A cryptographic -multilinear map is a kind of multilinear map, that is, a function such that for any integers and elements , , and which in addition is efficiently computable and satisfy 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, however, the problem of constructing such multilinear maps for seems much more difficult and the security of the proposed candidates is still unclear.