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The BGN (Boneh Goh Nissam) cryptosystem uses a Type-III bilinear pairing defined over composite order groups to have 1 homomorphic multiplication along with unlimited homomorphic additions. You may want to have a look at their paper: Evaluating 2-DNF Formulas on Ciphertexts, or any lecture on this cryptosystem. I hope this will help you.


Commonly, one uses multiplicative groups. When recall definition of a pairing, it must be bilinear, non degenerate, and easily computable. Note that the definition of e(x,y)=x.y can be interpreted as a external product. (x.y = y+...+y: n-times) and not as the internal group law. I agree with Maeher's answer. The first exemple is a bilinear pairing, but not ...


One specific example of a multiplicative group of a composite order is RSA. Group order could be not known to a proving party (hidden from). With this setup, one can prove relations over integers, not residues modulo group order. Proof based on Lagrange 4-squares theorem is a well-known example. No efficient implementation is known for a proof of "more" ...


What you present is a generalized version of the so called fixed-argument pairing inversion (FAPI) problem. The FAPI problem is given an element $z\in G_T$ and an element $h\in G$ to compute $f\in G$ such that $e(h,f)=z$. Note, that FAPI is implied by the computational Diffie Hellman problem: Given $(g,g^a,g^b)\in G^3$, call the FAPI oracle with $z\gets ...

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