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Assuming we have fixed pairing friendly elliptic curve groups $G_1$, $G_2$ and $G_T$ where for $a \in G_1$ and $b \in G_T$ it holds $e(a,b) \in G_T$.

Let's put some more context and we are working in bls12-381. what is a standard way to compute $e(a,b)*r$, where r is a random bit string. Putting in another way how do we map bitstrings to the target group $G_T$ ?

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Using bls12-381 the group $G_T$ us a multiplicative subgroup of $GF(p^{12})$ of order $\ell$ for a certain 381-bit prime $p$ and 256-bit prime $\ell$. We can convert a random bitstring to an element of $GF(p^{12})$ by hashing a to a uniform integer in $[0,p^{12}-1)$, expressing this integer in base $p$ and extracting the digits to use as coefficients in whatever basis we choose to represent $GF(p^{12})$. We can then map this element of $GF(p^{12})$ to our subgroup $G_T$ by raising to the power $(p^{12}-1)/\ell$.

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  • $\begingroup$ What if you want to recover back that r from its encoding to $[0,𝑝^{12}−1)$ ? $\endgroup$
    – curious
    Dec 30, 2023 at 16:55
  • $\begingroup$ @curious: That is not computationally easy using this method. Typically protocols only require the ability to hash into the group. $\endgroup$
    – Daniel S
    Dec 31, 2023 at 12:41

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