Paper 2019/458

Poseidon: A New Hash Function for Zero-Knowledge Proof Systems

Lorenzo Grassi
Dmitry Khovratovich
Christian Rechberger
Arnab Roy
Markus Schofnegger

The area of practical computational integrity proof systems, like SNARKs, STARKs, Bulletproofs, is seeing a very dynamic development with several constructions having appeared recently with improved properties and relaxed setup requirements. Many use cases of such systems involve, often as their most expensive part, proving the knowledge of a preimage under a certain cryptographic hash function, which is expressed as a circuit over a large prime field. A notable example is a zero-knowledge proof of coin ownership in the Zcash cryptocurrency, where the inadequacy of the SHA-256 hash function for such a circuit caused a huge computational penalty. In this paper, we present a modular framework and concrete instances of cryptographic hash functions which work natively with GF(p) objects. Our hash function Poseidon uses up to 8x fewer constraints per message bit than Pedersen Hash. Our construction is not only expressed compactly as a circuit, but can also be tailored for various proof systems using specially crafted polynomials, thus bringing another boost in performance. We demonstrate this by implementing a 1-out-of-a-billion membership proof with Merkle trees in less than a second by using Bulletproofs.

Note: In this updated version of Poseidon, we removed the inverse S-box x -> x^(-1) due to mistakes in the previous analysis. We do not recommend the Poseidon variant using this S-box anymore. Moreover, we made corrections to the bounds for algebraic attacks for the other Poseidon versions. This modification *does not change the round numbers* in any of the use cases we are aware of, in particular those mentioned in the paper.

Available format(s)
Cryptographic protocols
Publication info
Published elsewhere. USENIX Security '21
Zero-Knowledge Proof SystemsSNARKSTARKHash Function
Contact author(s)
L Grassi @ cs ru nl
khovratovich @ gmail com
christian rechberger @ tugraz at
arnab roy @ aau at
markus schofnegger @ gmail com
2023-03-04: last of 10 revisions
2019-05-10: received
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      author = {Lorenzo Grassi and Dmitry Khovratovich and Christian Rechberger and Arnab Roy and Markus Schofnegger},
      title = {Poseidon: A New Hash Function for Zero-Knowledge Proof Systems},
      howpublished = {Cryptology ePrint Archive, Paper 2019/458},
      year = {2019},
      note = {\url{}},
      url = {}
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