Why use $(r,s)$ instead of $(r,s^{-1})$ as DSA signature? A DSA signature consists of two scalars $(r,s)$.

When signing $s$ is generated as:


$s=k^{-1}(H(m)+xr) \mod q$
The signature is $(r,s)$


When verifying $s$ is only used to compute $w = s^{-1}$. So why does DSA store $s$ in the signature instead of $w$?

Using $w$ has no performance effect on signing since computing either $s$ or $w$ requires one modular inversion. $w = s^{-1} \mod q = k (H(m)+xr)^{-1} \mod q$.
It would speed up verification since the modular inversion to compute $w$ isn't necessary anymore.

The only advantage of $s$ I see is that the modular inversion can be computed before the hash is known. Did the DSA designers consider that pre-computation more important than the verification slowdown, or is there some other advantage of using $s$ that I don't see?
Well, it's been an entire day, and no one has given an authoritative answer; I'll throw in my guess as to why the people designing DSA made the choices they did.

With DSA, there are three operations that are relevent to this discussion:


A: do precomputation of a signature (without seeing the message being signed)
B: given a precomputed signature and a message, generate the actual signature
C: verify a signature.


Standard DSA and your variant DSA perform exactly the same operations, except that standard DSA computes a modular inverse during steps A and C, while your variant computes it at step B.

It is true that standard DSA does two modular inverse, while your variant does only one; however, that may not be the only factor.

If we look at step B, we see that the operations involved at quite cheap (a single modular multiplication and addition); if we were to include a modular multiplication operation there, we would increase the cost of that operation by a large percentage.  In contrast, steps A and C are already expensive (involving a modular exponentiation or point multiplication); including a modular inverse only slows down those operations by a small percentage.

That is, DSA has the property that, with precomputation, generating a signature is extremely fast.  Your variant significantly reduces this advantage; the DSA designers designed to keep this advantage, even at the cost of increasing the total computation required.

Now, you may disagree with the above logic; in fact, people have proposed a DSA-variant exactly like what you suggest.