The Groth16 SNARK prover is not such a complicated algorithm, but it does involve several steps, each of which builds on the next. The prover must be implemented for two settings of parameters: the parameters associated with the curve MNT4-753 and those associated with the curve MNT6-753.

Once a collection of parameters (r, q, e, G_1, G_2) is fixed, the sub-problems making up the SNARK prover are

1. 1. Addition and multiplication mod q.
   2. Addition and multiplication mod r.
   It is possible for the "modulus" (here either q or r) to be a parameter in one's code, so typically 1a and 1b will share an implementation.
2. Arithmetic in the extension field \mathbb{F}_{q^e}. This is pretty straightforward to implement once step 1a is complete, though the implementation will differ between MNT4 and MNT6.
3. 1. The group operation in G_1. This is easy to implement after completing 1a.
   2. The group operation in G_2. This is easy to implement after completing 2 and typically can share an implementation with the G_1 group operation if working in a language with generics or templates. The only difference between the two being which field operations (i.e., multiplication and addition) are used: those of \mathbb{F}_q for G_1 and those of \mathbb{F}_{q^e} for G_2.
4. 1. Multi-exponentiation in G_1. There are many techniques for computing this efficiently, but they all essentially rely on the group operation from 3a.
   2. Multi-exponentiation in G_2. Again, this is similar to 4a and could share an implementation which was generic over the group operation.
5. The fast-fourier-transform (FFT) over the field \mathbb{F}_r. This only depends on 1b.

Finally, the Groth16 SNARK prover itself simply performs 4 multi-exponentiations in G_1, 1 multi-exponentiation in G_2, and a few FFTs, combining the results together in a [simple way].

The colored boxes illustrate when one can use a generic implementation to implement both algorithms. Thus, there are 5 essentially different parts of the prover implementation.