Incremental Proofs of Sequential Work

  • Nico Döttling
  • Russell W. F. LaiEmail author
  • Giulio Malavolta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11477)


A proof of sequential work allows a prover to convince a verifier that a certain amount of sequential steps have been computed. In this work we introduce the notion of incremental proofs of sequential work where a prover can carry on the computation done by the previous prover incrementally, without affecting the resources of the individual provers or the size of the proofs.

To date, the most efficient instance of proofs of sequential work [Cohen and Pietrzak, Eurocrypt 2018] for N steps require the prover to have \(\sqrt{N}\) memory and to run for \(N + \sqrt{N}\) steps. Using incremental proofs of sequential work we can bring down the prover’s storage complexity to \(\log N\) and its running time to N.

We propose two different constructions of incremental proofs of sequential work: Our first scheme requires a single processor and introduces a poly-logarithmic factor in the proof size when compared with the proposals of Cohen and Pietrzak. Our second scheme assumes \(\log N\) parallel processors but brings down the overhead of the proof size to a factor of 9. Both schemes are simple to implement and only rely on hash functions (modelled as random oracles).


  1. 1.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM (JACM) 45(1), 70–122 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKS and proof-carrying data. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, Palo Alto, CA, USA, 1–4 June, pp. 111–120. ACM Press (2013)Google Scholar
  3. 3.
    Bitansky, N., Goldwasser, S., Jain, A., Paneth, O., Vaikuntanathan, V., Waters, B.: Time-lock puzzles from randomized encodings. In: Sudan, M. (ed.) ITCS 2016, Cambridge, MA, USA, 14–16 January, pp. 345–356. ACM (2016)Google Scholar
  4. 4.
    Boneh, D., Bonneau, J., Bünz, B., Fisch, B.: Verifiable delay functions. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part I. LNCS, vol. 10991, pp. 757–788. Springer, Cham (2018). Scholar
  5. 5.
    Cohen, B., Pietrzak, K.: Simple proofs of sequential work. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part II. LNCS, vol. 10821, pp. 451–467. Springer, Cham (2018). Scholar
  6. 6.
    Dwork, C., Naor, M.: Pricing via processing or combatting junk mail. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 139–147. Springer, Heidelberg (1993). Scholar
  7. 7.
    Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). Scholar
  8. 8.
    Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: 24th ACM STOC, Victoria, British Columbia, Canada, 4–6 May, pp. 723–732. ACM Press (1992)Google Scholar
  9. 9.
    Mahmoody, M., Moran, T., Vadhan, S.P.: Time-lock puzzles in the random oracle model. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 39–50. Springer, Heidelberg (2011). Scholar
  10. 10.
    Mahmoody, M., Moran, T., Vadhan, S.P.: Publicly verifiable proofs of sequential work. In: Kleinberg, R.D. (ed.) ITCS 2013, Berkeley, CA, USA, 9–12 January, pp. 373–388. ACM (2013)Google Scholar
  11. 11.
    Micali, S.: CS proofs (extended abstracts). In: 35th FOCS, Santa Fe, New Mexico, 20–22 November, pp. 436–453. IEEE Computer Society Press (1994)Google Scholar
  12. 12.
    Pietrzak, K.: Simple verifiable delay functions. Cryptology ePrint Archive, Report 2018/627 (2018).
  13. 13.
    Rivest, R.L., Shamir, A., Wagner, D.A.: Time-lock puzzles and timed-release crypto (1996)Google Scholar
  14. 14.
    Valiant, P.: Incrementally verifiable computation or proofs of knowledge imply time/space efficiency. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 1–18. Springer, Heidelberg (2008). Scholar
  15. 15.
    Wesolowski, B.: Efficient verifiable delay functions. Cryptology ePrint Archive, Report 2018/623 (2018).

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  • Nico Döttling
    • 1
  • Russell W. F. Lai
    • 2
    Email author
  • Giulio Malavolta
    • 3
  1. 1.CISPA Helmholtz Center for Information SecuritySaarbrückenGermany
  2. 2.Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  3. 3.Carnegie Mellon UniversityPittsburghUSA

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