Enhancements are Blackbox Non-trivial: Impossibility of Enhanced Trapdoor Permutations from Standard Trapdoor Permutations
Trapdoor permutations (TDP) are a fundamental primitive in cryptography. Several variants of this notion have emerged as a result of different applications. However, it is not clear whether these variants can be based on the standard notion of TDPs.
We study the question of whether enhanced trapdoor permutations can be based on classical trapdoor permutations. The main motivation of our work is in the context of existing TDP-based constructions of oblivious transfer and non-interactive zero knowledge protocols, which require enhancements to the classical TDP notion. We prove that these enhancements are non-trivial, in the sense that there does not exist fully blackbox constructions of enhanced TDPs from classical TDPs.
On the technical side, we show that the enhanced TDP security of any construction in the random TDP oracle world can be broken via a polynomial number of queries to the TDP oracle as well as a weakening oracle, which provides inversion with respect to randomness. We also show that the standard one-wayness of the random TDP oracle stays intact in the presence of this weakening oracle.
I am grateful to the anonymous reviewers for their useful comments, and especially to one reviewer for their very elaborate comments. I would also like to thank Bruce Kapron for commenting on an earlier draft of the paper.
- [BPR+08]Boneh, D., Papakonstantinou, P.A., Rackoff, C., Vahlis, Y., Waters, B.: On the impossibility of basing identity based encryption on trapdoor permutations. In: 49th FOCS, 25–28 October 2008, Philadelphia, PA, USA, pp. 283–292. IEEE Computer Society Press (2008)Google Scholar
- [CL17]Canetti, R., Lichtenberg, A.: Certifying trapdoor permutations, revisited. Cryptology ePrint Archive, Report 2017/631 (2017). http://eprint.iacr.org/2017/631
- [FLS90]Feige, U., Lapidot, D., Shamir, A.: Multiple non-interactive zero knowledge proofs based on a single random string (extended abstract). In: 31st FOCS, 22–24 October 1990, St. Louis, Missouri, pp. 308–317. IEEE Computer Society Press (1990)Google Scholar
- [GKM+00]Gertner, Y., Kannan, S., Malkin, T., Reingold, O., Viswanathan, M.: The relationship between public key encryption and oblivious transfer. In: 41st FOCS, 12–14 November 2000, Redondo Beach, CA, USA, pp. 325–335. IEEE Computer Society Press (2000)Google Scholar
- [GMR01]Gertner, Y., Malkin, T., Reingold, O.: On the impossibility of basing trapdoor functions on trapdoor predicates. In: 42nd FOCS, 14–17 October 2001, pp. 126–135, Las Vegas, NV, USA. IEEE Computer Society Press (2001)Google Scholar
- [GMW87]Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: Aho, A. (ed.) 19th ACM STOC, pp. 218–229, 25–27 May 1987, New York City, NY, USA. ACM Press, New York (1987)Google Scholar
- [Gol11]Goldreich, O.: Basing non-interactive zero-knowledge on (enhanced) trapdoor permutations: the state of the art. In: Goldreich, O. (ed.) Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation. LNCS, vol. 6650, pp. 406–421. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22670-0_28CrossRefGoogle Scholar
- [HHRS07]Haitner, I., Hoch, J.J., Reingold, O., Segev, G.: Finding collisions in interactive protocols - a tight lower bound on the round complexity of statistically-hiding commitments. In: 48th FOCS, pp. 669–679, 20–23 October 2007, Providence, RI, USA. IEEE Computer Society Press (2007)Google Scholar
- [IR89]Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: 21st ACM STOC, 15–17 May, Seattle, WA, USA, pp. 44–61. ACM Press (1989)Google Scholar
- [Rab79]Rabin, M.O.: Digital signatures and public key functions as intractable as factorization. Technical report MIT/LCS/TR-212, Massachusetts Institute of Technology, January 1979Google Scholar
- [Yao82]Yao, A.C.-C.: Theory and applications of trapdoor functions (extended abstract). In: 23rd FOCS, pp. 80–91, 3–5 November, Chicago, Illinois. IEEE Computer Society Press (1982)Google Scholar