# Why are these specific values used to initialise the hash buffer in SHA-512?

I'm reading the book Network Security Essentials written by William Stallings.

To create a message digest with SHA-512, we have to go through some steps:

2. append length
3. initialize hash buffer
4. ... so on.

In the book is written for step 3:

A 512 bit buffer is used to hold intermediate & final results of the hash function. The buffer can be represented as eight 64-bit registers (a,b,c,d,e,f,g,h). These registers are initialized to the following 64-bit integers (here in hexadecimal values):

 a = 6A09E667F3BCC908
b = BB67AE8584CAA73B
...


up to register h.

Why are these specific hexadecimal values used, is there any particular reason behind this? I know why they are necessary, but why these specific ones?

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The initial hash values for SHA-512 are the 64-bit binary expansion of the fractional part of the square root of the 9th through 16th primes (23, 29, 31, ..., 53). That is:

$$I_0 = \left \lfloor \mathrm{frac} \left (\sqrt{23} \right ) · 2^{64} \right \rfloor$$ $$I_1 = \left \lfloor \mathrm{frac} \left (\sqrt{29} \right ) · 2^{64} \right \rfloor$$ $$\cdots$$

And obviously the reason they use primes is that their square root is always irrational, which makes for a nice, random-looking number. Taking the square root of four would be less interesting, as it would give 0. It also means the constants are relatively independent from each other, as none of them can be easily expressed in terms of any other, which might be a weakness - but that probably doesn't matter as much.

They are chosen this way because while they generally don't need to be anything in particular, as long as the chosen numbers don't have some special properties (i.e. setting them to zero might be a bad choice), the designers of SHA-512 wanted people to know that the numbers weren't chosen so as to stealthily introduce a backdoor in the algorithm (for instance, if a certain combination of initial values were chosen, it may have been possible for the NSA to break the hash function more easily) which would be very difficult to confirm or refute as it would require months, perhaps years of cryptanalysis.

Using publicly available numbers that anyone can recreate, here the fractional parts of some prime square roots, solves this problem - there is no way anyone can change these numbers (nobody can change the square root of 23, for instance), and it is considerably harder to design an algorithm to have a backdoor under set constants, than to design the algorithm first and work out those constants later on.

The first few digits of $\pi$ or $e$ are popular choices too, but only if you use the first digits. If you were using, say, the digits starting at position 94858991, I would get very suspicious.

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In addition: the design of SHA-2 hashes (and SHA-1, MD5..) pads the end of the message using two redundant methods (padding with a 1 bit then some number of 0 bits; and padding with the message length). Using only the first of these two methods would leave these hashes open to an attack by the party that decided the constant (possibility to insert a secret prefix without changing the hash). This shows that it is sound to use "nothing up my sleeve numbers" here. – fgrieu Nov 12 '12 at 11:55
SHA-512 uses the first 8 primes. 6A09E667F3BCC908 is $\lfloor\sqrt(2)-1*2^{64}\rfloor$. – otus Jun 9 '14 at 17:07