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There is a generalisation of the Diffie-Hellman problem to multi-linear groups known as the multi-linear Diffie-Hellman problem (MDHP, see this paper for example). Specifically, for a group $G_1$ endowed with an $n$-multi-linear map $e$ to a group $G_2$ the MDHP is: given $g, g^{a_1}, g^{a_2},\ldots, g^{a_{n+1}}\in G_1$ compute $e(g,g,\ldots,g)^{a_1a_2\cdots ... 3 The problem with "why" "Why" is generally an unfortunate question. It is often very hard or impossible to answer. The reasoning goes like this: if you ask "why" a (reasonably complex) thing is like it is, any meaningful answer usually breaks the issue down into subcomponents. Then, you can and need to ask "why" for ... 2 Maybe I can give another answer from the perspective of Multiparty Computation (MPC), which studies the problem of enabling multiple parties to securely compute a function on sensitive data while revealing only the outputs. A very important tool for solving the problem stated above is secret-sharing, which enables distribution of a secret$s$into$n$shares ... 3 I think you got it backwards: Algebraic structures like rings and groups and fields are the underlying concept of all commonly used types of numbers like the integers, rationals, reals and complex numbers. In algebra it is quite common to do theorems and proofs in the structure with the minimum requirements - so they are valid in a wide range of structures, ... 7 Groups have properties which are useful for many cryptographic operations When you multiply 2 numbers in a cryptographic operation you want the result of the multiplication also to be in the same set. For e.g. if you are multiplying something which fits in a byte (or n bytes) by something similar, you also want the result also to fit in a byte (or n bytes). ... 15 Kindly, let me know what was the actual problem which leads us to use groups in cyptogrpahy? Well, we use groups and other similar mathematical constructs because: We found there are problems that appeared to be difficult to solve with those groups We found ways to translate the difficulty of solving those problems into the cryptographical strength of ... 2 Lynn's advice was bad at the time, and is very outdated now. It was never secure to use 170-bit curves; for a start that would result in Pollard rho security of only 85 bits (assuming a prime-order curve). Worse, for an embedding degree of 6 you have a target extension field$G_T\$ of only 1020 bits which is (as of 2021) probably breakable in practice. The ...