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BinaryField: speed up multiplication for GCM and smaller curves

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This commit is contained in:
terrafrost 2019-01-07 06:33:11 -06:00
parent 835b1207fa
commit 4ae33f9bde
1 changed files with 131 additions and 6 deletions
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137
phpseclib/Math/BinaryField/Integer.php
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@ -8,7 +8,8 @@
* These 1's or 0's represent the coefficients of the x**n, where n is the
* location of the given bit. When you add numbers over a binary finite field
* the result should have a coefficient of 1 or 0 as well. Hence addition
* and subtraction become the same operation as XOR, etc.
* and subtraction become the same operation as XOR.
* eg. 1 + 1 + 1 == 3 % 2 == 1 or 0 - 1 == -1 % 2 == 1
*
* PHP version 5 and 7
*
@ -186,12 +187,12 @@ class Integer extends Base
}
/**
* Perform polynomial multiplation
* Perform polynomial multiplation in the traditional way
*
* @return string[]
* @return string
* @link https://en.wikipedia.org/wiki/Finite_field_arithmetic#Multiplication
*/
private static function polynomialMultiply($x, $y)
private static function regularPolynomialMultiply($x, $y)
{
$precomputed = [ltrim($x, "\0")];
$x = strrev(BinaryField::base256ToBase2($x));
@ -222,6 +223,128 @@ class Integer extends Base
return $result;
}
/**
* Perform polynomial multiplation
*
* Uses karatsuba multiplication to reduce x-bit multiplications to a series of 32-bit multiplications
*
* @return string
* @link https://en.wikipedia.org/wiki/Karatsuba_algorithm
*/
private static function polynomialMultiply($x, $y)
{
if (strlen($x) == strlen($y)) {
$length = strlen($x);
} else {
$length = max(strlen($x), strlen($y));
$x = str_pad($x, $length, "\0", STR_PAD_LEFT);
$y = str_pad($y, $length, "\0", STR_PAD_LEFT);
}
switch (true) {
case PHP_INT_SIZE == 8 && $length <= 4:
return $length != 4 ?
self::subMultiply(str_pad($x, 4, "\0", STR_PAD_LEFT), str_pad($y, 4, "\0", STR_PAD_LEFT)) :
self::subMultiply($x, $y);
case PHP_INT_SIZE == 4 || $length > 32:
return self::regularPolynomialMultiply($x, $y);
}
$m = $length >> 1;
$x1 = substr($x, 0, -$m);
$x0 = substr($x, -$m);
$y1 = substr($y, 0, -$m);
$y0 = substr($y, -$m);
$z2 = self::polynomialMultiply($x1, $y1);
$z0 = self::polynomialMultiply($x0, $y0);
$z1 = self::polynomialMultiply(
self::subAdd2($x1, $x0),
self::subAdd2($y1, $y0)
);
$z1 = self::subAdd3($z1, $z2, $z0);
$xy = self::subAdd3(
$z2 . str_repeat("\0", 2 * $m),
$z1 . str_repeat("\0", $m),
$z0
);
return ltrim($xy, "\0");
}
/**
* Perform polynomial multiplication on 2x 32-bit numbers, returning
* a 64-bit number
*
* @param string $x
* @param string $y
* @return string
* @link https://www.bearssl.org/constanttime.html#ghash-for-gcm
*/
private static function subMultiply($x, $y)
{
$x = unpack('N', $x)[1];
$y = unpack('N', $y)[1];
$x0 = $x & 0x11111111;
$x1 = $x & 0x22222222;
$x2 = $x & 0x44444444;
$x3 = $x & 0x88888888;
$y0 = $y & 0x11111111;
$y1 = $y & 0x22222222;
$y2 = $y & 0x44444444;
$y3 = $y & 0x88888888;
$z0 = ($x0 * $y0) ^ ($x1 * $y3) ^ ($x2 * $y2) ^ ($x3 * $y1);
$z1 = ($x0 * $y1) ^ ($x1 * $y0) ^ ($x2 * $y3) ^ ($x3 * $y2);
$z2 = ($x0 * $y2) ^ ($x1 * $y1) ^ ($x2 * $y0) ^ ($x3 * $y3);
$z3 = ($x0 * $y3) ^ ($x1 * $y2) ^ ($x2 * $y1) ^ ($x3 * $y0);
$z0&= 0x1111111111111111;
$z1&= 0x2222222222222222;
$z2&= 0x4444444444444444;
$z3&= -8608480567731124088; // 0x8888888888888888 gets interpreted as a float
$z = $z0 | $z1 | $z2 | $z3;
return pack('J', $z);
}
/**
* Adds two numbers
*
* @param string $x
* @param string $y
* @return string
*/
private static function subAdd2($x, $y)
{
$length = max(strlen($x), strlen($y));
$x = str_pad($x, $length, "\0", STR_PAD_LEFT);
$y = str_pad($y, $length, "\0", STR_PAD_LEFT);
return $x ^ $y;
}
/**
* Adds three numbers
*
* @param string $x
* @param string $y
* @return string
*/
private static function subAdd3($x, $y, $z)
{
$length = max(strlen($x), strlen($y), strlen($z));
$x = str_pad($x, $length, "\0", STR_PAD_LEFT);
$y = str_pad($y, $length, "\0", STR_PAD_LEFT);
$z = str_pad($z, $length, "\0", STR_PAD_LEFT);
return $x ^ $y ^ $z;
}
/**
* Adds two BinaryFieldIntegers.
*
@ -239,6 +362,7 @@ class Integer extends Base
return new static($this->instanceID, $x ^ $y);
}
/**
* Subtracts two BinaryFieldIntegers.
*
@ -285,13 +409,14 @@ class Integer extends Base
// row n-2 and the product of the quotient and the auxiliary in row
// n-1
$temp = static::polynomialMultiply($aux1, $q);
$aux = str_pad($aux0, strlen($temp), "\0", STR_PAD_LEFT) ^ $temp;
$aux = str_pad($aux0, strlen($temp), "\0", STR_PAD_LEFT) ^
str_pad($temp, strlen($aux0), "\0", STR_PAD_LEFT);
$aux0 = $aux1;
$aux1 = $aux;
}
$temp = new static($this->instanceID);
$temp->value = $aux1;
$temp->value = ltrim($aux1, "\0");
return $temp;
}