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What role does a circuit play in ZK-SNARKS? Is it executed on both the prover's and the verifier's machine? If so, how would you program it to check if one value is greater than another value?

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Technically neither party necessarily evaluates the circuit.

The two parties share the description of a circuit $C$, an output value $y$ and potentially some input values $x_1\dots,x_\ell$. the prover than proves to the verifier that they know additional secret inputs $s_1,\dots,s_n$, such that if one were to evaluate the circuit $C$ on inputs $x_1\dots,x_\ell,s_1,\dots,s_n$ the result would be $y$. I.e., $$C(x_1\dots,x_\ell,s_1,\dots,s_n)=y.$$

The argument of knowledge property guarantees that the prover must actually know said inputs in order to convince the verifier. The zero-knowledge property guarantees that the verifier learns nothing from the exchange except that the above statement is true. In particular the verifier learn no information about $s_1,\dots,s_n$.

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