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Why the definition of bilinearity property is different in cryptography compared to mathematics?

Background:

In Wikipedia (bilinear map definition), a condition listed as the following:

For any $lambda in F, {displaystyle B(lambda v,w)=B(v,lambda w)=lambda B(v,w)}$


In a math.stackexchange post, i saw this comment:

which defines bilinearity in the same way as the Wikipedia link. So I assume, the above definition of bilinearity is correct.


Now, the problem:

in the pairing-based cryptography setting:

the bilinearity is defined as such:
${displaystyle forall a,bin F_{q}^{*}, forall Pin G_{1},Qin G_{2}: eleft(aP,bQright)=eleft(P,Qright)^{ab}}$

The question:

in math: bilinearity is: $eleft(aP,bQright)=eleft(P,Qright)ab$ (a and b have become coefficients)

in crypto: bilinearity is: $eleft(aP,bQright)=eleft(P,Qright)^{ab}$ (a and b have become exponents)

Why is this difference? Aren’t the definitions conflicting with each other? In wicktionary I saw the following explanation:

Linear (preserving linear combinations) in each variable.

How come $eleft(aP,bQright)=eleft(P,Qright)^{ab}$ is preserving the linear combinations? Isn’t exponent and coefficients are very different and not linear with each other?

Obviously, math is correct, and crypto is also correct. So, I really appreciate if someone can point out to me where am I missing the details or misinterpreting them?

P.S: someone in crypto.stackexchange defined the bilinearity as:

Bilinear: for all g āˆˆ šŗ1 and a; b āˆˆ $Z_{q}^{*}, e(g^a, g^b) = e(g,g)^{a,b}$

I haven’t seen this notation at all. Is the definition of bilinearity loose? So there can be multiple definitions? If so, why and how?

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We want to build a wallet application, when user registers we want to open a specific wallet to user but there is a problem when user deposits crypto we want to transfer balance (erc20 transfer) to cold wallet then show it on database, when user want to withdraw we have to transfer erc20 also, that::Listen

We want to build a wallet application, when user registers we want to open a specific wallet to user but there is a problem when user deposits crypto we want to transfer balance (erc20 transfer) to cold wallet then show it on database, when user want to withdraw we have to transfer erc20 also, that means for each user we need to do 2 transfers at least, is there more efficent way as solution of this problem to lower transfer counts?

Why the definition of bilinearity property is different in cryptography compared to mathematics?

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