# Rate-1, Linear Time and Additively Homomorphic UC Commitments

## Abstract

We construct the first UC commitment scheme for binary strings with the optimal properties of rate approaching 1 and linear time complexity (in the amortised sense, using a small number of seed OTs). On top of this, the scheme is additively homomorphic, which allows for applications to maliciously secure 2-party computation. As tools for obtaining this, we make three contributions of independent interest: we construct the first (binary) linear time encodable codes with non-trivial distance and rate approaching 1, we construct the first almost universal hash function with small seed that can be computed in linear time, and we introduce a new primitive called interactive proximity testing that can be used to verify whether a string is close to a given linear code.

## Notes

### Acknowledgements

A major part of this work was done while Ignacio Cascudo and Nico Döttling were also 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.

Nico Döttling gratefully acknowledges support by the DAAD (German Academic Exchange Service) under the postdoctoral program (57243032). While at Aarhus University, he was supported by European Research Council Starting Grant 279447. His research is also supported in part from a DARPA/ARL SAFEWARE award, AFOSR Award FA9550-15-1-0274, and NSF CRII Award 1464397. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government.

Jesper Buus Nielsen was supported by European Research Council Starting Grant 279447.

The authors thank the anonymous reviewers of CRYPTO 2016 for their comments, which contributed to improve the paper.

## Supplementary material

## References

- [AHMR15]Afshar, A., Hu, Z., Mohassel, P., Rosulek, M.: How to efficiently evaluate RAM programs with malicious security. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 702–729. Springer, Heidelberg (2015)Google Scholar
- [BCPV13]Blazy, O., Chevalier, C., Pointcheval, D., Vergnaud, D.: Analysis and improvement of Lindell’s UC-secure commitment schemes. In: Jacobson Jr., M.J., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds.) ACNS 2013. LNCS, vol. 7954, pp. 534–551. Springer, Heidelberg (2013)CrossRefGoogle Scholar
- [Bra16]Brandão, L.T.A.N.: Very-efficient simulatable flipping of many coins into a well. In: Cheng, C.M., et al. (eds.) PKC 2016. LNCS, vol. 9615, pp. 297–326. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49387-8_12 CrossRefGoogle Scholar
- [Can01]Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: FOCS, pp. 136–145. IEEE Computer Society (2001)Google Scholar
- [CDD+15]Cascudo, I., Damgård, I., David, B.M., Giacomelli, I., Nielsen, J.B., Trifiletti, R.: Additively homomorphic UC commitments with optimal amortized overhead. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 495–515. Springer, Heidelberg (2015)Google Scholar
- [CF01]Canetti, R., Fischlin, M.: Universally composable commitments. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, p. 19. Springer, Heidelberg (2001)CrossRefGoogle Scholar
- [CLOS02]Canetti, R., Lindell, Y., Ostrovsky, R., Sahai, A.: Universally composable two-party and multi-party secure computation. In: STOC, pp. 494–503 (2002)Google Scholar
- [CRVW02]Capalbo, M.R., Reingold, O., Vadhan, S.P., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: Proceedings on 34th Annual ACM Symposium on Theory of Computing, 19–21 May 2002, Montréal, Québec, Canada, pp. 659–668 (2002)Google Scholar
- [DDGN14]Damgård, I., David, B., Giacomelli, I., Nielsen, J.B.: Compact VSS and efficient homomorphic UC commitments. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 213–232. Springer, Heidelberg (2014)Google Scholar
- [DI14]Druk, E., Ishai, Y.: Linear-time encodable codes meeting the gilbert-varshamov bound and their cryptographic applications. In: Naor, M. (ed.) Innovations in Theoretical Computer Science, ITCS 2014, Princeton, NJ, USA, 12–14 January 2014, pp. 169–182. ACM (2014)Google Scholar
- [FJN+13]Frederiksen, T.K., Jakobsen, T.P., Nielsen, J.B., Nordholt, P.S., Orlandi, C.: MiniLEGO: efficient secure two-party computation from general assumptions. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 537–556. Springer, Heidelberg (2013)CrossRefGoogle Scholar
- [FJNT16]Frederiksen, T.K., Jakobsen, T.P., Nielsen, J.B., Trifiletti, R.: On the complexity of additively homomorphic uc commitments. In: Kushilevitz, E., et al. (eds.) TCC 2016-A. LNCS, vol. 9562, pp. 542–565. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49096-9_23 CrossRefGoogle Scholar
- [GI02]Guruswami, V., Indyk, P.: Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets. In: Reif, J.H. (ed.) Proceedings on 34th Annual ACM Symposium on Theory of Computing, 19–21 May 2002, Montréal, Québec, Canada, pp. 812–821. ACM (2002)Google Scholar
- [GI03]Guruswami, V., Indyk, P.: Linear time encodable and list decodable codes. In: Larmore and Goemans [LG03], pp. 126–135Google Scholar
- [GI05]Guruswami, V., Indyk, P.: Linear-time encodable/decodable codes with near-optimal rate. IEEE Trans. Inf. Theor.
**51**(10), 3393–3400 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - [GIKW14]Garay, J.A., Ishai, Y., Kumaresan, R., Wee, H.: On the complexity of UC commitments. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 677–694. Springer, Heidelberg (2014)CrossRefGoogle Scholar
- [IKOS08]Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Cryptography with constant computational overhead. In: Dwork, C. (ed.) STOC, pp. 433–442. ACM (2008)Google Scholar
- [IPS08]Ishai, Y., Prabhakaran, M., Sahai, A.: Founding cryptography on oblivious transfer – efficiently. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 572–591. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- [IPS09]Ishai, Y., Prabhakaran, M., Sahai, A.: Secure arithmetic computation with no honest majority. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 294–314. Springer, Heidelberg (2009)CrossRefGoogle Scholar
- [LG03]Larmore, L.L., Goemans, M.X. (eds.) Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 9–11 June 2003, San Diego, CA, USA. ACM (2003)Google Scholar
- [Lin11]Lindell, Y.: Highly-efficient universally-composable commitments based on the DDH assumption. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 446–466. Springer, Heidelberg (2011)CrossRefGoogle Scholar
- [Nao91]Naor, M.: Bit commitment using pseudorandomness. J. Cryptol.
**4**(2), 151–158 (1991)CrossRefzbMATHGoogle Scholar - [PVW08]Peikert, C., Vaikuntanathan, V., Waters, B.: A framework for efficient and composable oblivious transfer. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 554–571. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- [Spi96]Spielman, D.A.: Linear-time encodable and decodable error-correcting codes. IEEE Trans. Inf. Theor.
**42**(6), 1723–1731 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - [VZ12]Vadhan, S., Zheng, C.J.: Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In: Proceedings of the 44th Symposium on Theory of Computing, pp. 817–836. ACM (2012)Google Scholar