I will present Zinc, a proof system that addresses the problem of high arithmetization costs in proof systems. Arithmetization is the step where one turns the computation/relation one wants to prove into a constraint that is in a suitable form for the proof system. In many natural applications this step is very costly, in the sense that the final constraints is easily of ‘size’ 16x or 32x than the original computation. Such applications include: hash computations, CPU instructions (arithmetic modulo 2^n and bitwise operations), lattice-based operations (as in, e.g. vFHE operations), legacy cryptography, machine learning, etc. Zinc (https://eprint.iacr.org/2025/316) is a proof system for constraints expressed over the integers. These, as we will see, allow to arithmetize many of the above operations cheaply. Additionally, for the same number of constraints, Zinc’s running time is similar to current state-of-the-art hash based proof system.I will also discuss our ongoing work towards a proof system where all the above operations are cheaply arithmetizable. During the talk I will focus especially on the problem of proving lattice-based operations, with a view of applying our techniques to vFHE