PHP Montgomery Modular Exponentiation Engine with interleaved multiplication
| author | Jim Wigginton terrafrost@php.net |
|---|---|
| package | Default |
__construct(integer|\phpseclib3\Math\BigInteger\Engines\numeric-string $x,integer $base = 10)
integer|\phpseclib3\Math\BigInteger\Engines\numeric-stringinteger Base-10 number or base-$base number if $base set.
integer
__debugInfo(): array
Will be called, automatically, when print_r() or var_dump() are called
array __sleep(): array
Will be called, automatically, when serialize() is called on a BigInteger object.
array __toString(): string
string __wakeup(): void
Will be called, automatically, when unserialize() is called on a BigInteger object.
abs(): \phpseclib3\Math\BigInteger\Engines\PHP
addHelper(array $x_value,boolean $x_negative,array $y_value,boolean $y_negative): array
array
boolean
array
boolean
array array_repeat(integer $input,integer $multiplier): array
integer
integer
array base256_lshift(string &$x,integer $shift): void
Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
string
integer
baseSquare(array $value): array
Squaring can be done faster than multiplying a number by itself can be. See HAC 14.2.4 / MPM 5.3 for more information.
array
array bitwise_leftRotate(integer $shift): \phpseclib3\Math\BigInteger\Engines\Engine
Instead of the top x bits being dropped they're appended to the shifted bit string.
integer
\phpseclib3\Math\BigInteger\Engines\Engine bitwise_leftShift(integer $shift): \phpseclib3\Math\BigInteger\Engines\PHP
Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
integer
\phpseclib3\Math\BigInteger\Engines\PHP bitwise_not(): \phpseclib3\Math\BigInteger\Engines\Engine|string
bitwise_rightRotate(integer $shift): \phpseclib3\Math\BigInteger\Engines\Engine
Instead of the bottom x bits being dropped they're prepended to the shifted bit string.
integer
\phpseclib3\Math\BigInteger\Engines\Engine bitwise_rightShift(integer $shift): \phpseclib3\Math\BigInteger\Engines\PHP
Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
integer
\phpseclib3\Math\BigInteger\Engines\PHP bitwise_small_split(integer $split): \phpseclib3\Math\BigInteger\Engines\list<int>
integer
\phpseclib3\Math\BigInteger\Engines\list bitwise_split(integer $split): array<mixed,\phpseclib3\Math\BigInteger\Engines\Engine>
Splits BigInteger's into chunks of $split bits
integer
array<mixed,\phpseclib3\Math\BigInteger\Engines\Engine> bitwiseAndHelper(\phpseclib3\Math\BigInteger\Engines\Engine $x): \phpseclib3\Math\BigInteger\Engines\Engine
bitwiseOrHelper(\phpseclib3\Math\BigInteger\Engines\Engine $x): \phpseclib3\Math\BigInteger\Engines\Engine
bitwiseXorHelper(\phpseclib3\Math\BigInteger\Engines\Engine $x): \phpseclib3\Math\BigInteger\Engines\Engine
compareHelper(array $x_value,boolean $x_negative,array $y_value,boolean $y_negative): integer
| see | static::compare() |
|---|
array
boolean
array
boolean
integer convertToObj(array $arr): static
array
static createRecurringModuloFunction(): callable
Sometimes it may be desirable to do repeated modulos with the same number outside of modular exponentiation
callable divide_digit(array $dividend,integer $divisor): array
abc / x = a00 / x + b0 / x + c / x
array
integer
array extendedGCDHelper(\phpseclib3\Math\BigInteger\Engines\Engine $n): \phpseclib3\Math\BigInteger\Engines\array{gcd:
\phpseclib3\Math\BigInteger\Engines\array{gcd:Engine, x: Engine, y: Engine}
getLength(): integer
integer getLengthInBytes(): integer
integer getPrecision(): integer
Returns the precision if it exists, -1 if it doesn't
integer initialize(integer $base)
| see | \phpseclib3\Math\BigInteger\Engines\parent::__construct() |
|---|---|
integer
int2bytes(integer $x): string
integer
string isNegative(): boolean
boolean isOdd(): boolean
boolean isPrime(integer|boolean $t = false): boolean
Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the $t parameter is distributability. BigInteger::randomPrime() can be distributed across multiple pageloads on a website instead of just one.
integer|boolean
boolean isValidEngine(): boolean
boolean jsonSerialize()
karatsuba(array $x_value,array $y_value): array
karatsubaSquare(array $value): array
lshift(integer $shift)
Shifts BigInteger's by $shift bits.
integer
make_odd()
If the current number is odd it'll be unchanged. If it's even, one will be added to it.
| see | self::randomPrime() |
|---|---|
maxHelper(array $nums): \phpseclib3\Math\BigInteger\Engines\Engine
minHelper(array $nums): \phpseclib3\Math\BigInteger\Engines\Engine
minMaxBits(integer $bits): \phpseclib3\Math\BigInteger\Engines\array{min:
integer
\phpseclib3\Math\BigInteger\Engines\array{min:static, max: static}
modInverse67108864(array $x,string $class): integer
Based off of the bnpInvDigit function implemented and justified in the following URL:
http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js
The following URL provides more info:
http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85
As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to 40 bits, which only 64-bit floating points will support.
Thanks to Pedro Gimeno Fortea for input!
array
string
integer modInverseHelper(\phpseclib3\Math\BigInteger\Engines\Engine $n): static|false
Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses.
{@internal See HAC 14.64 for more information.}
static|false multiplyHelper(array $x_value,boolean $x_negative,array $y_value,boolean $y_negative): array
array
boolean
array
boolean
array multiplyReduce(array $x,array $y,array $n,string $class): array
| see | self::slidingWindow() |
|---|
array
array
array
string
array negate(): static
Given $k, returns -$k
static normalize(\phpseclib3\Math\BigInteger\Engines\PHP $result): static
Removes leading zeros and truncates (if necessary) to maintain the appropriate precision
static pad(string $str): string
string
string powHelper(\phpseclib3\Math\BigInteger\Engines\PHP $n): \phpseclib3\Math\BigInteger\Engines\PHP
powModHelper(\phpseclib3\Math\BigInteger\Engines\PHP $x,\phpseclib3\Math\BigInteger\Engines\PHP $e,\phpseclib3\Math\BigInteger\Engines\PHP $n,string $class): \phpseclib3\Math\BigInteger\Engines\PHP
The most naive approach to modular exponentiation has very unreasonable requirements, and and although the approach involving repeated squaring does vastly better, it, too, is impractical for our purposes. The reason being that division - by far the most complicated and time-consuming of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
Modular reductions resolve this issue. Although an individual modular reduction takes more time then an individual division, when performed in succession (with the same modulo), they're a lot faster.
The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction, although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because the product of two odd numbers is odd), but what about when RSA isn't used?
In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however, uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and the other, a power of two - and recombine them, later. This is the method that this modPow function uses. Montgomery Reduction with Even Modulus elaborates.
string
\phpseclib3\Math\BigInteger\Engines\PHP powModInner(\phpseclib3\Math\BigInteger\Engines\PHP $e,\phpseclib3\Math\BigInteger\Engines\PHP $n): \phpseclib3\Math\BigInteger\Engines\PHP
powModOuter(\phpseclib3\Math\BigInteger\Engines\Engine $e,\phpseclib3\Math\BigInteger\Engines\Engine $n): static|false
static|false prepareReduce(array $x,array $n,string $class): array
| see | self::slidingWindow() |
|---|
array
array
string
array random(integer $size): \phpseclib3\Math\BigInteger\Engines\Engine
Bit length is equal to $size
integer
\phpseclib3\Math\BigInteger\Engines\Engine randomPrime(integer $size): \phpseclib3\Math\BigInteger\Engines\Engine
Bit length is equal to $size
integer
\phpseclib3\Math\BigInteger\Engines\Engine randomRangeHelper(\phpseclib3\Math\BigInteger\Engines\Engine $min,\phpseclib3\Math\BigInteger\Engines\Engine $max): \phpseclib3\Math\BigInteger\Engines\Engine
Returns a random number between $min and $max where $min and $max can be defined using one of the two methods:
BigInteger::randomRange($min, $max) BigInteger::randomRange($max, $min)
\phpseclib3\Math\BigInteger\Engines\Engine randomRangePrimeInner(\phpseclib3\Math\BigInteger\Engines\Engine $x,\phpseclib3\Math\BigInteger\Engines\Engine $min,\phpseclib3\Math\BigInteger\Engines\Engine $max): static|false
static|false randomRangePrimeOuter(\phpseclib3\Math\BigInteger\Engines\Engine $min,\phpseclib3\Math\BigInteger\Engines\Engine $max): static|false
static|false reduce(array $x,array $n,string $class): array
Interleaves the montgomery reduction and long multiplication algorithms together as described in HAC 14.36
array
array
string
array regularMultiply(array $x_value,array $y_value): array
Modeled after 'multiply' in MutableBigInteger.java.
array
array
array root(integer $n = 2): \phpseclib3\Math\BigInteger\Engines\Engine
rootHelper(integer $n): \phpseclib3\Math\BigInteger\Engines\Engine
rootInner(integer $n): \phpseclib3\Math\BigInteger\Engines\Engine
Returns the nth root of a positive biginteger, where n defaults to 2
{@internal This function is based off of this page and this stackoverflow question.}
integer
\phpseclib3\Math\BigInteger\Engines\Engine rshift(integer $shift)
Shifts BigInteger's by $shift bits.
integer
safe_divide(integer $x,integer $y): integer
Even if int64 is being used the division operator will return a float64 value if the dividend is not evenly divisible by the divisor. Since a float64 doesn't have the precision of int64 this is a problem so, when int64 is being used, we'll guarantee that the dividend is divisible by first subtracting the remainder.
integer
integer
integer scan1divide(\phpseclib3\Math\BigInteger\Engines\PHP $r): integer
ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));
| see | self::isPrime() |
|---|
integer setBitmask(integer $bits): static
| see | \phpseclib3\Math\BigInteger\Engines\Engine::setPrecision() |
|---|
integer
static setModExpEngine(\phpseclib3\Math\BigInteger\Engines\class-string<Engine> $engine)
Throws an exception if the type is invalid
\phpseclib3\Math\BigInteger\Engines\class-string
setPrecision(integer $bits)
Some bitwise operations give different results depending on the precision being used. Examples include left shift, not, and rotates.
integer
setupIsPrime(): integer
integer slidingWindow(\phpseclib3\Math\BigInteger\Engines\Engine $x,\phpseclib3\Math\BigInteger\Engines\Engine $e,\phpseclib3\Math\BigInteger\Engines\Engine $n,\phpseclib3\Math\BigInteger\Engines\class-string<T> $class): \phpseclib3\Math\BigInteger\Engines\T
Based on HAC 14.85 / MPM 7.7. In a departure from those algorithims, however, this function performs a modular reduction after every multiplication and squaring operation. As such, this function has the same preconditions that the reductions being used do.
| template | T of Engine |
|---|
\phpseclib3\Math\BigInteger\Engines\class-string
\phpseclib3\Math\BigInteger\Engines\T square(\phpseclib3\Math\BigInteger\Engines\list<static> $x): \phpseclib3\Math\BigInteger\Engines\list<static>
\phpseclib3\Math\BigInteger\Engines\list
\phpseclib3\Math\BigInteger\Engines\list squareReduce(array $x,array $n,string $class): array
| see | self::slidingWindow() |
|---|
array
array
string
array subtractHelper(array $x_value,boolean $x_negative,array $y_value,boolean $y_negative): array
array
boolean
array
boolean
array testBit( $x): boolean
boolean testPrimality(integer $t): boolean
Uses the Miller-Rabin primality test. See HAC 4.24 for more info.
integer
boolean testSmallPrimes()
| see | self::isPrime() |
|---|---|
toBits(boolean $twos_compliment = false): string
Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're saved as two's compliment.
boolean
string toBytes(boolean $twos_compliment = false): string
boolean
string toBytesHelper(): string
Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're saved as two's compliment.
string toHex(boolean $twos_compliment = false): string
boolean
string toString(): string
string trim(\phpseclib3\Math\BigInteger\Engines\list<static> $value): \phpseclib3\Math\BigInteger\Engines\list<static>
Removes leading zeros
\phpseclib3\Math\BigInteger\Engines\list
\phpseclib3\Math\BigInteger\Engines\list VARIABLE
$cache[self::VARIABLE] tells us whether or not the cached data is still valid.
DATA
VALUE
SIGN
KARATSUBA_CUTOFF
At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
FAST_BITWISE
| see | |
|---|---|
ENGINE_DIR
| see | |
|---|---|
PRIMES
zero :\phpseclib3\Math\BigInteger\Engines\array<class-string<static>,
| var | static> |
|---|
\phpseclib3\Math\BigInteger\Engines\array, one :\phpseclib3\Math\BigInteger\Engines\array<class-string<static>,
| var | static> |
|---|
\phpseclib3\Math\BigInteger\Engines\array, two :\phpseclib3\Math\BigInteger\Engines\array<class-string<static>,
| var | static> |
|---|
\phpseclib3\Math\BigInteger\Engines\array, modexpEngine :\phpseclib3\Math\BigInteger\Engines\array<class-string<static>,
| var | class-string |
|---|
\phpseclib3\Math\BigInteger\Engines\array, isValidEngine :\phpseclib3\Math\BigInteger\Engines\array<class-string<static>,
| var | bool> |
|---|
\phpseclib3\Math\BigInteger\Engines\array, value :\GMP|string|array|integer
| var |
|---|
\GMP|string|array|integer is_negative :boolean
| var |
|---|
boolean precision :integer
| see | |
|---|---|
| var |
integer bitmask :static|false
| see | |
|---|---|
| var |
static|false reduce :callable
| var |
|---|
callable hex :string
| see | |
|---|---|
| var |
string