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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 ...


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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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