# Oblivious Transfer from Any Non-trivial Elastic Noisy Channel via Secret Key Agreement

## Abstract

A \((\gamma ,\delta )\)-elastic channel is a binary symmetric channel between a sender and a receiver where the error rate of an honest receiver is \(\delta \) while the error rate of a dishonest receiver lies within the interval \([\gamma , \delta ]\). In this paper, we show that from *any* non-trivial elastic channel (i.e., \(0<\gamma<\delta <\frac{1}{2}\)) we can implement oblivious transfer with information-theoretic security. This was previously (Khurana et al., Eurocrypt 2016) only known for a subset of these parameters. Our technique relies on a new way to exploit protocols for information-theoretic key agreement from noisy channels. We also show that information-theoretically secure commitments where the receiver commits follow from any non-trivial elastic channel.

## Keywords

Oblivious transfer Elastic channels Key agreement Commitments## Notes

### Acknowledgments

Part of this work was carried out while Ignacio Cascudo was with Aarhus University. The authors acknowledge support from the Danish National Research Foundation and The National Science Foundation of China (under the grant 61361136003) for the Sino-Danish Center for the Theory of Interactive Computation and from the Center for Research in Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council. In addition, Ignacio Cascudo acknowledges support from the Danish Council for Independent Research, grant no. DFF-4002-00367, Ivan Damgrd was also supported by the advanced ERC grant MPCPRO and Samuel Ranellucci was supported by European Research Council Starting Grant 279447. We thank Jesper Buus Nielsen, Maciej Obremski and the anonymous reviewers for their helpful comments.

## Supplementary material

## References

- [BCC88]Brassard, G., Chaum, D., Crépeau, C.: Minimum disclosure proofs of knowledge. J. Comput. Syst. Sci.
**37**(2), 156–189 (1988)MathSciNetCrossRefMATHGoogle Scholar - [BCS96]Brassard, G., Crépeau, C., Santha, M.: Oblivious transfers and intersecting codes. IEEE Trans. Inf. Theory
**42**(6), 1769–1780 (1996)MathSciNetCrossRefMATHGoogle Scholar - [BS94]Brassard, G., Salvail, L.: Secret-key reconciliation by public discussion. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 410–423. Springer, Heidelberg (1994). doi: 10.1007/3-540-48285-7_35 Google Scholar
- [Can01]Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: Proceedings of 42nd IEEE Symposium on Foundations of Computer Science, pp. 136–145. IEEE (2001)Google Scholar
- [CK88]Crépeau, C., Kilian, J.: Achieving oblivious transfer using weakened security assumptions (Extended Abstract). In: 29th Annual Symposium on Foundations of Computer Science, White Plains, New York, USA, 24–26 October 1988, pp. 42–52 (1988)Google Scholar
- [CMW05]Crépeau, C., Morozov, K., Wolf, S.: Efficient unconditional oblivious transfer from almost any noisy channel. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 47–59. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30598-9_4 CrossRefGoogle Scholar
- [Cré97]Crépeau, C.: Efficient cryptographic protocols based on noisy channels. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 306–317. Springer, Heidelberg (1997). doi: 10.1007/3-540-69053-0_21 Google Scholar
- [CS06]Crépeau, C., Savvides, G.: Optimal reductions between oblivious transfers using interactive hashing. In: Proceedings of Advances in Cryptology - EUROCRYpPT, 25th Annual International Conference on the Theory and Applications of Cryptographic Techniques, St. Petersburg, Russia, May 28–June 1, pp. 201–221 (2006)Google Scholar
- [CvdGT95]Crépeau, C., Graaf, J., Tapp, A.: Committed oblivious transfer and private multi-party computation. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 110–123. Springer, Heidelberg (1995). doi: 10.1007/3-540-44750-4_9 CrossRefGoogle Scholar
- [DFMS04]Damgård, I., Fehr, S., Morozov, K., Salvail, L.: Unfair noisy channels and oblivious transfer. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 355–373. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24638-1_20 CrossRefGoogle Scholar
- [DKS99]Damgård, I., Kilian, J., Salvail, L.: On the (im)possibility of basing oblivious transfer and bit commitment on weakened security assumptions. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 56–73. Springer, Heidelberg (1999). doi: 10.1007/3-540-48910-X_5 Google Scholar
- [DORS08]Dodis, Y., Ostrovsky, R., Reyzin, L., Smith, A.D.: Fuzzy extractors: how to generate strong keys from biometrics and other noisy data. SIAM J. Comput.
**38**(1), 97–139 (2008)MathSciNetCrossRefMATHGoogle Scholar - [Est04]Estren, G.: Universally composable committed oblivious transfer and multi-party computation assuming only basic black-box primitives. Ph.D. thesis, McGill University (2004)Google Scholar
- [GMW86]Goldreich, O., Micali, S., Wigderson, A.: How to prove all NP statements in zero-knowledge and a methodology of cryptographic protocol design (Extended Abstract). In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 171–185. Springer, Heidelberg (1987). doi: 10.1007/3-540-47721-7_11 CrossRefGoogle Scholar
- [HKN+05]Harnik, D., Kilian, J., Naor, M., Reingold, O., Rosen, A.: On robust combiners for oblivious transfer and other primitives. In: Proceedings of Advances in Cryptology - EUROCRYpPT, 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Aarhus, Denmark, pp. 96–113, 22–26 May 2005Google Scholar
- [IKO+11]Ishai, Y., Kushilevitz, E., Ostrovsky, R., Prabhakaran, M., Sahai, A., Wullschleger, J.: Constant-rate oblivious transfer from noisy channels. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 667–684. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22792-9_38 CrossRefGoogle Scholar
- [Kil88]Kilian, J.: Founding cryptography on oblivious transfer. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 20–31. ACM (1988)Google Scholar
- [Kil92]Kilian, J.: A note on efficient zero-knowledge proofs and arguments. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 723–732. ACM (1992)Google Scholar
- [KMS16]Khurana, D., Maji, H.K., Sahai, A.: Secure computation from elastic noisy channels. In: Proceedings of Advances in Cryptology - EUROCRYpPT - 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, Austria, Part II, pp. 184–212, 8–12 May 2016Google Scholar
- [Mau93]Maurer, U.M.: Secret key agreement by public discussion from common information. IEEE Trans. Inf. Theory
**39**(3), 733–742 (1993)MathSciNetCrossRefMATHGoogle Scholar - [PDMN11]Pinto, A.C.B., Dowsley, R., Morozov, K., Nascimento, A.C.A.: Achieving oblivious transfer capacity of generalized erasure channels in the malicious model. IEEE Trans. Inf. Theory
**57**(8), 5566–5571 (2011)MathSciNetCrossRefGoogle Scholar