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$e:G_1 \times G_1 \rightarrow G_2$, where $g$ is a generator of $G_1$. $\text{H}: \lbrace 0,1\rbrace ^* \rightarrow G_1$.

Is $e(\text{H}(D)g^a,g) == e(\text{H}(D)g,g)^a$ ? where $D$ is a string

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Does the following equation hold in bilinear pairings?

No, not in general.

$g$ is posited to be a generator, and so (for any fixed $D$), we have $H(D) = g^b$, for some integer $b$ (which is hard to compute, but we're not going to have to compute it).

So, we have $e(H(D)g^a, g) = e(g^{a+b}, g)$.

If we move the $a$ exponent out, we then get $e(g^{1 + b/a}, g) = e( g \cdot g^{b/a}, g)^a$

Doing the same thing one the other side, we would get $e(H(D)g, g)^a = e(g \cdot g^b, g)^a$

It should be clear that these two terms are not (in general) identical.

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  • $\begingroup$ $e(\text{H}(D)g^a,g) = e(\text{H}(D),g).e(g^a,g) = e(\text{H}(D),g).e(g,g)^a = e(\text{H}(D)g,g)^a$. Is this correct? $\endgroup$ Commented Feb 3, 2021 at 7:15
  • $\begingroup$ @RabindraMoirangthem: how'd you get from $e(H(D), g) \cdot e(g,g)^a$ to $e(H(D)g, g)^a$? How did $H(D)$ get managed to be raised to the power of $a$? $\endgroup$
    – poncho
    Commented Feb 3, 2021 at 12:50
  • $\begingroup$ crypto.stackexchange.com/questions/19653/… what about this one? $\endgroup$ Commented Feb 4, 2021 at 4:37

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