Thursday, December 17, 2015

There's Math.random(), and then there's Math.random()

Randomness visualization before PRNG update (V8 4.7) and after (V8 4.9) (visualization by Mike Malone, http://bl.ocks.org/mmalone/bf59aa2e44c44dde78ac, CC-BY-4.0)

Math.random()
Returns a Number value with positive sign, greater than or equal to 0 but less than 1, chosen randomly or pseudo randomly with approximately uniform distribution over that range, using an implementation-dependent algorithm or strategy. This function takes no arguments.


Math.random() is the most well-known and frequently-used source of randomness in Javascript. In V8 and most other Javascript engines, it is implemented using a pseudo-random number generator (PRNG). As with all PRNGs, the random number is derived from an internal state, which is altered by a fixed algorithm for every new random number. So for a given initial state, the sequence of random numbers is deterministic. Since the bit size n of the internal state is limited, the numbers that a PRNG generates will eventually repeat themselves. The upper bound for the period length of this permutation cycle is of course 2n.

There are many different PRNG algorithms; among the most well-known ones are Mersenne-Twister and LCG. Each has its particular characteristics, advantages, and drawbacks. Ideally, it would use as little memory as possible for the initial state, be quick to perform, have a large period length, and offer a high quality random distribution. While memory usage, performance, and period length can easily be measured or calculated, the quality is harder to determine. There is a lot of math behind statistical tests to check the quality of random numbers. The de-facto standard PRNG test suite, TestU01, implements many of these tests.

Until recently (up to version 4.9.40), V8’s choice of PRNG was MWC1616 (multiply with carry, combining two 16-bit parts). It uses 64 bits of internal state and looks roughly like this:
uint32_t state0 = 1;
uint32_t state1 = 2;
uint32_t mwc1616() {
  state0 = 18030 * (state0 & 0xffff) + (state0 >> 16);
  state1 = 30903 * (state1 & 0xffff) + (state1 >> 16);
  return state0 << 16 + (state1 & 0xffff);
The 32-bit value is then turned into a floating point number between 0 and 1 in agreement with the specification.

MWC1616 uses little memory and is pretty fast to compute, but unfortunately offers sub-par quality:

  • The number of random values it can generate is limited to 232 as opposed to the 252 numbers between 0 and 1 that double precision floating point can represent.
  • The more significant upper half of the result is almost entirely dependent on the value of state0. The period length would be at most 232, but instead of few large permutation cycles, there are many short ones. With a badly chosen initial state, the cycle length could be less than 40 million.
  • It fails many statistical tests in the TestU01 suite.

This has been pointed out to us, and having understood the problem and after some research, we decided to reimplement Math.random based on an algorithm called xorshift128+. It uses 128 bits of internal state, has a period length of 2128 - 1, and passes all tests from the TestU01 suite.
uint64_t state0 = 1;
uint64_t state1 = 2;
uint64_t xorshift128plus() {
  uint64_t s1 = state0;
  uint64_t s0 = state1;
  state0 = s0;
  s1 ^= s1 << 23;
  s1 ^= s1 >> 17;
  s1 ^= s0;
  s1 ^= s0 >> 26;
  state1 = s1;
  return state0 + state1;
}
The new implementation landed in V8 4.9.41.0 within a few days of us becoming aware of the issue. It will become available with Chrome 49. Both Firefox and Safari switched to xorshift128+ as well.

Make no mistake however: even though xorshift128+ is a huge improvement over MWC1616, it still is not cryptographically secure. For use cases such as hashing, signature generation, and encryption/decryption, ordinary PRNGs are unsuitable. The Web Cryptography API introduces window.crypto.getRandomValues, a method that returns cryptographically secure random values, at a performance cost.

Please keep in mind, if you find areas of improvement in V8 and Chrome, even ones that—like this one—do not directly affect spec compliance, stability, or security, please file an issue on our bug tracker.

Posted by Yang Guo, Software Engineer and dice designer