## Stream ciphers: a practical solution for efficient homomorphic-ciphertext compression.(English)Zbl 1400.94132

Summary: In typical applications of homomorphic encryption, the first step consists for Alice of encrypting some plaintext $$m$$ under Bob’s public key $$\mathsf {pk}$$ and of sending the ciphertext $$c = \mathsf {HE}_{\mathsf {pk}}(m)$$ to some third-party evaluator Charlie. This paper specifically considers that first step, i.e., the problem of transmitting $$c$$ as efficiently as possible from Alice to Charlie. As others suggested before, a form of compression is achieved using hybrid encryption. Given a symmetric encryption scheme $$\mathsf {E}$$, Alice picks a random key $$k$$ and sends a much smaller ciphertext $$c^\prime = (\mathsf {HE}_{\mathsf {pk}}(k), \mathsf {E}_k(m))$$ that Charlie decompresses homomorphically into the original $$c$$ using a decryption circuit $$\mathcal {C}_{{\mathsf {E}^{-1}}}$$. In this paper, we revisit that paradigm in light of its concrete implementation constraints, in particular $$\mathsf {E}$$ is chosen to be an additive IV-based stream cipher. We investigate the performances offered in this context by Trivium, which belongs to the eSTREAM portfolio, and we also propose a variant with 128-bit security: Kreyvium. We show that Trivium, whose security has been firmly established for over a decade, and the new variant Kreyvium has excellent performance. We also describe a second construction, based on exponentiation in binary fields, which is impractical but sets the lowest depth record to $$8$$ for $$128$$-bit security.

### MSC:

 94A60 Cryptography

### Keywords:

stream ciphers; homomorphic cryptography; trivium

### Software:

Trivium; PRINCE; KTANTAN; eSTREAM; FHEW; HElib; KATAN; SHIELD
Full Text:

### References:

 [1] Adj, G; Menezes, A; Oliveira, T; Rodríguez-Henríquez, F, Computing discrete logarithms in $${\mathbb{F}_{3^{6*137}}}$$ using magma, IACR Cryptol. ePrint Arch., 2014, 57, (2014) · Zbl 1400.11161 [2] M. Albrecht, C. Rechberger, T. Schneider, T. Tiessen, M. Zohner, Ciphers for MPC and FHE, in EUROCRYPT, Part I. LNCS, vol. 9056 (Springer, 2015), pp. 430-454 [3] Algorithms, key size and parameters report 2014. Technical report, ENISA (2014) · Zbl 1402.94055 [4] F. Armknecht, V. Mikhalev, On lightweight stream ciphers with shorter internal states, in FSE. LNCS, vol. 9054, (Springer, 2015), pp. 451-470 · Zbl 1382.94050 [5] J. Aumasson, I. Dinur, W. Meier, A. Shamir, Cube testers and key recovery attacks on reduced-round MD6 and Trivium, in FSE. LNCS, vol. 5665 (Springer, 2009), pp. 1-22 · Zbl 1291.94051 [6] S. Babbage, A space/time trade-off in exhaustive search attacks on stream ciphers, in European Convention on Security and Detection, vol. 408, (IEEE, 1995) [7] R. Barbulescu, P. Gaudry, A. Joux, E. Thomé, A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic, in EUROCRYPT. LNCS, vol. 8441 (Springer, 2014), pp. 1-16 · Zbl 1326.11080 [8] M. Bellare, A. Desai, E. Jokipii, P. Rogaway, A concrete security treatment of symmetric encryption, inFOCS, (IEEE Computer Society, 1997), pp. 394-403 [9] C. Berbain, H. Gilbert, On the security of IV dependent stream ciphers, in FSE. LNCS, vol. 4593 (Springer, 2007), pp. 254-273 · Zbl 1186.94423 [10] A. Biryukov, A. Shamir, Cryptanalytic time/memory/data tradeoffs for stream ciphers, in ASIACRYPT. LNCS, vol. 1976 (Springer, 2000), pp. 1-13 · Zbl 0980.94013 [11] M. Bodrato, Towards optimal toom-cook multiplication for univariate and multivariate polynomials in characteristic 2 and 0, in WAIFI. LNCS, vol. 4547 (Springer, 2007), pp. 116-133 · Zbl 1213.68715 [12] J. Borghoff, A. Canteaut, T. Güneysu, E.B. Kavun, M. Knezevic, L.R. Knudsen, G. Leander, V. Nikov, C. Paar, C. Rechberger, P. Rombouts, S.S. Thomsen, T. Yalçin, PRINCE—a low-latency block cipher for pervasive computing applications, in ASIACRYPT. LNCS, vol. 7658 (Springer, 2012), pp. 208-225 · Zbl 1292.94035 [13] J.W. Bos, K.E. Lauter, J. Loftus, M. Naehrig, Improved security for a ring-based fully homomorphic encryption scheme, in IMACC. LNCS, vol. 8308 (Springer, 2013), pp. 45-64 · Zbl 1317.94088 [14] Z. Brakerski, Fully homomorphic encryption without modulus switching from classical GapSVP, in CRYPTO. LNCS, vol. 7417 (Springer, 2012), pp. 868-886 · Zbl 1296.94091 [15] Brakerski, Z; Gentry, C; Vaikuntanathan, V, (leveled) fully homomorphic encryption without bootstrapping, TOCT, 6, 13, (2014) · Zbl 1347.68121 [16] C. Carlet, P. Méaux, Y. Rotella, Boolean functions with restricted input and their robustness; application to the FLIP cipher. IACR Trans. Symmetric Cryptol. 2017(3), 192-227 (2017) [17] S. Carpov, P. Dubrulle, R. Sirdey, Armadillo: a compilation chain for privacy preserving applications, in ACM CCSW (2015) [18] A. Chakraborti, A. Chattopadhyay, M. Hassan, M. Nandi, TriviA: a fast and secure authenticated encryption scheme, in CHES. LNCS, vol. 9293 (Springer, 2015), pp. 330-353 · Zbl 1380.94077 [19] M. Chenal, Q. Tang, On key recovery attacks against existing somewhat homomorphic encryption schemes, in LATINCRYPT. LNCS, vol. 8895 (Springer, 2015), pp. 239-258 · Zbl 1370.94495 [20] J.H. Cheon, J. Coron, J. Kim, M.S. Lee, T. Lepoint, M. Tibouchi, A. Yun, Batch fully homomorphic encryption over the integers, in EUROCRYPT. LNCS, vol. 7881 (Springer, 2013), pp. 315-335 · Zbl 1306.94040 [21] I. Chillotti, N. Gama, M. Georgieva, M. Izabachène, Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds, in ASIACRYPT. LNCS, vol. 10031 (Springer, 2016), pp. 3-33 · Zbl 1384.94044 [22] J. Coron, T. Lepoint, M. Tibouchi, Scale-invariant fully homomorphic encryption over the integers, in PKC. LNCS, vol. 8383 (Springer, 2014), pp. 311-328 · Zbl 1335.94041 [23] N. Courtois, W. Meier, Algebraic attacks on stream ciphers with linear feedback, in EUROCRYPT. LNCS, vol. 2656 (Springer, 2003), pp. 345-359 · Zbl 1038.94525 [24] Cramer, R; Shoup, V, Design and analysis of practical public-key encryption schemes secure against adaptive chosen ciphertext attack, SIAM J. Comput., 33, 167-226, (2003) · Zbl 1045.94013 [25] C. De Cannière, O. Dunkelman, M. Knezevic, KATAN and KTANTAN—a family of small and efficient hardware-oriented block ciphers, in CHES. LNCS, vol. 5747 (Springer, 2009), pp. 272-288 [26] C. De Cannière, J. Lano, B. Preneel, Comments on the rediscovery of time memory data tradeoffs. Technical report, eSTREAM—ECRYPT Stream Cipher Project (2005). www.ecrypt.eu.org/stream/papersdir/040.pdf. Accessed 21 Dec 2017 [27] C. De Cannière, B. Preneel, Trivium, in New Stream Cipher Designs—The eSTREAM Finalists. LNCS, vol. 4986 (Springer, 2008), pp. 244-266 · Zbl 1285.94054 [28] Dinur, I; Liu, Y; Meier, W; Wang, Q, Optimized interpolation attacks on lowmc, IACR Cryptol. ePrint Arch., 2015, 418, (2015) · Zbl 1382.94092 [29] I. Dinur, A. Shamir, Cube attacks on tweakable black box polynomials, in EUROCRYPT. LNCS, vol. 5479 (Springer, 2009), pp. 278-299 · Zbl 1239.94045 [30] Doröz, Y; Hu, Y; Sunar, B, Homomorphic AES evaluation using the modified LTV scheme, Des. Codes Cryptogr., 80, 333-358, (2016) · Zbl 1402.94055 [31] Y. Doröz, A. Shahverdi, T. Eisenbarth, B. Sunar, Toward practical homomorphic evaluation of block ciphers using Prince, in WAHC. LNCS, vol. 8438 (Springer, 2014), pp. 208-220 [32] L. Ducas, D. Micciancio, FHEW: bootstrapping homomorphic encryption in less than a second, in EUROCRYPT. LNCS, vol. 9056 (Springer, 2015), pp. 617-640 · Zbl 1370.94509 [33] S. Duval, V. Lallemand, Y. Rotella, Cryptanalysis of the FLIP family of stream ciphers, in CRYPTO. LNCS, vol. 9814 (Springer, 2016), pp. 457-475 · Zbl 1384.94059 [34] ECRYPT—European network of excellence in cryptology: the eSTREAM stream cipher project (2005). http://www.ecrypt.eu.org/stream/. Accessed 21 Dec 2017 [35] Fan, J; Vercauteren, F, Somewhat practical fully homomorphic encryption, IACR Cryptol. ePrint Arch., 2012, 144, (2012) [36] S. Fau, R. Sirdey, C. Fontaine, C. Aguilar, G. Gogniat, Towards practical program execution over fully homomorphic encryption schemes, in IEEE International Conference on P2P, Parallel, Grid, Cloud and Internet Computing, (2013), pp. 284-290 [37] P. Fouque, T. Vannet, Improving key recovery to 784 and 799 rounds of Trivium using optimized cube attacks, in FSE. LNCS, vol. 8424 (Springer, 2013), pp. 502-517 · Zbl 1321.94058 [38] T. Fuhr, B. Minaud, Match box meet-in-the-middle attack against KATAN, inFSE. LNCS, vol. 8540 (Springer, 2014), pp. 61-81 · Zbl 1382.94106 [39] C. Gentry, Fully homomorphic encryption using ideal lattices, in STOC, (ACM, 2009), pp. 169-178 · Zbl 1304.94059 [40] C. Gentry, S. Halevi, N.P. Smart, Homomorphic evaluation of the AES circuit, in CRYPTO. LNCS, vol. 7417 (Springer, 2012), pp. 850-867 · Zbl 1296.94117 [41] C. Gentry, A. Sahai, B. Waters, Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based, in CRYPTO. LNCS, vol. 8042 (Springer, 2013), pp. 75-92 · Zbl 1310.94148 [42] J.D. Golic, Cryptanalysis of alleged A5 stream cipher, in EUROCRYPT. LNCS, vol. 1233 (Springer, 1997), pp. 239-255 [43] T. Graepel, K.E. Lauter, M. Naehrig, ML confidential: machine learning on encrypted data, in ICISC. LNCS, vol. 7839 (Springer, 2012), pp. 1-21 · Zbl 1293.68110 [44] R. Granger, T. Kleinjung, J. Zumbrägel, Breaking ‘128-bit secure’ supersingular binary curves—(or how to solve discrete logarithms in $${\mathbb{F}_{2^{4 · 1223}}}$$ and $${\mathbb{F}_{2^{12 · 367}}}$$), in CRYPTO, Part II. LNCS, vol. 8617 (Springer, 2014), pp. 126-145 · Zbl 1334.94080 [45] S. Halevi, V. Shoup, Algorithms in HElib, in CRYPTO, Part I. LNCS, vol. 8616 (Springer, 2014), pp. 554-571 · Zbl 1343.94061 [46] S. Halevi, V. Shoup, Bootstrapping for HElib, in EUROCRYPT. LNCS, vol. 9056 (Springer, 2015), pp. 641-670 · Zbl 1370.94516 [47] Herranz, J; Hofheinz, D; Kiltz, E, Some (in)sufficient conditions for secure hybrid encryption, Inf. Comput., 208, 1243-1257, (2010) · Zbl 1205.68148 [48] J. Hong, P. Sarkar, New applications of time memory data tradeoffs, in ASIACRYPT. LNCS, vol. 3788 (Springer, 2005), pp. 353-372 · Zbl 1154.68395 [49] T. Iwata, New block cipher modes of operation with beyond the birthday bound security, in FSE. LNCS, vol. 4047 (Springer, 2006), pp. 310-327 · Zbl 1234.94049 [50] T. Jakobsen, L.R. Knudsen, The interpolation attack on block ciphers, in FSE. LNCS, vol. 1267 (Springer, 1997), pp. 28-40 · Zbl 1385.94047 [51] A. Joux, C. Pierrot, Improving the polynomial time precomputation of Frobenius representation discrete logarithm algorithms—simplified setting for small characteristic finite fields, in ASIACRYPT, Part I. LNCS, vol. 8873 (Springer, 2014), pp. 378-397 · Zbl 1306.94064 [52] J. Katz, Y. Lindell, Introduction to Modern Cryptography, 2nd edition. Chapman and Hall/CRC Press, Boca Raton (2014) · Zbl 1143.94001 [53] A. Khedr, G. Gulak, V. Vaikuntanathan, SHIELD: scalable homomorphic implementation of encrypted data-classifiers. IEEE Trans. Comput. 65(9), 2848-2858 (2016) · Zbl 1360.68435 [54] S. Knellwolf, W. Meier, M. Naya-Plasencia, conditional differential cryptanalysis of NLFSR-based cryptosystems, inASIACRYPT. LNCS, vol. 6477 (Springer, 2010), pp. 130-145 · Zbl 1253.94056 [55] S. Knellwolf, W. Meier, M, Naya-Plasencia, Conditional differential cryptanalysis of Trivium and KATAN, in SAC. LNCS, vol. 7118 (Springer, 2011), pp. 200-212 · Zbl 1292.94095 [56] K. Lauter, A. López-Alt, M. Naehrig, Private computation on encrypted genomic data, in LATINCRYPT. LNCS (2014) · Zbl 1378.94053 [57] T. Lepoint, M. Naehrig, A comparison of the homomorphic encryption schemes FV and YASHE, in AFRICACRYPT. LNCS, vol. 8469 (Springer, 2014), pp. 318-335 · Zbl 1318.94071 [58] T. Lepoint, P. Paillier, On the minimal number of bootstrappings in homomorphic circuits, in WAHC. LNCS, vol. 7862 (Springer, 2013), pp. 189-200 [59] M. Liu, Degree evaluation of NFSR-based cryptosystems, in CRYPTO. LNCS, vol. 10402 (Springer, 2017) · Zbl 1406.94073 [60] P. Méaux, A. Journault, F.X. Standaert, C. Carlet, Towards stream ciphers for efficient FHE with low-noise ciphertexts, in EUROCRYPT. LNCS, vol. 9665 (Springer, 2016), pp. 311-343 · Zbl 1384.94086 [61] A. Maximov, A. Biryukov, Two trivial attacks on Trivium, in SAC, vol. 4876 (Springer, 2007), pp. 36-55 · Zbl 1154.94418 [62] M. Naehrig, K.E. Lauter, V. Vaikuntanathan, Can homomorphic encryption be practical? in ACM CCSW, (ACM, 2011), pp. 113-124 [63] National Institute of Standards and Technology, Recommendation for block cipher modes of operation. NIST Special Publication 800-38A (2001) [64] M. Paindavoine, B. Vialla, Minimizing the number of bootstrappings in fully homomorphic encryption, in SAC 2015. LNCS, vol. 9566 (Springer, 2016), pp. 25-43 · Zbl 1339.94057 [65] Pincin, A, A new algorithm for multiplication in finite fields, IEEE Trans. Comput., 38, 1045-1049, (1989) · Zbl 0693.12015 [66] C. Rechberger, The FHEMPCZK-cipher zoo. Presented at the FSE 2016 rump session (2016). http://fse.2016.rump.cr.yp.to/. Accessed 21 Dec 2017 [67] P. Rogaway, Evaluation of some block cipher modes of operation. Cryptrec (2011). http://web.cs.ucdavis.edu/ rogaway/papers/modes.pdf. Accessed 21 Dec 2017 [68] Smart, NP; Vercauteren, F, Fully homomorphic SIMD operations, Des. Codes Cryptogr., 71, 57-81, (2014) · Zbl 1323.94140 [69] Y. Todo, T. Isobe, Y. Hao, W. Meier, Cube attacks on non-blackbox polynomials based on division property, in CRYPTO. LNCS, vol. 10402 (Springer, 2017) · Zbl 1406.94081 [70] K. Yasuda, A new variant of PMAC: beyond the birthday bound, in CRYPTO. LNCS, vol. 6841 (Springer, 2011), pp. 596-609 · Zbl 1290.94139
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