Which properties of a group are used in the steps of Diffie Hellman?

I’m trying to understand which properties of a group are used in DHKE at each step.

For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e.g. $$k_{pubA}$$ = $$A^{k_{prA}}$$ (mod p)?

When creating the shared key Alice and Bob appear to also use the associative property of a group $$k_{AB}$$ = $$B^{k_{prA}}$$ (mod p)?

So to perform both main steps of DHKE the multiplicative inverse property does not seem to be used at all?

I’m trying to understand which properties of a group are used in DHKE at each step.

Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $$A^3 = A\times A \times A = (A \times A) \times A = A \times (A \times A)$$ is well defined), but other than that, you really don't need anything. You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).

We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.

Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?" We do have assumptions such as the CDH assumption ("given $$g, g^a, g^b$$, it's hard to compute $$g^{ab}$$), however we don't know what semigroup properties ensure that...

• I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here? Jan 22, 2019 at 21:39
• The selection of group is critical for security. Using DH with the group $\mod p$ for many primes $p$ with group operation $\times$ is secure as far as we know, yet using DH with the group $\mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic. Jan 22, 2019 at 22:58
• @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible? Jan 22, 2019 at 23:31
• @JohnGalt For $\mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $\mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $\mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well. Jan 23, 2019 at 0:21
• @JohnGalt I was showing how the choice of group representation matters for security, not just the group order. So while choosing any group for DH will make the key exchange work, only some choices of representation and order are secure. Jan 23, 2019 at 1:08