Boneh–Franklin scheme

The Boneh–Franklin scheme is an identity-based encryption system proposed by Dan Boneh and Matthew K. Franklin in 2001. This article refers to the protocol version called BasicIdent. It is an application of pairings (Weil pairing) over elliptic curves and finite fields.

As the scheme bases upon pairings, all computations are performed in two groups, ${\displaystyle \textstyle G_{1}}$ and ${\displaystyle \textstyle G_{2}}$:

For ${\displaystyle \textstyle G_{1}}$, let ${\displaystyle \textstyle p}$ be prime, ${\displaystyle \textstyle p\equiv 2\mod 3}$ and consider the elliptic curve ${\displaystyle \textstyle E:y^{2}=x^{3}+1}$ over ${\displaystyle \textstyle \mathbb {Z} /p\mathbb {Z} }$. Note that this curve is not singular as ${\displaystyle \textstyle 4a^{3}+27b^{2}=27=3^{3}}$ only equals ${\displaystyle \textstyle 0}$ for the case ${\displaystyle \textstyle p=3}$ which is excluded by the additional constraint.

...
Wikipedia