Let us see an example of how cryptographic hash functions are used in Zero-Knowledge Proof Systems. Following code written in Zokrates DSL Toolbox is an example of computing a Hash using Zero-Knowledge Proof systems. The programming instructions are compiled first. Then we will proceed to the setup of the arithmetic circuit through the setup. Then we export the verifier and will compute the proof.
import "hashes/sha256/512bitPacked" as sha256packed
def main(private field a, private field b, private field c, private field d) -> (field):
h = sha256packed([a, b, c, d])
Let us also analyse another code example to construct a Pedersen Hash in the construction of a Zero-Knowledge Proof using Zokrates Toolbox.
import "hashes/pedersen/6bit" as pedersen
def main() -> (field):
field e = [1, 1, 0, 0, 0, 0]
field d = pedersen(e)
5483803361072598088157572477433311028290255512997784196805059543720485966024 == d
8712718144085345152615259409576985937188455136179509057889474614313734076278 == d
As we can see in these two examples, proving the knowledge of a preimage under a cryptographic hash function expressed as a circuit over a large prime field becomes one of the most computationally expensive parts in the arithmetic circuit construction in a zero-knowledge proof system. Hence all the pursuits for inventing SNARK and STARK friendly cryptographic hash functions are greeted with much of enthusiasms!
The paper mentioned in this question claims a reduction in the number of constraints per message bit than Pedersen Hash which would potentially improve the performance in the construction of polynomial commitments for ZKSNARK. Likewise, the proposed binary hash function STARKAD could be beneficial in the construction of ZKSTARK based systems in the near future.
Our hash-function Poseidon uses up to 8x fewer constraints per message bit than Pedersen Hash, whereas our binary hash-function Starkad wins by a substantial margin over the other recent designs