplyPoint($this->p, $lambdas[0])[0];
$rhs = $this->p[0]->multiply($this->beta);
$lambda = $lhs->equals($rhs) ? $lambdas[0] : $lambdas[1];
$this->basis = static::extendedGCD($lambda->toBigInteger(), $this->order);
///*
foreach ($this->basis as $basis) {
echo strtoupper($basis['b']->toHex(true)) . "\n\n"; }
//*/
}
$npoints = $nscalars = [];
for ($i = 0; $i < count($points); $i++) { $p = $points[$i];
$k = $scalars[$i]->toBigInteger();
// begin split
list($v1, $v2) = $this->basis;
$c1 = $v2['b']->multiply($k);
list($c1, $r) = $c1->divide($this->order); if ($this->order->compare($r->multiply($two)) <= 0) {
$c1 = $c1->add($one);
}
$c2 = $v1['b']->negate()->multiply($k);
list($c2, $r) = $c2->divide($this->order); if ($this->order->compare($r->multiply($two)) <= 0) {
$c2 = $c2->add($one);
}
$p1 = $c1->multiply($v1['a']);
$p2 = $c2->multiply($v2['a']);
$q1 = $c1->multiply($v1['b']);
$q2 = $c2->multiply($v2['b']);
$k1 = $k->subtract($p1)->subtract($p2);
$k2 = $q1->add($q2)->negate();
// end split
$beta = [
$p[0]->multiply($this->beta),
$p[1],
clone $this->one
];
return [
$p[0]->multiply($this->beta),
$p[1],
clone $this->one
];
}, $p['naf']);
$beta['nafwidth'] = $p['nafwidth'];
}
if ($k1->isNegative()) {
$k1 = $k1->negate();
$p = $this->negatePoint($p);
}
if ($k2->isNegative()) {
$k2 = $k2->negate();
$beta = $this->negatePoint($beta);
}
$pos = 2 * $i;
$npoints[$pos] = $p;
$nscalars[$pos] = $this->factory->newInteger($k1);
$pos++;
$npoints[$pos] = $beta;
$nscalars[$pos] = $this->factory->newInteger($k2);
}
return parent::multiplyAddPoints($npoints, $nscalars);
}
/**
* Returns the numerator and denominator of the slope
*
* @return FiniteField[]
*/
protected function doublePointHelper
(array $p) {
$numerator = $this->three->multiply($p[0])->multiply($p[0]);
$denominator = $this->two->multiply($p[1]);
return [$numerator, $denominator];
}
/**
* Doubles a jacobian coordinate on the curve
*
* See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
*
* @return FiniteField[]
*/
protected function jacobianDoublePoint
(array $p) {
list($x1, $y1, $z1) = $p; $a = $x1->multiply($x1);
$b = $y1->multiply($y1);
$c = $b->multiply($b);
$d = $x1->add($b);
$d = $d->multiply($d)->subtract($a)->subtract($c)->multiply($this->two);
$e = $this->three->multiply($a);
$f = $e->multiply($e);
$x3 = $f->subtract($this->two->multiply($d));
$y3 = $e->multiply($d->subtract($x3))->subtract(
$this->eight->multiply($c)
);
$z3 = $this->two->multiply($y1)->multiply($z1);
return [$x3, $y3, $z3];
}
/**
* Doubles a "fresh" jacobian coordinate on the curve
*
* See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-mdbl-2007-bl
*
* @return FiniteField[]
*/
protected function jacobianDoublePointMixed
(array $p) {
$xx = $x1->multiply($x1);
$yy = $y1->multiply($y1);
$yyyy = $yy->multiply($yy);
$s = $x1->add($yy);
$s = $s->multiply($s)->subtract($xx)->subtract($yyyy)->multiply($this->two);
$m = $this->three->multiply($xx);
$t = $m->multiply($m)->subtract($this->two->multiply($s));
$x3 = $t;
$y3 = $s->subtract($t);
$y3 = $m->multiply($y3)->subtract($this->eight->multiply($yyyy));
$z3 = $this->two->multiply($y1);
return [$x3, $y3, $z3];
}
/**
* Tests whether or not the x / y values satisfy the equation
*
* @return boolean
*/
public function verifyPoint
(array $p) {
$lhs = $y->multiply($y);
$temp = $x->multiply($x)->multiply($x);
$rhs = $temp->add($this->b);
return $lhs->equals($rhs);
}
/**
* Calculates the parameters needed from the Euclidean algorithm as discussed at
* http://diamond.boisestate.edu/~liljanab/MATH308/GuideToECC.pdf#page=148
*
* @param BigInteger $u
* @param BigInteger $v
* @return BigInteger[]
*/
protected static function extendedGCD(BigInteger $u, BigInteger $v)
{
$one = new BigInteger(1);
$zero = new BigInteger();
$a = clone $one;
$b = clone $zero;
$c = clone $zero;
$d = clone $one;
$stop = $v->bitwise_rightShift($v->getLength() >> 1);
$a1 = clone $zero;
$b1 = clone $zero;
$a2 = clone $zero;
$b2 = clone $zero;
$postGreatestIndex = 0;
while (!$v->equals($zero)) {
list($q) = $u->divide($v);
$temp = $u;
$u = $v;
$v = $temp->subtract($v->multiply($q));
$temp = $a;
$a = $c;
$c = $temp->subtract($a->multiply($q));
$temp = $b;
$b = $d;
$d = $temp->subtract($b->multiply($q));
if ($v->compare($stop) > 0) {
$a0 = $v;
$b0 = $c;
} else {
$postGreatestIndex++;
}
if ($postGreatestIndex == 1) {
$a1 = $v;
$b1 = $c->negate();
}
if ($postGreatestIndex == 2) {
$rhs = $a0->multiply($a0)->add($b0->multiply($b0));
$lhs = $v->multiply($v)->add($b->multiply($b));
if ($lhs->compare($rhs) <= 0) {
$a2 = $a0;
$b2 = $b0->negate();
} else {
$a2 = $v;
$b2 = $c->negate();
}
break;
}
}
return [
['a' => $a1, 'b' => $b1],
['a' => $a2, 'b' => $b2]
];
}
}