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.


1 Answer 1


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

  • 1
    $\begingroup$ Note that It is not limited to elliptic curves. It is a general map. $\endgroup$
    – kelalaka
    Sep 7, 2020 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, 2020 at 13:42
  • $\begingroup$ @kiukige is it better? $\endgroup$
    – Ievgeni
    Sep 7, 2020 at 13:48

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