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I would like to design a hardware-based accelerator for a pairing algorithm to make it faster. I know that I need to do arithmetic over a base and an extension field.

Could anyone would suggest which of the algorithms I have to use for arithmetic over base and extension fields for addition, subtraction and most importantly for multiplication and exponentiation? I am planning to implement “Tate pairing over BN curves”.

The second part of my question is: how to do field extension like from a base field $F_p$ to $F_p^{12}$?

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