Commutative Groups but with an extra operation.
Unit is a neutral element of the Commutative Ring. Multiplicative unit is 1 in integer ring set. addition unit is 0 in integer ring set.
Set of integers with unit 1 is a commutative ring.
This shows that we must think of rings and groups as algebraic structures. Think solely of the structure and rules given, like ditributive property and the mappings. Here we are given a table of all computations, we traditionally are used to computing (1+1=2). But here we don’t compute, we merely abstract it away and look at the simple structure of a ring.
Ring structure $(Z_n, +,*)$ corresponds to group G in the following way:
$g^{x+y} = g^x *g^y$
This mapping or Correspondence helps polynomials with coefficients in $Z_n$ to be evaluated in the exponent of the group generator. We know this will be inside G since we defined its generator as g.
This is simply amazing since now we have obfuscated the actual values of the coefficients of the polynomials, hiding them inside this group element that is assumed to be difficult since its using the exponential map. AKA a DL Group
Beauty of cyclic groups: prover can hide secret parameters s inside the group. Then, when verifier tries to verify, they will use elements of that group to perform operations with that obfuscated value, knowing it belongs to group G. This is homomorphic hiding/encryption: you can perform operations on encrypted values that would work. The resultant value would also belong in the group.
binary string → value in ring $Z_n$
Then, using a hash function H, you can hash each bit and multiply with the element 2^x