Cryptographic multilinear map

A cryptographic ${\displaystyle n}$-multilinear map is a kind of multilinear map, that is, a function ${\displaystyle e:G_{1}\times \cdots \times G_{n}\rightarrow G_{T}}$ such that for any integers ${\displaystyle a_{1},\ldots ,a_{n}}$ and elements ${\displaystyle g_{i}\in G_{i}}$, ${\displaystyle e(g_{1}^{a_{1}},\ldots ,g_{n}^{a_{n}})=e(g_{1},\ldots ,g_{n})^{\prod _{i=1}^{n}a_{i}}}$, and which in addition is efficiently computable and satisfies some security properties. It has several applications on cryptography, as key exchange protocols, identity-based encryption, and broadcast encryption. There exist constructions of cryptographic 2-multilinear maps, known as bilinear maps,^[1] however, the problem of constructing such multilinear^[1] maps for ${\displaystyle n>2}$ seems much more difficult^[2] and the security of the proposed candidates is still unclear.^[3]