# Large FHE Gates from Tensored Homomorphic Accumulator

## Abstract

The main bottleneck of all known Fully Homomorphic Encryption schemes lies in the bootstrapping procedure invented by Gentry (STOC’09). The cost of this procedure can be mitigated either using Homomorphic SIMD techniques, or by performing larger computation per bootstrapping procedure.

In this work, we propose new techniques allowing to perform more operations per bootstrapping in FHEW-type schemes (EUROCRYPT’13). While maintaining the quasi-quadratic \(\tilde{O}(n^2)\) complexity of the whole cycle, our new scheme allows to evaluate gates with \(\varOmega (\log n)\) input bits, which constitutes a quasi-linear speed-up. Our scheme is also very well adapted to large threshold gates, natively admitting up to \(\varOmega (n)\) inputs. This could be helpful for homomorphic evaluation of neural networks.

Our theoretical contribution is backed by a preliminary prototype implementation, which can perform 6-to-6 bit gates in less than 10 s on a single core, as well as threshold gates over 63 input bits even faster.

## Keywords

Fully Homomorphic Encryption Large gates Threshold gates Ideal lattices## Supplementary material

## References

- 1.Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) 41st ACM STOC, Bethesda, MD, USA, 31 May–2 June 2009, pp. 169–178. ACM Press (2009)Google Scholar
- 2.Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009). crypto.stanford.edu/craig
- 3.Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_25CrossRefGoogle Scholar
- 4.Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Ostrovsky, R. (ed.) 52nd FOCS, Palm Springs, CA, USA, 22–25 October 2011, pp. 97–106. IEEE Computer Society Press (2011)Google Scholar
- 5.Gentry, C., Halevi, S., Smart, N.P.: Fully homomorphic encryption with polylog overhead. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 465–482. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_28CrossRefGoogle Scholar
- 6.Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS 2012, Cambridge, MA, USA, 8–10 January 2012, pp. 309–325. ACM (2012)Google Scholar
- 7.Halevi, S., Shoup, V.: Bootstrapping for HElib. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part I. LNCS, vol. 9056, pp. 641–670. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_25Google Scholar
- 8.Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5CrossRefGoogle Scholar
- 9.Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in \(\text{NC}^1\). In: 18th ACM STOC, Berkeley, CA, USA, 28–30 May 1986, pp. 1–5. ACM Press (1986)Google Scholar
- 10.Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: Naor, M. (ed.) ITCS 2014, Princeton, NJ, USA, 12–14 January 2014, pp. 1–12. ACM (2014)Google Scholar
- 11.Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_17CrossRefGoogle Scholar
- 12.Ducas, L., Micciancio, D.: FHEW: bootstrapping homomorphic encryption in less than a second. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part I. LNCS, vol. 9056, pp. 617–640. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_24Google Scholar
- 13.Biasse, J.-F., Ruiz, L.: FHEW with efficient multibit bootstrapping. In: Lauter, K., Rodríguez-Henríquez, F. (eds.) LATINCRYPT 2015. LNCS, vol. 9230, pp. 119–135. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22174-8_7CrossRefGoogle Scholar
- 14.Gama, N., Izabachène, M., Nguyen, P.Q., Xie, X.: Structural lattice reduction: generalized worst-case to average-case reductions and homomorphic cryptosystems. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 528–558. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_19CrossRefGoogle Scholar
- 15.Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016, Part I. LNCS, vol. 10031, pp. 3–33. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_1CrossRefGoogle Scholar
- 16.Riordan, J., Shannon, C.E.: The number of two-terminal series-parallel networks. Stud. Appl. Math.
**21**(1–4), 83–93 (1942)MathSciNetGoogle Scholar - 17.Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054868CrossRefGoogle Scholar
- 18.Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_1CrossRefGoogle Scholar
- 19.Halevi, S., Halevi, T., Shoup, V., Stephens-Davidowitz, N.: Implementing BP-obfuscation using graph-induced encoding. Cryptology ePrint Archive, Report 2017/104 (2017). http://eprint.iacr.org/2017/104
- 20.Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Improving TFHE: faster packed homomorphic operations and efficient circuit bootstrapping. Cryptology ePrint Archive, Report 2017/430 (2017). http://eprint.iacr.org/2017/430
- 21.Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Eldar, Y., Kutyniok, G. (eds.) Compressed Sensing, Theory and Applications, pp. 210–268. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
- 22.Rivasplata, O.: Subgaussian Random Variables: An Expository Note (2012). https://sites.ualberta.ca/~omarr/publications/subgaussians.pdf
- 23.Hoffstein, J., Howgrave-Graham, N., Pipher, J., Silverman, J.H., Whyte, W.: NTRUSign: digital signatures using the NTRU lattice. In: Joye, M. (ed.) CT-RSA 2003. LNCS, vol. 2612, pp. 122–140. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36563-X_9CrossRefGoogle Scholar
- 24.Blum, A., Furst, M., Kearns, M., Lipton, R.J.: Cryptographic primitives based on hard learning problems. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 278–291. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48329-2_24CrossRefGoogle Scholar
- 25.Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) 37th ACM STOC, Baltimore, MA, USA, 22–24 May 2005, pp. 84–93. ACM Press (2005)Google Scholar
- 26.Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast cryptographic primitives and circular-secure encryption based on hard learning problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_35CrossRefGoogle Scholar
- 27.Gentry, C., Halevi, S., Peikert, C., Smart, N.P.: Ring switching in BGV-style homomorphic encryption. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 19–37. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32928-9_2CrossRefGoogle Scholar
- 28.Gentry, C., Halevi, S., Peikert, C., Smart, N.P.: Field switching in BGV-style homomorphic encryption. Cryptology ePrint Archive, Report 2012/240 (2012). http://eprint.iacr.org/2012/240
- 29.Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE
**93**(2), 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”CrossRefGoogle Scholar - 30.Albrecht, M.R., Player, R., Scott, S.: On the concrete hardness of learning with errors. Cryptology ePrint Archive, Report 2015/046 (2015). http://eprint.iacr.org/2015/046
- 31.Albrecht, M.R.: On dual lattice attacks against small-secret LWE and parameter choices in HElib and SEAL. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part II. LNCS, vol. 10211, pp. 103–129. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56614-6_4CrossRefGoogle Scholar
- 32.Castryck, W., Iliashenko, I., Vercauteren, F.: Provably weak instances of ring-LWE revisited. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part I. LNCS, vol. 9665, pp. 147–167. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_6CrossRefGoogle Scholar
- 33.Ducas, L., Durmus, A.: Ring-LWE in polynomial rings. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 34–51. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30057-8_3CrossRefGoogle Scholar
- 34.Lyubashevsky, V., Peikert, C., Regev, O.: A toolkit for ring-LWE cryptography. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 35–54. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_3CrossRefGoogle Scholar
- 35.Peikert, C.: An efficient and parallel Gaussian sampler for lattices. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 80–97. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_5CrossRefGoogle Scholar
- 36.Peikert, C.: How (not) to instantiate ring-LWE. In: Zikas, V., De Prisco, R. (eds.) SCN 2016. LNCS, vol. 9841, pp. 411–430. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44618-9_22Google Scholar