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Sequentially Composable Rational Proofs

Part of the Lecture Notes in Computer Science book series (LNSC,volume 9406)


We show that Rational Proofs do not satisfy basic compositional properties in the case where a large number of “computation problems” are outsourced. We show that a “fast” incorrect answer is more remunerable for the prover, by allowing him to solve more problems and collect more rewards. We present an enhanced definition of Rational Proofs that removes the economic incentive for this strategy and we present a protocol that achieves it for some uniform bounded-depth circuits.


  • Sequential Composability
  • Arithmetic Circuit
  • Input Gate
  • Threshold Gate
  • Input Wire

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This work was supported by NSF grant CNS-1545759

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  1. 1.

    If we think of cost as time, then in the same time interval in which P solves one problem, \(\widetilde{P}\) can solve up to n problems, earning a lot more money, by answering fast and incorrectly.

  2. 2.

    We point out that the Prover can provide the Verifier with the requested gate and then the Verifier can use the uniformity of the circuit to check that the Prover has given him the correct gate at each level in time O(T(n)).


  1. Azar, P.D., Micali, S.: Rational proofs. In: 2012 ACM Symposium on Theory of Computing, pp. 1017–1028 (2012)

    Google Scholar 

  2. Azar, P.D., Micali, S.: Super-efficient rational proofs. In: 2013 ACM Conference on Electronic Commerce, pp. 29–30 (2013)

    Google Scholar 

  3. Belenkiy, M., Chase, M., Erway, C.C., Jannotti, J., Küpçü, A., Lysyanskaya, A.: Incentivizing outsourced computation. In: NetEcon 2008, pp. 85–90 (2008)

    Google Scholar 

  4. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press (2001)

    Google Scholar 

  5. Dwork, C., Naor, M., Sahai, A.: Concurrent zero-knowledge. J. ACM 51(6), 851–898 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Guo, S., Hubacek, P., Rosen, A., Vald, M.: Rational arguments: single round delegation with sublinear verification. In: 2014 Innovations in Theoretical Computer Science Conference (2014)

    Google Scholar 

  7. Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof-systems. In: Proceedings of the seventeenth Annual ACM Symposium on Theory of computing. ACM (1985)

    Google Scholar 

  8. Walfish, M., Blumberg, A.J.: Verifying computations without reexecuting them. Commun. ACM 58(2), 74–84 (2015)

    CrossRef  Google Scholar 

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Correspondence to Rosario Gennaro .

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Campanelli, M., Gennaro, R. (2015). Sequentially Composable Rational Proofs. In: Khouzani, M., Panaousis, E., Theodorakopoulos, G. (eds) Decision and Game Theory for Security. GameSec 2015. Lecture Notes in Computer Science(), vol 9406. Springer, Cham.

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25593-4

  • Online ISBN: 978-3-319-25594-1

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