# pairing-based cryptography

Recently, a lot of attention has been paid on how to construct cryptographic primitives using pairings, like identity-based encrytion, one-round tripartite Diffie-Hellman, etc. (which will be disscussed in later posts).

However what is pairing?

Pairing is a map, $$e:G_1\times G_2\rightarrow G_T$$, where $$G_1,G_2$$ are both additive cyclic groups of prime order $$q$$, $$G_T$$ is another multiplicative cyclic group of order $$q$$, satisfies the following properties:

Bilinearity:

$\forall a,b\in F_q^\ast, \forall P\in G_1,Q\in G_2:e(aP, bQ)=e(P,Q)^{ab}$

Non-degeneracy:

$e\neq 1$

Computability:

$There\ exists\ efficient\ algorithms\ to\ compute\ e.$

# 1 intractable problems

## 1.1 BDHP

Bilinear Diffie-Hellman Problem (BDHP) is believed to be intractable, which means it is hard to compute $$e(P,P)^{abc}$$ given $$<P,aP,bP,cP>$$, $$a,b,c\in \mathbb{Z}_q^\ast$$, $$P$$ is a generator of $$\mathbb{G}_1$$.

In cryptography.

# 3 Tate pairing

Tate pairing is another common used instantiation.

# References

 D. Boneh, M. Franklin, Identity-based encryption from the weil pairing, in: Annual International Cryptology Conference, Springer, 2001: pp. 213–229.

 A. Joux, A one round protocol for tripartite diffie-hellman, in: International Algorithmic Number Theory Symposium, Springer, 2000: pp. 385–393.

Tags: cryptography.