I am not at all knowledgeable in elliptic curve cryptography. So, here lies a couple of questions that I failed to find answers for to my satisfaction.

  1. Is there any known Type-III bilinear pairing defined over composite order groups?
  2. Do DLIN/SXDH assumptions hold in composite order groups?
  • $\begingroup$ I suggest you stick to one question per question. This site format doesn't work so well when you have more than one question in your question. $\endgroup$ – D.W. Jan 30 '15 at 6:06

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.

  • $\begingroup$ BGN uses a symmetric pairing over composite order bilinear groups. $\endgroup$ – DrLecter Jan 28 '15 at 11:28
  • $\begingroup$ It works as well with asymmetric pairings, and this lets you choose your ciphertext which can be allowed on the right hand and on the left hand of the multiplication. That being said, I still don't know if there has been implementation on this in the asymetric case, but it's a start... $\endgroup$ – Florian Bourse Jan 28 '15 at 12:38
  • $\begingroup$ Yeah, and many other protocols would also work in this setting. But I do not think that this helps the OP as it is not discussed in this paper. $\endgroup$ – DrLecter Jan 28 '15 at 12:41

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