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Shweta Aggrawal avatar Crypto

A doubt in pairing based cryptography

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Shweta Aggrawal

  1. I have seen authors taking $G_1=G_2=G_T=G$ to be the same group of prime order $q$.

  2. What I know is that for pairing of type $$e:G_1\times G_2\rightarrow G_T,$$ size of the element in the target group is $kn$ where $n$ is the size of an element in $G_1$ and $k$ is the embedding degree.

Source: A New Family of Pairing-Friendly elliptic curves by Michael Scott and Aurore Guillevic. and this question

I am confused as it looks like these two points are contradicting each other.

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meshcollider avatar
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meshcollider
Where is $G_1 = G_T$ stated? Generally $G_1$ and $G_2$ are subgroups of elliptic curves (and may be equal, i.e. a Type 1 pairing), and $G_T$ is a subgroup of the multiplicative group of a finite field, so $G_1 \neq G_T$.
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Daniel S avatar
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Daniel S
@meshcollider I know of [some proposals](https://ijpam.uniud.it/online_issue/201738/38-Kumar-Pal-Arvind.pdf) to use pairings on singular curves. In such cases, there is a natural isomorphism between the curve group and $\mathbb F_p^\times$ which can then be taken as $G_1$, $G_2$ and $G_T$. I don't know if this is what OP refers to.
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