Let $G_1, G_2$ be additive groups and $G_T$ a multiplicative group, all of prime order $p$. Let $P $ in $G_1, Q $ in $G_2$ be generators of $G_1$ and $G_2$ respectively. A pairing is a map:
$e:( G_1, G_2 )-->G_T $

can anyone explain it?

  • $\begingroup$ What reading have you done? What effort have you made? Have you looked in Wikipedia? Have you looked in modern textbooks? Have you searched via Google? Have you read course notes on the topic? We expect you to do some investigation on your own before asking here: ask only questions that you actually care about -- and if you care about it, do a little research on your own. In this case, it is very easy to find basic information on pairings (e.g., on Wikipedia). $\endgroup$ – D.W. Aug 23 '13 at 18:16
  • $\begingroup$ Explain what? What is your question? Have you been reading some books or other materials about the subject? Which? $\endgroup$ – Gilles 'SO- stop being evil' Aug 24 '13 at 16:54

A pairing is a non degenerate and bilinear map from $G_1\times G_2$ to $G_T$. This means that if $g_1$, $g_2$ are generators of $G_1$ and $G_2$ then:

  • By non-degeneracy, $e(g_1,g_2)\neq 1$ and, in fact, $g_T=e(g_1,g_2)$ is a generator of $G_T$
  • By bilinearity, for any $h_1=a_1g_1$ and $h_2=a_2g_2$, we have $e(h_1,h_2)=g_T^{a_1a_2}$. Note that you don't need to know $a_1$ and $a_2$ to compute the pairing, thus $e$ can be used to transport a discrete logarithm problem from $G_1$ (or $G_2$) to $G_T$.

In addition, you might be interested to know that $G_1$ and $G_2$ are denoted additively and $G_T$ multiplicatively because the only known construction for cryptographic pairing as an elliptic curve on the left (for $G_1$ and $G_2$) and a finite field on the right (for $G_T$). The notations follow from standard usage in these groups.

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