Cryptographic multilinear map

A cryptographic formula_1-multilinear map is a kind of multilinear map, that is, a function formula_2 such that for any integers formula_3 and elements formula_4, formula_5, 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 formula_6 seems much more difficult and the security of the proposed candidates is still unclear.
Definition.
For "n" = 2.
In this case, multilinear maps are mostly known as bilinear maps or parings, and they are usually defined as follows: Let formula_7 be two additive cyclic groups of prime order formula_8, and formula_9 another cyclic group of order formula_8 written multiplicatively. A pairing is a map: formula_11, which satisfies the following properties:
In addition, for security purposes, the discrete logarithm problem is required to be hard in both formula_15 and formula_16.
General case (for any "n").
We say that a map formula_2 is a formula_1-multilinear map if it satisfies the following properties:
In addition, for security purposes, the discrete logarithm problem is required to be hard in formula_31.
Candidates.
All the candidates multilinear maps are actually slightly generalizations of multilinear maps known as graded-encoding systems, since they allow the map formula_19 to be applied partially: instead of being applied in all the formula_1 values at once, which would produce a value in the target set formula_9, it is possible to apply formula_19 to some values, which generates values in intermediate target sets. For example, for formula_40, it is possible to do formula_41 then formula_42.
The three main candidates are GGH13, which is based on ideals of polynomial rings; CLT13, which is based approximate GCD problem and works over integers, hence, it is supposed to be easier to understand than GGH13 multilinear map; and GGH15, which is based on graphs.