# Can we use HMQV in an asynchronous setting?

Following the HMQV paper, to perform a key-exchange, Alice ($$\hat{A}$$) and Bob ($$\hat{B}$$) perform the following:

• $$\hat{A}$$ generates the long-term key pair ($$sk_A= a$$, $$pk_A =g^a$$) and the ephemeral key pair ($$x, X=g^x$$). Then $$\hat{A}$$ sends [$$\hat{A},\hat{B}, X, \mathrm{sig}_\hat{A}(X,\hat{B})$$] to $$\hat{B}$$
• $$\hat{B}$$ generates the long-term key pair ($$sk_B= b$$, $$pk_B =g^b$$) and the ephemeral key pair ($$y, X=g^y$$). Then $$\hat{B}$$ sends [$$\hat{B},\hat{A}, Y, \mathrm{sig}_\hat{B}(Y,X,\hat{A})$$] to $$\hat{B}$$

Then both compute the session key $$\sigma=g^{(x+da)(y+eb)}$$ as $$\sigma_\hat{A}=(YB^e)^{(x+da)}$$ and $$\sigma_\hat{B}=(XA^d)^{(y+eb)}$$ respectively, with $$d=\overline{H}(X,\hat{B})$$ and $$e=\overline{H}(Y,\hat{A})$$

Following are my questions:

1. if $$X$$ and $$Y$$ are pre-computed (before the key-exchange) and available on a TTP, can HMQV be used as an asynchronous key-exchange?
2. In terms of performance, is HMQV better than X3DH? X3DH computes the session key as $$\sigma=\mathrm{KDF}(g^{ay}||g^{bx}||g^{xy})$$
3. is $$\sigma = (x+da)(y+eb)G$$ the right transformation of HMQV final result in Elliptic curve cryptography?
4. in the hash function $$\overline{H}$$, can we discard $$\hat{A}$$ and $$\hat{B}$$ without affecting the robustness of HMQV?