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If you can use something more lightweight than pairings then you should skip pairings. There is a technical property of pairings than is not the case with other crypto primitives. And this is the evaluation of multiplications on the exponent that cannot be achieved in other cases. If you need some sort of summation then you encode in the exponent and then ...


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Pairings in cryptography is a very important tool, the introduction of which has developed a new field, that is pairing-based cryptography. After the independent pioneering work by Joux and by Sakai et al.("Cryptosystems based on pairing"), many pairing-based crypto-systems emerged. In cryptography, pairings are often treated as "black-box", and then we ...


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Surely bilinearity helps and and you have to be aware of the fact that $e(g^a,g^b)\cdot e(g^a,g^{-b})=e(g^a,g^b)\cdot e(g^a,g^b)^{-1}=1$. You see that from your last equation $$=e(g^{a\lambda_i}\cdot H(x)^{-v_ir_i},g^{u})\cdot e(g^{r_i}, g^{u't}\cdot H(x)^{v_i(u+\gamma)})\cdot e(g^{r_i}, g^{-tu'})\cdot e(g, H(x)^{v_i\gamma r_i})^{-1}$$ you get ...



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