# How to Mathematically Prove the Bilinear Pairing Properties [closed]

I am currently working on Bilinear Pairing.To start my work i need to find the mathematically prove of three properties of bilinear pairing.

Let $G_{1}$ and $G_{T}$be a cyclic multiplicative group with the same prime order q, that is, $|G_{1}|$ =$|G_{2}|$ = q. Let g be a generator of G1. An efficient bilinear map $e$: $G_{1}$ × $G_{1}$ → $G_{T}$, with the following properties:

1. Bilinear: for all g ∈ $G_{1}$ and a; b ∈ $Z_{q}^{^{*}}$ , $e(g^{a},g^{b})=e(g,g)^{^{ab}}$.
2. Non-degenerate: there exists g ∈ G1 such that e(g, g) ̸= 1.

I need to find some mathematical prove for Bilinear map e. I choose a cyclic multiplicative group $G_{5}$ having generator g=2 and (a=3,b=4) then how to prove the first property ie $e(2^{3},2^{4})= e(2,2)^{^{3.4}}$

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## closed as unclear what you're asking by DrLecter, fgrieu, e-sushi, figlesquidge, rathJan 15 '14 at 23:30

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

What map(s) do you need to prove satisfy(ies) those properties? $\;$ – Ricky Demer Jan 15 '14 at 11:20
What you give is a definition. There is nothing to prove for a definition :/ If you have a concrete map $e$ then you could ask proving that it satisfies the definition. – DrLecter Jan 15 '14 at 11:32
@DrLecter i did not understand how the bilinear equation is proved. Say we take G1=(0,1,2,3,4) and g=(2) ,a=3 and b=4 then how we will prove e(2^3,2^4)=e(2,2)^3.4 – Raginisingh Jan 15 '14 at 11:55
@Raginisingh Since you ask the question here, I assume that you are looking for pairings for cryptographic use? This also requires that the discrete logarithm problem in $G1$ and $GT$ is hard. Such pairings are only known to exist on certain elliptic curve groups for $G1$ ($G2$) and related multiplicative groups of finite fields (for $GT$). In your example you use the group $G1=(Z_5,+)$ and what would be $e(2,2)^{3\cdot 4}$, since you have not specified $GT$. For a more general (non crypto) treatment of bilinear maps you may look here. – DrLecter Jan 15 '14 at 12:21
@DrLecter sir actually i am unable to understand how $G_{T}$ is calculated .can u help me out taking some real time example with mathematical equations. – Raginisingh Jan 15 '14 at 12:31