Taek • yesterday at 8:15 PM

I ran it 500,000 times, discarding the 10% most entropic results ... in the hopes of arriving at a relatively conservative estimate for the amount of entropy you actually get from each iteration. Here's the prompt I used to generate the code: https://chatgpt.com/share/6ab2df4a-7f94-83ea-aecf-1bb57c4838...

And here are the results of running that code:

  === No hashing ===
  Clock resolution: 0.000000001 seconds
  Clock reads:                       500,000
  Second-difference outcomes:        499,998
  Retained outcomes:                 449,998 (90.000%)
  Average Shannon information:       1.755579 bits/retained outcome
  Marginal min-entropy estimate:      1.339460 bits/retained outcome
  Lag-1 conditional min-entropy:      0.960079 bits/retained adjacent outcome
  Conservative descriptive proxy:    0.960079 bits/retained outcome
  Proxy scaled per clock iteration:  0.864067 bits/iteration
  These are empirical timing statistics, not a proven entropy rate.

  === One SHA-256 between clock reads ===
  Clock resolution: 0.000000001 seconds
  Clock reads:                       500,000
  Second-difference outcomes:        499,998
  Retained outcomes:                 449,998 (90.000%)
  Average Shannon information:       4.205076 bits/retained outcome
  Marginal min-entropy estimate:      3.610848 bits/retained outcome
  Lag-1 conditional min-entropy:      3.351217 bits/retained adjacent outcome
  Conservative descriptive proxy:    3.351217 bits/retained outcome
  Proxy scaled per clock iteration:  3.016082 bits/iteration
  These are empirical timing statistics, not a proven entropy rate.

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As GPT helpfully points out, this isn't a proven guarantee, but a reasonable estimate is somewhere between 3 and 4 bits of entropy per hash. That means 50 is actually enough, though if you want to be conservative I don't think there's any harm in doing 500 or even 5,000 instead of 50. And, if you are going to be using this in a hostile environment, it doesn't hurt to also add a fortuna-like accumulator that resets your entropy every once in a while.

I said this in another reply as well, but the reason that you get 3-4 bits of entropy per hash is because of the fundamental nature of CPUs. In addition to having considerable professional experience with cryptography, I also have considerable professional experience with hardware; hardware is fickle as hell, especially when your transistors are tens of nanometers large. Every time you flip a bit, you expend some energy, which heats up the chip, and the heat changes the timing of the next clock cycle. Chips are composed of literally billions of transistors, and each one is going to have a different temperature, because clock cycles last less than a nanosecond (well, embedded hardware is slower but the same idea still applies reliably) and that's not enough time for temperature deltas to dissipate across the chip.

Hashing is particularly chaotic because it lights up a different set of transistors on each clock cycle, which means the hotspots on the chip are being jerked around. Some transistors are going to light up 5-10 times in a row, and others are going to be idle 5-10 times in a row, and then randomly that changes. And all of this changes the number of picoseconds that it takes for a clock cycle to complete, which means that each clock cycle is genuinely going to take a different amount of time to complete, and stuff like temperature throttling is completely not at play whatsoever, because we're not talking about chip-wide temperatures, we're literally talking about temperature deltas between transistor a and transistor b.

That makes it a really wonderful source of entropy for cryptographic applications, because the CPU clock is so critical that it's almost never buggy (especially relative to other components that provide entropy), it's also almost impossible to manipulate reliably by an attacker (unless the attacker has an exploit that allows them to set the value of the clock directly - which is possible, but it's a very narrow surface area relative to other entropy sources), and you can completely take advantage of this entropy entirely in userspace, which once again heavily minimizes attack surface area and exposure to bugs.