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Consider the rational functions $f_P^k$ and $f_P'$. Since $\operatorname{div}$ is a homomorphism of semigroups (i.e. $\operatorname{div}(fg)=\operatorname{div}f+\operatorname{div}g$), we have $$\operatorname{div}(f_P^k)=k\cdot\operatorname{div}f_P=k\cdot(m[P]-m[\mathcal O])=km[P]-km[\mathcal O]=\operatorname{div}f_P'\text.$$ Now with theorem 5.36 of "An ...


If you can express your operation well using plain groups, go with normal elliptic curves. But the pairing adds additional mathematical structure, which enables new algorithms. Some examples: BLS signatures They're verifiably deterministic and small, 2x security level, instead of 3 to 4 times, with DSA/Schnorr/ElGamal signatures. These use the pairing ...

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