SPARKs: Succinct Parallelizable Arguments of Knowledge

Naomi Ephraim, Cody Freitag, Ilan Komargodski, Rafael Pass

Research output: Contribution to journalArticlepeer-review


We introduce the notion of a Succinct Parallelizable Argument of Knowledge (SPARK). This is an argument of knowledge with the following three efficiency properties for computing and proving a (non-deterministic, polynomial time) parallel RAM computation that can be computed in parallel time T with at most p processors: — The prover’s (parallel) running time is T + polylog(T · p). (In other words, the prover’s running time is essentially T for large computation times!) — The prover uses at most p · polylog(T · p) processors. — The communication and verifier complexity are both polylog(T · p). The combination of all three is desirable, as it gives a way to leverage a moderate increase in parallelism in favor of near-optimal running time. We emphasize that even a factor two overhead in the prover’s parallel running time is not allowed. Our main contribution is a generic construction of SPARKs from any succinct argument of knowledge where the prover’s parallel running time is T · polylog(T · p) when using p processors, assuming collision-resistant hash functions. When suitably instantiating our construction, we achieve a four-round SPARK for any parallel RAM computation assuming only collision resistance. Additionally assuming the existence of a succinct non-interactive argument of knowledge (SNARK), we construct a non-interactive SPARK that also preserves the space complexity of the underlying computation up to polylog(T · p) factors. We also show the following applications of non-interactive SPARKs. First, they immediately imply delegation protocols with near optimal prover (parallel) running time. This, in turn, gives a way to construct verifiable delay functions (VDFs) from any sequential function. When the sequential function is also memory-hard, this yields the first construction of a memory-hard VDF.

Original languageAmerican English
Article number31
JournalJournal of the ACM
Issue number5
StatePublished - 1 Oct 2022

Bibliographical note

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© 2022 Association for Computing Machinery.


  • Succinct arguments
  • non-interactive
  • parallelization


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