I read a cryptography scheme that it include the following operation:
$$c= H(e(g_1,g_n)^t)$$ where H is a hash function. I need to know what the operation $e$ means.
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2$\begingroup$ Actually, as far as I know, all articles have some explanation. Boneh et al's 2DNF explains very well. Evaluating 2-DNF Formulas on Ciphertexts. Where did you read? $\endgroup$– kelalakaSep 7, 2020 at 13:33
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It's the pairing function. This bilinear map which takes as input an element from the set $\mathbb{G}\times\hat{\mathbb{G}}$ (in the common case, elliptic curves), and outputs a group element in $\mathbb{G}_T$, the target group.
In the symmetric case (Type 1), because $e$ is bilinear, you can deduce that $$e(g_1, g_n)^t= e(g, g)^{x_1x_nt}$$
with $x_1,x_n$ respectively the discrete logarithms of $g_1, g_n$ in base $g$.
For more details check: https://en.wikipedia.org/wiki/Pairing-based_cryptography
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1$\begingroup$ Note that It is not limited to elliptic curves. It is a general map. $\endgroup$– kelalakaSep 7, 2020 at 13:34
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$\begingroup$ Could you please explain what exactly the bilinear map do in this computation if we suppose that e is a bilinear map $G$ X $G$ -> $G_T$ and $G , G_T$ are two cyclic groups of prime order p and g is a random generator for G $\endgroup$– kiukigeSep 7, 2020 at 13:42
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