# Notes on GGH13 Without the Presence of Ideals

## Abstract

We investigate the merits of altering the Garg, Gentry and Halevi (GGH13) graded encoding scheme to remove the presence of the ideal \(\langle g \rangle \). In particular, we show that we can alter the form of encodings so that effectively a new \(g_i\) is used for each source group \({\mathbb {G}}_i\), while retaining correctness. This would appear to prevent all known attacks on IO candidates instantiated using GGH13. However, when analysing security in a simplified branching program model, we present an IO distinguishing attack that does not use \(\langle g \rangle \). This result opens a counterpoint with the work of Halevi (EPRINT 2015) which stated that the core computational hardness problem underpinning GGH13 is computing a basis of this ideal. Our attempts seem to suggest that there is a structural vulnerability in the way that GGH13 encodings are constructed that lies deeper than the presence of \(\langle g \rangle \). Tangentially, we observe that our attack is prevented when considering all the added machinery of IO candidates.

### References

- [ABD16]Albrecht, M., Bai, S., Ducas, L.: A subfield lattice attack on overstretched NTRU assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 153–178. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_6 CrossRefGoogle Scholar
- [ADGM16]Apon, D., Döttling, N., Garg, S., Mukherjee, P.: Cryptanalysis of indistinguishability obfuscations of circuits over GGH13. Cryptology ePrint Archive, Report 2016/1003 (2016). http://eprint.iacr.org/2016/1003
- [AGIS14]Ananth, P.V., Gupta, D., Ishai, Y., Sahai, A.: Optimizing obfuscation: avoiding Barrington’s theorem. In: Ahn, G.-J., Yung, M., Li, N. (eds.), ACM CCS 2014, pp. 646–658. ACM Press, November 2014Google Scholar
- [Bar89]Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in nc\({^1}\). J. Comput. Syst. Sci.
**38**(1), 150–164 (1989)MathSciNetCrossRefMATHGoogle Scholar - [BGK+14]Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 221–238. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_13 CrossRefGoogle Scholar
- [BLR+15]Boneh, D., Lewi, K., Raykova, M., Sahai, A., Zhandry, M., Zimmerman, J.: Semantically secure order-revealing encryption: multi-input functional encryption without obfuscation. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 563–594. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_19 Google Scholar
- [BMSZ15]Badrinarayanan, S., Miles, E., Sahai, A., Zhandry, M.: Post-zeroizing obfuscation: the case of evasive circuits. Cryptology ePrint Archive, Report 2015/167 (2015). http://eprint.iacr.org/2015/167
- [BS83]Baur, W., Strassen, V.: The complexity of partial derivatives. Theor. Comput. Sci.
**22**, 317–330 (1983)MathSciNetCrossRefMATHGoogle Scholar - [BWZ14]Boneh, D., Waters, B., Zhandry, M.: Low overhead broadcast encryption from multilinear maps. In: Garay and Gennaro [GG14], pp. 206–223Google Scholar
- [CGH17]Chen, Y., Gentry, C., Halevi, S.: Cryptanalyses of candidate branching program obfuscators. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 278–307. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_10 CrossRefGoogle Scholar
- [CIV16]Castryck, W., Iliashenko, I., Vercauteren, F.: Provably weak instances of ring-LWE revisited. In: Fischlin and Coron [FC16], pp. 147–167Google Scholar
- [CJL16]Cheon, J.H., Jeong, J., Lee, C.: An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero. LMS J. Comput. Math.
**19**(A), 255–266 (2016)MathSciNetCrossRefMATHGoogle Scholar - [CLLT16]Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Cryptanalysis of GGH15 multilinear maps. In: Robshaw and Katz [RK16], pp. 607–628Google Scholar
- [CLLT17]Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Zeroizing attacks on indistinguishability obfuscation over CLT13. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10174, pp. 41–58. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54365-8_3 CrossRefGoogle Scholar
- [CLR15]Cheon, J.H., Lee, C., Ryu, H.: Cryptanalysis of the new CLT multilinear maps. Cryptology ePrint Archive, Report 2015/934 (2015). http://eprint.iacr.org/2015/934
- [CLT13]Coron, J.-S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_26 CrossRefGoogle Scholar
- [CLT14]Coron, J.-S., Lepoint, T., Tibouchi, M.: Cryptanalysis of two candidate fixes of multilinear maps over the integers. Cryptology ePrint Archive, Report 2014/975 (2014). http://eprint.iacr.org/2014/975
- [FC16]Fischlin, M., Coron, J.-S. (eds.): EUROCRYPT 2016, Part I. LNCS, vol. 9665. Springer, Heidelberg (2016)MATHGoogle Scholar
- [FRS16]Fernando, R., Rasmussen, P.M.R., Sahai, A.: Preventing CLT zeroizing attacks on obfuscation. Cryptology ePrint Archive, Report 2016/1070 (2016). http://eprint.iacr.org/2016/1070
- [GG14]Garay, J.A., Gennaro, R. (eds.): CRYPTO 2014, Part I. LNCS, vol. 8616. Springer, Heidelberg (2014)MATHGoogle Scholar
- [GGH13a]Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_1 CrossRefGoogle Scholar
- [GGH+13b]Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th FOCS, pp. 40–49. IEEE Computer Society Press, October 2013Google Scholar
- [GGH+13c]Garg, S., Gentry, C., Halevi, S., Sahai, A., Waters, B.: Attribute-based encryption for circuits from multilinear maps. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 479–499. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_27 CrossRefGoogle Scholar
- [GGH15]Gentry, C., Gorbunov, S., Halevi, S.: Graph-induced multilinear maps from lattices. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 498–527. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_20 CrossRefGoogle Scholar
- [GMM+16]Garg, S., Miles, E., Mukherjee, P., Sahai, A., Srinivasan, A., Zhandry, M.: Secure obfuscation in a weak multilinear map model. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 241–268. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_10 CrossRefGoogle Scholar
- [GMS16]Garg, S., Mukherjee, P., Srinivasan, A.: Obfuscation without the vulnerabilities of multilinear maps. Cryptology ePrint Archive, Report 2016/390 (2016). http://eprint.iacr.org/2016/390
- [Hal15]Halevi, S.: Graded encoding, variations on a scheme. Cryptology ePrint Archive, Report 2015/866 (2015). http://eprint.iacr.org/2015/866
- [HJ16]Hu, Y., Jia, H.: Cryptanalysis of GGH map. In: Fischlin and Coron [FC16], pp. 537–565Google Scholar
- [Kay09]Kayal, N.: The complexity of the annihilating polynomial. In: Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15–18 July 2009, pp. 184–193. IEEE Computer Society (2009)Google Scholar
- [KF17]Kirchner, P., Fouque, P.-A.: Revisiting lattice attacks on overstretched NTRU parameters. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 3–26. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_1 CrossRefGoogle Scholar
- [Kil88]Kilian, J.: Zero-knowledge with log-space verifiers. In: 29th FOCS, pp. 25–35. IEEE Computer Society Press, October 1988Google Scholar
- [LTV12]López-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Karloff, H.J., Pitassi, T. (eds.), 44th ACM STOC, pp. 1219–1234. ACM Press, May 2012Google Scholar
- [MR04]Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. In: 45th FOCS, pp. 372–381. IEEE Computer Society Press, October 2004Google Scholar
- [MSW14]Miles, E., Sahai, A., Weiss, M.: Protecting obfuscation against arithmetic attacks. Cryptology ePrint Archive, Report 2014/878 (2014). http://eprint.iacr.org/2014/878
- [MSZ16a]Miles, E., Sahai, A., Zhandry, M.: Annihilation attacks for multilinear maps: cryptanalysis of indistinguishability obfuscation over GGH13. In: Robshaw and Katz [RK16], pp. 629–658Google Scholar
- [MSZ16b]Miles, E., Sahai, A., Zhandry, M.: Secure obfuscation in a weak multilinear map model: a simple construction secure against all known attacks. Cryptology ePrint Archive, Report 2016/588 (2016). http://eprint.iacr.org/2016/588
- [PST14]Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation from semantically-secure multilinear encodings. In: Garay and Gennaro [GG14], pp. 500–517Google Scholar
- [RK16]Robshaw, M., Katz, J. (eds.): CRYPTO 2016, Part II. LNCS, vol. 9815. Springer, Heidelberg (2016)MATHGoogle Scholar