How to implement a random float function so that it does not lose entropy? (PHP)

Question

Hi,

I am trying to generate random floats using nothing but bytes I get from /dev/urandom. Currently my best idea is to get the platform precision doing something like:

$maximumPrecision = strlen('' . 1/3) - 2;

and then construct a string of 0-9 in a loop the number of times $maximumPrecision tells us. For examle, if the precision is 12, I would generate 12 random numbers and concatenate them. I think it's an ugly idea.

Update: Does this make sense?

$bytes =getRandomBytes(7); // Just a function that returns random bytes.
$bytes[6] = $bytes[6] & chr(15); // Get rid off the other half
$bytes .= chr(0); // Add a null byte

$parts = unpack('V2', $bytes);

$number = $parts[1] + pow(2.0, 32) * $parts[2];
$number /= pow(2.0, 52);

Answer 1

PHP's float type typically is implemented as IEEE double. This format has 52 bit precision of mantissa, so in principle it should be able to generate 2⁵² different uniform numbers in [0, 1).

Therefore, you could extract 52 bits from the /dev/urandom, interpret as an integer, and divide by 2⁵². For example:

// assume we have 52 bits of data, little endian.
$bytes = "\x12\x34\x56\x78\x9a\xbc\x0d\x00";
//                                  ^   ^^ 12 bits of padding.

$parts = unpack('V2', $bytes);

$theNumber = $parts[1] + pow(2.0, 32) * $parts[2];  // <-- this is precise.
$theNumber /= pow(2.0, 52);                         // <-- this is precise.

Answer 2

The problem here is that the IEEE double precision number is defined in terms of exponents of base 2:

n = 2^exponent * 1.mantissa

Since you want an exponent -1, and there's no integer number n such that 2^n = 0.1, it gets complicated.

This generates a number between 1 and 2. You can subtract 1, but you'll lose a tiny amount of entropy if you do that (KennyTM's answer yields a number in that range, though, and uses all entropy -- this answer attempts to create the representation directly):

$source = fopen("/dev/urandom", "rb");

//build big-endian double
//take only 32 bits because of 32-bit platforms
$byte_batch_1 = fread($source, 4); //32-bit
$byte_batch_2 = fread($source, 4); //32-bit, we only need 20

$offset = (1 << 10) -1;

$first_word = unpack("N", $byte_batch_2);
$first_word = reset($first_word);
$first_word &= 0xFFFFF; //leave only 20 lower bits
$first_word |= $offset << 20;

$str = pack("N", $first_word) . $byte_batch_1;

//convert to little endian if necessary
if (pack('s', 1) == "\x01\x00") { //little-endian
    $str = strrev($str);
}

$float = unpack("d", $str);
$float = reset($float);