/**
* Calculates the greatest common divisor and Bezout's identity.
*
* Say you have 693 and 609. The GCD is 21. Bezout's identity states that there exist integers x and y such that
* 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which
* combination is returned is dependant upon which mode is in use. See
* {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bezout's identity - Wikipedia} for more information.
*
* Here's an example:
* <code>
* <?php
* include 'Math/BigInteger.php';
*
* $a = new Math_BigInteger(693);
* $b = new Math_BigInteger(609);
*
* extract($a->extendedGCD($b));
*
* echo $gcd->toString() . "\r\n"; // outputs 21
* echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
* ?>
* </code>
*
* @param Math_BigInteger $n
* @return Math_BigInteger
* @access public
* @internal Calculates the GCD using the binary xGCD algorithim described in
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,
* the more traditional algorithim requires "relatively costly multiple-precision divisions".
*/
function extendedGCD($n)
{
switch (MATH_BIGINTEGER_MODE) {
case MATH_BIGINTEGER_MODE_GMP:
extract(gmp_gcdext($this->value, $n->value));
return array('gcd' => $this->_normalize(new Math_BigInteger($g)), 'x' => $this->_normalize(new Math_BigInteger($s)), 'y' => $this->_normalize(new Math_BigInteger($t)));
case MATH_BIGINTEGER_MODE_BCMATH:
// it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
// best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,
// the basic extended euclidean algorithim is what we're using.
$u = $this->value;
$v = $n->value;
$a = '1';
$b = '0';
$c = '0';
$d = '1';
while (bccomp($v, '0', 0) != 0) {
$q = bcdiv($u, $v, 0);
$temp = $u;
$u = $v;
$v = bcsub($temp, bcmul($v, $q, 0), 0);
$temp = $a;
$a = $c;
$c = bcsub($temp, bcmul($a, $q, 0), 0);
$temp = $b;
$b = $d;
$d = bcsub($temp, bcmul($b, $q, 0), 0);
}
return array('gcd' => $this->_normalize(new Math_BigInteger($u)), 'x' => $this->_normalize(new Math_BigInteger($a)), 'y' => $this->_normalize(new Math_BigInteger($b)));
}
$y = $n->copy();
$x = $this->copy();
$g = new Math_BigInteger();
$g->value = array(1);
while (!($x->value[0] & 1 || $y->value[0] & 1)) {
$x->_rshift(1);
$y->_rshift(1);
$g->_lshift(1);
}
$u = $x->copy();
$v = $y->copy();
$a = new Math_BigInteger();
$b = new Math_BigInteger();
$c = new Math_BigInteger();
$d = new Math_BigInteger();
$a->value = $d->value = $g->value = array(1);
$b->value = $c->value = array();
while (!empty($u->value)) {
while (!($u->value[0] & 1)) {
$u->_rshift(1);
if (!empty($a->value) && $a->value[0] & 1 || !empty($b->value) && $b->value[0] & 1) {
$a = $a->add($y);
$b = $b->subtract($x);
}
$a->_rshift(1);
$b->_rshift(1);
}
while (!($v->value[0] & 1)) {
$v->_rshift(1);
if (!empty($d->value) && $d->value[0] & 1 || !empty($c->value) && $c->value[0] & 1) {
$c = $c->add($y);
$d = $d->subtract($x);
}
$c->_rshift(1);
$d->_rshift(1);
}
if ($u->compare($v) >= 0) {
$u = $u->subtract($v);
$a = $a->subtract($c);
$b = $b->subtract($d);
} else {
$v = $v->subtract($u);
$c = $c->subtract($a);
$d = $d->subtract($b);
}
}
return array('gcd' => $this->_normalize($g->multiply($v)), 'x' => $this->_normalize($c), 'y' => $this->_normalize($d));
}
echo gmp_strval($div1) . "\n";
// gmp_fact
$fact1 = gmp_fact(5);
// 5 * 4 * 3 * 2 * 1
echo gmp_strval($fact1) . "\n";
$fact2 = gmp_fact(50);
// 50 * 49 * 48, ... etc
echo gmp_strval($fact2) . "\n";
// gmp_gcd
$gcd = gmp_gcd("12", "21");
echo gmp_strval($gcd) . "\n";
// gmp_gcdext
$a = gmp_init(12);
$b = gmp_init(21);
$g = gmp_gcd($a, $b);
$r = gmp_gcdext($a, $b);
$check_gcd = gmp_strval($g) == gmp_strval($r['g']);
$eq_res = gmp_add(gmp_mul($a, $r['s']), gmp_mul($b, $r['t']));
$check_res = gmp_strval($g) == gmp_strval($eq_res);
if ($check_gcd && $check_res) {
$fmt = "Solution: %d*%d + %d*%d = %d\n";
printf($fmt, gmp_strval($a), gmp_strval($r['s']), gmp_strval($b), gmp_strval($r['t']), gmp_strval($r['g']));
} else {
echo "Error while solving the equation\n";
}
// gmp_hamdist
$ham1 = gmp_init("1001010011", 2);
$ham2 = gmp_init("1011111100", 2);
echo gmp_hamdist($ham1, $ham2) . "\n";
echo gmp_popcount(gmp_xor($ham1, $ham2)) . "\n";
// gmp_init (although that's probably tested well by now)
/**
* Finds the greatest common denominator of two numbers using the extended
* Euclidean algorithm.
*
* The returned array is ( a0, b0, gcd( a, b ) ), where
* a0 * a + b0 * b = gcd( a, b )
*
* @param resource $a The first number
* @param resource $b The second number
* @return array(resource)
*/
public function gcd($a, $b)
{
$result = gmp_gcdext($a, $b);
return array($result['s'], $result['t'], $result['g']);
}
<?php
$n = gmp_init("34293864345");
$n1 = gmp_init("23434293864345");
$a = array(array(123, 45), array(4341, 9734), array(23487, 333), array(-234234, -123123), array(-100, -2234), array(345, "34587345"), array(345, "0"), array("345556456", 345873), array("34545345556456", "323432445873"), array($n, $n1));
foreach ($a as $val) {
$r = gmp_gcdext($val[0], $val[1]);
var_dump(gmp_strval($r['g']));
var_dump(gmp_strval($r['s']));
var_dump(gmp_strval($r['t']));
}
var_dump(gmp_gcdext($val[0], array()));
var_dump(gmp_gcdext(array(), array()));
var_dump(gmp_gcdext(array(), array(), 1));
var_dump(gmp_gcdext(array()));
var_dump(gmp_gcdext());
echo "Done\n";
public function extendedGCD($n)
{
switch (MATH_BIGINTEGER_MODE) {
case MATH_BIGINTEGER_MODE_GMP:
$_gmp_gcdext = gmp_gcdext($this->value, $n->value);
$g = $_gmp_gcdext['g'];
$s = $_gmp_gcdext['s'];
$t = $_gmp_gcdext['t'];
return array('gcd' => $this->_normalize(new Math_BigInteger($g)), 'x' => $this->_normalize(new Math_BigInteger($s)), 'y' => $this->_normalize(new Math_BigInteger($t)));
case MATH_BIGINTEGER_MODE_BCMATH:
$u = $this->value;
$v = $n->value;
$a = '1';
$b = '0';
$c = '0';
$d = '1';
while (bccomp($v, '0', 0) != 0) {
$q = bcdiv($u, $v, 0);
$temp = $u;
$u = $v;
$v = bcsub($temp, bcmul($v, $q, 0), 0);
$temp = $a;
$a = $c;
$c = bcsub($temp, bcmul($a, $q, 0), 0);
$temp = $b;
$b = $d;
$d = bcsub($temp, bcmul($b, $q, 0), 0);
}
return array('gcd' => $this->_normalize(new Math_BigInteger($u)), 'x' => $this->_normalize(new Math_BigInteger($a)), 'y' => $this->_normalize(new Math_BigInteger($b)));
}
$y = $n->copy();
$x = $this->copy();
$g = new Math_BigInteger();
$g->value = array(1);
while (!($x->value[0] & 1 || $y->value[0] & 1)) {
$x->_rshift(1);
$y->_rshift(1);
$g->_lshift(1);
}
$u = $x->copy();
$v = $y->copy();
$a = new Math_BigInteger();
$b = new Math_BigInteger();
$c = new Math_BigInteger();
$d = new Math_BigInteger();
$a->value = $d->value = $g->value = array(1);
$b->value = $c->value = array();
while (!empty($u->value)) {
while (!($u->value[0] & 1)) {
$u->_rshift(1);
if (!empty($a->value) && $a->value[0] & 1 || !empty($b->value) && $b->value[0] & 1) {
$a = $a->add($y);
$b = $b->subtract($x);
}
$a->_rshift(1);
$b->_rshift(1);
}
while (!($v->value[0] & 1)) {
$v->_rshift(1);
if (!empty($d->value) && $d->value[0] & 1 || !empty($c->value) && $c->value[0] & 1) {
$c = $c->add($y);
$d = $d->subtract($x);
}
$c->_rshift(1);
$d->_rshift(1);
}
if (0 <= $u->compare($v)) {
$u = $u->subtract($v);
$a = $a->subtract($c);
$b = $b->subtract($d);
} else {
$v = $v->subtract($u);
$c = $c->subtract($a);
$d = $d->subtract($b);
}
}
return array('gcd' => $this->_normalize($g->multiply($v)), 'x' => $this->_normalize($c), 'y' => $this->_normalize($d));
}
public static function add(\fpoirotte\Pssht\ECC\Curve $curve, \fpoirotte\Pssht\ECC\Point $P, \fpoirotte\Pssht\ECC\Point $Q)
{
$mod = $curve->getModulus();
$xP = $P->coordinates['x'];
$yP = $P->coordinates['y'];
$xQ = $Q->coordinates['x'];
$yQ = $Q->coordinates['y'];
if (!gmp_cmp($xP, $xQ) && !gmp_cmp($yP, $yQ)) {
$alphanum = gmp_add(gmp_mul('3', gmp_pow($xP, '2')), $curve->getA());
$alphaden = gmp_mul('2', $yP);
} else {
$alphanum = gmp_sub($yQ, $yP);
$alphaden = gmp_sub($xQ, $xP);
}
$bezout = gmp_gcdext($alphaden, $mod);
$alpha = gmp_mod(gmp_mul($alphanum, $bezout['s']), $mod);
$xR = gmp_sub(gmp_sub(gmp_pow($alpha, '2'), $xP), $xQ);
$yR = gmp_sub(gmp_mul($alpha, gmp_sub($xP, $xR)), $yP);
return new static(gmp_mod(gmp_add($xR, $mod), $mod), gmp_mod(gmp_add($yR, $mod), $mod));
}
