# In zkSNARKS, does R1CS require every step of the calculation, or just statements which confirm the calculation was performed correctly?

I was attempting to figure out a way to implement the modulo operation as a set of gates in an Rank-1 Constraint System, detailed by Vitalik Buterin here

However, it occurred to me that maybe we don't actually need to break down each step of the calculation into an R1CS gate, as long we sufficiently implement gates which verify the prover performed the calculation correctly.

So instead of a complicated series of gates, the following calculations can be performed "behind the scenes" (assuming unsigned ints):

m = x % y
d = x // y


And the gates in our R1CS only need to consist of:

x = d*y + m
m < y


(I am aware that m < y is a tad complicated, but not nearly as complicated as actually performing the modulo operation itself)

Am I correct that we should only include the bare minimum constraints? Or are there some sort of security flaws when doing this and it is better to have a more complicated system of gates to validate every step of the calculation?

More precisely, given $$g^x$$, if there is an efficient MOD gate, then it means that you can obtain $$x \pmod{p_{0}}$$ and $$x \pmod{p_{1}}$$ efficiently - for two arbitrarily chosen primes $$p_{0},p_{1}$$ - from $$g^x$$. Then by applying the Chinese Remainder Theorem, it is trivial to get $$x$$, hence Discrete Logarithm Problem could be solved efficiently.
Note: in your second approach, namely creating MOD gate with the following constraints: $$x = d*y + m \land m < y$$ would also require a single MOD, since a malicious prover could make $$d*y$$ overflow modulo the field modulus. This needs to be checked in the circuit, otherwise a false remainder could be proven correct.
• Wrt. to your note: "would also require a single mod", but it's not an arbitrary modulus, it's just the field congruency; not an explicit modulus gate. Second, an honest prover would not overflow $d\cdot y$, since $d = x//y$. Commented Oct 12, 2019 at 8:27
• Is it possible to come up with two sets $(d,m)$ s.t. $d\cdot y + m=x \pmod p \wedge m<y$? Something inside tells me that should be unique, but that got me wondering. Commented Oct 12, 2019 at 18:33