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fix typo
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ajgara committed Jan 22, 2024
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Expand Up @@ -35,7 +35,7 @@ $$e(A, B)=t$$

Do not get confused, we have two different finite fields. $\mathbb{F}_p$ is the field where the coordinates of the points live. $\mathbb{F}_q$ is the field where things end up after the pairing, this field has the same order as the group $\mathbb{G}$ of the curve. The pairing has a property called bilinearity:

$$e(xA, yB)=t^{ab}$$
$$e(xA, yB)=t^{xy}$$

This property is crazy. It states that adding $A$ with itself one more time, is equivalent to multiplying $t$ with itself in $\mathbb{F}_q$ one more time. It mixes two different worlds. The main algorithm for computing pairings is divided in two steps the **Miller loop** and the **final exponentiation**. Most optimizations efforts use properties of specific curves in order to reduce the size of the loop or the exponentiation.

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