# Is security of zk-SNARKS related to the size of the arithmetic circuit they evaluate?

In the context of zk-SNARKS, we are given an arithmetic circuit $$C$$, public outputs $$y_1, \ldots, y_n = \mathbf{y}$$ and some public inputs $$x_1, \ldots, x_ℓ$$. The prover wants to prove knowledge of some extra secret inputs $$s_1, \ldots, s_m$$ such that if one were to evaluate the circuit $$C$$ on inputs $$x_1…,x_ℓ,s_1,…,s_m$$ the result would be $$\mathbf{y}$$.

In some application like in HALO, the recursion property depends on the size of the arithmetic circuit being evaluated. However it does not seem to be related to the security of the scheme, fixed at roughly 128bits. If so, given a certain security level $$\lambda$$, can one create a zk-SNARK with any circuit size? (granted that the circuit does verify the "valid assignments" we want provers to have)

We have polynomials identity $$$$A(x) \cdot B(x) - C(x) \equiv h(x) Z(x)$$$$ with $$\deg(Z(x)) = n$$, $$\deg(A(x)) = \deg(B(x)) = \deg(C(x)) = n-1$$. It follows, there are at most $$2n-2$$ roots. It also follows, soundness error is at most $$\frac{2n-2}{q}$$, to be compared to $$\frac{1}{q}$$ of Schnorr protocol, for $$q$$ being group order.