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I am in the middle of FPGA based Hardware architecture design for the computation of Pairing (particularly R-ate Pairing) over BN curves. Where, the point addition, and point doubling should be computed over $\mathbb{F}_{p^2}$, and line function, miller variable have to performed over $\mathbb{F}_{p^{12}}$. The chosen Towering strategies is $\mathbb{F}_{p}$->$\mathbb{F}_{p^2}$->$\mathbb{F}_{p^4}$->$\mathbb{F}_{p^{12}}$. I have design hardware architecture for finite field primitives like addition, subtraction and multiplication over $\mathbb{F}_{p}$. The same unit can also perform extension field operation as a series of $\mathbb{F}_{p}$ operations.

Now I need to devise an efficient scheduling policy to execute point addition, point doubling, line function and final exponentiation operations through the suggested unit.

I need to find a suitable combined formulae for point addition and line function, point doubling and line function in projective coordinates, a reference would be a great help.

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Whoa... that sounds really interesting. I only know epsilon about pairings, but do you have like a project page or a description of what you're trying to accomplish? – pg1989 Aug 8 '14 at 19:39
@user4602 You could take a look at the bnparings library. Its a Java implementation of optimal ate pairing in BN curves. – DrLecter Aug 9 '14 at 13:59
@user4602, I've edited the lecture request out of your question, because it would be off topic. I hope someone more knowledgeable will be able to help with the formulas. – otus Aug 11 '14 at 12:31

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