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.


It's the pairing function. This bilinear map which takes as Input the set $\mathbb{G}\times\hat{\mathbb{G}}$ (in the common case, eliptic curves), and output a group element in $\mathbb{G}_T$ the group target.

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 be the discrete logarithms of $g_1, g_n$ in base $g$.

For more details check: https://en.wikipedia.org/wiki/Pairing-based_cryptography

| improve this answer | |
  • $\begingroup$ Note that It is not limited to elliptic curves. It is a general map. $\endgroup$ – kelalaka Sep 7 at 13:34
  • $\begingroup$ 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 $\endgroup$ – kiukige Sep 7 at 13:42
  • $\begingroup$ @kiukige is it better? $\endgroup$ – Ievgeni Sep 7 at 13:48

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