# How are Elliptic Curve Cryptography and Pairing Based Cryptography related?

I have been doing a project that uses the PBC library developed by Ben Lynn. But I am still not clear on how PBC is related to ECC. I know that this is a site for complex crypto QA, but I did not know where else my question would be answered. I have tried to learn from online sites, but I don't understand most of them.

If my questions merits a good answer, please help me. Otherwise please give me some pointers in the right direction.

## 1 Answer

Most pairing-based cryptography (PBC) schemes are based in elliptic curve cryptography (ECC). The main function in PBC is the pairing, which is a function $e$ with two parameters, e.g. $r = e(P, Q)$. The relationship with ECC is that $P$ and $Q$ are points in elliptic curves over finite fields. The value $r$ is an element of a certain finite field (related to the fields used in the elliptic curves).

Some PBC protocols also need to compute point multiplication, e.g. $kP$, which is also the central algorithm of ECC and therefore can use the same existing techniques and optimizations.

• Thanks for the answer. Well, does this mean that PBC can also be implemented over other schemes? – Ajoy Mar 6 '13 at 14:51
• Yes, but the only alternative I know is PBC using elliptic nets, which are related but different to ECC. – Conrado Mar 6 '13 at 18:51