On the hardness of addition when the elements of a field is represented by the powers of generator and possible any existant scheme

We can represent elements of a finite field $$F$$ in various ways polynomial basis and normal basis. There is one other; generator-based representation and this is based on the fact that the multiplicative group of a finite field ($$F^*$$) is cyclic so there is a generator of this cyclic group, call $$g$$ and we write $$\langle g \rangle = F^*$$. Now every element of $$F^*$$ can be written as powers of the generator as $$x = g^i \in F$$ for some positive integer $$i$$. The identity is $$g^0$$. In this way, the multiplication is easy

$$\text{let } a,b\in F, \text{ and } a = g^x, b = g^y \text{ then } a\cdot b = g^{x+y}.$$ The bonus is the fact that the discrete log is easy since we tack the elements by the generator's power but the addition seems hard. $$a +b = g^x + g^y = g^z, \quad z =?$$ Questions;

2. Are there any cryptographic scheme that uses this hardness?

The term used for the function $$Z(\cdot)$$ defined by $$g^{x}+g^{y}=g^{x}(1+g^{y-x})=g^x g^{Z(y-x)},$$ in coding theory literature is Zech's logarithm. So for your formulation $$z=x+Z(y-x)=y+Z(x-y).$$ It is no more than the DL of $$(1+g^{x-y})$$ and is believed to be as hard as DL.

In coding theory it is normally considered for even characteristic groups $$G=GF(2^m)^\ast$$ in which case it is actually just the DL composed with a permutation $$\pi$$ of order 2 in the additive exponent group $$\mathbb{Z}_{2^m-1}.$$

I seem to recall Odlyzko mentioning it in a survey on DL in early 1990's. In any case, I don't know of it being directly used in cryptographic schemes, but others might.