# Hashing to the target group of bilinear pairing

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 $$G_T$$ ?

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 $$(p^{12}-1)/\ell$$.
• What if you want to recover back that r from its encoding to $[0,𝑝^{12}−1)$ ? Dec 30, 2023 at 16:55