Lesson 08 · The eight initial hash values - Build SHA-256 from Scratch

The eight initial hash values

SHA-256 starts every hash from eight fixed 32-bit words, derived from the fractional parts of the square roots of the first eight primes. Today you pin those eight constants and see where the first one comes from.

The goal

Provide the eight initial hash values H0 through H7 as a fixed array of words.

Start here - the target

TO DO

Scenario: The initial hash state matches the standard

Giventhe eight SHA-256 initial hash values, the fractional parts of the square roots of the first eight primes (2, 3, 5, 7, 11, 13, 17, 19), scaled by 2^32

WhenInitialHash() returns them in order as H[0] through H[7]

ThenH[0] is 0x6a09e667, H[4] is 0x510e527f, and H[7] is 0x5be0cd19

AndH[0] equals floor((sqrt(2) - 1) * 2^32), which is 0x6a09e667

Background

SHA-256 begins from a fixed starting state: eight 32-bit words, H[0] through H[7], that every hash of every message starts from. They are not arbitrary magic - each is the first 32 bits of the fractional part of the square root of one of the first eight primes. Using an irrational constant like this is a “nothing up my sleeve” number: it demonstrably was not hand-picked to hide a weakness, because anyone can rederive it.

You can check the first one yourself. sqrt(2) is 1.41421356...; drop the integer part to get the fraction 0.41421356..., multiply by 2^32, and take the floor - the result is 0x6a09e667, exactly H[0]. The other seven follow the same recipe with primes 3, 5, 7, 11, 13, 17, and 19. You do not need to recompute them at runtime (the standard just lists the words), but pinning H[0], the middle H[4] = 0x510e527f, and the last H[7] = 0x5be0cd19 confirms you have transcribed the table correctly - these are the words the compression function starts from.

Make it work

// the fractional bits of the square roots of the first 8 primes
func InitialHash() [8]uint32 {
  return [8]uint32{
    0x6a09e667, 0xbb67ae85, 0x3c6ef372, 0xa54ff53a,
    0x510e527f, 0x9b05688c, 0x1f83d9ab, 0x5be0cd19,
  }
}

CheckpointDONE

You have the eight initial hash values. Commit and stop here.