Abstract
Trapdoor functions (TDFs) are one of the fundamental building blocks in cryptography. Studying the underlying assumptions and the efficiency of the resulting instantiations is therefore of both theoretical and practical interest. In this work we improve the input-to-image rate of TDFs based on the Diffie-Hellman problem. Specifically, we present:
-
(a)
A rate-1 TDF from the computational Diffie-Hellman (CDH) assumption, improving the result of Garg, Gay, and Hajiabadi [EUROCRYPT 2019], which achieved linear-size outputs but with large constants. Our techniques combine non-binary alphabets and high-rate error-correcting codes over large fields.
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(b)
A rate-1 deterministic public-key encryption satisfying block-source security from the decisional Diffie-Hellman (DDH) assumption. While this question was recently settled by Döttling et al. [CRYPTO 2019], our scheme is conceptually simpler and concretely more efficient. We demonstrate this fact by implementing our construction.
S. Garg—Supported in part from DARPA/ARL SAFEWARE Award W911NF15C0210, AFOSR Award FA9550-15-1-0274, AFOSR Award FA9550-19-1-0200, AFOSR YIP Award, NSF CNS Award 1936826, DARPA and SPAWAR under contract N66001-15-C-4065, a Hellman Award and research grants by the Okawa Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). The views expressed are those of the author and do not reflect the official policy or position of the funding agencies.
G. Malavolta—“Part of this work was done while the author was at Carnegie Mellon University.
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Notes
- 1.
We mention that building rate-1 TDFs satisfying one-wayness alone is trivial. If a TDF \(\mathsf {TDF} \) maps n-bit inputs to \(n^c\)-bit outputs, then define a second TDF whose input is of the form \((\mathsf {x}\in \{0,1\}^n, \mathsf {x}' \in \{0,1\}^{n^{c+1}})\), and the output is \((\mathsf {TDF} (\mathsf {x}), \mathsf {x}')\). While this trivial construction achieves rate-1, it destroys stronger properties such as deterministic-encryption security.
- 2.
By increasing the the size of \(\varSigma \) to e.g. the next power of 2, the bit representation of each symbol in \(\varSigma \) grows by at most one bit, i.e., the rate of such an encoding is \(1 - 1/\lambda \).
- 3.
We can also prove security just by sampling a single r, but the proof will be more complicated.
References
Bellare, M., et al.: Hedged public-key encryption: how to protect against bad randomness. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 232–249. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_14
Bellare, M., Boldyreva, A., O’Neill, A.: Deterministic and efficiently searchable encryption. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 535–552. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74143-5_30
Birrell, E., Chung, K.-M., Pass, R., Telang, S.: Randomness-dependent message security. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 700–720. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36594-2_39
Boldyreva, A., Fehr, S., O’Neill, A.: On notions of security for deterministic encryption, and efficient constructions without random oracles. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 335–359. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_19
Bellare, M., Fischlin, M., O’Neill, A., Ristenpart, T.: Deterministic encryption: definitional equivalences and constructions without random oracles. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 360–378. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_20
Boyle, E., Gilboa, N., Ishai, Y.: Breaking the circuit size barrier for secure computation under DDH. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part I. LNCS, vol. 9814, pp. 509–539. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_19
Bellare, M., Hofheinz, D., Yilek, S.: Possibility and impossibility results for encryption and commitment secure under selective opening. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 1–35. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_1
Brakerski, Z., Lombardi, A., Segev, G., Vaikuntanathan, V.: Anonymous IBE, leakage resilience and circular security from new assumptions. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part I. LNCS, vol. 10820, pp. 535–564. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_20
Döttling, N., Garg, S.: From selective IBE to full IBE and selective HIBE. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part I. LNCS, vol. 10677, pp. 372–408. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_13
Döttling, N., Garg, S.: Identity-based encryption from the diffie-hellman assumption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 537–569. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_18
Döttling, N., Garg, S., Hajiabadi, M., Masny, D.: New constructions of identity-based and key-dependent message secure encryption schemes. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part I. LNCS, vol. 10769, pp. 3–31. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76578-5_1
Döttling, N., Garg, S., Ishai, Y., Malavolta, G., Mour, T., Ostrovsky, R.: Trapdoor hash functions and their applications. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019, Part III. LNCS, vol. 11694, pp. 3–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_1
Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)
Freeman, D.M., Goldreich, O., Kiltz, E., Rosen, A., Segev, G.: More constructions of lossy and correlation-secure trapdoor functions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 279–295. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_17
Garg, S., Gay, R., Hajiabadi, M.: New techniques for efficient trapdoor functions and applications. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019, Part III. LNCS, vol. 11478, pp. 33–63. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_2
Garg, S., Hajiabadi, M.: Trapdoor functions from the computational diffie-hellman assumption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 362–391. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_13
Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: 21st ACM STOC, Seattle, WA, USA, 15–17 May 1989, pp. 25–32. ACM Press (1989)
Impagliazzo, R., Levin, L.A., Luby, M.: Pseudo-random generation from one-way functions (extended abstracts). In: 21st ACM STOC, Seattle, WA, USA, 15–17 May 1989, pp. 12–24. ACM Press (1989)
Lombardi, A., Quach, W., Rothblum, R.D., Wichs, D., Wu, D.J.: New constructions of reusable designated-verifier NIZKs. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019, Part III. LNCS, vol. 11694, pp. 670–700. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_22
Mol, P., Yilek, S.: Chosen-ciphertext security from slightly lossy trapdoor functions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 296–311. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_18
Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: 40th ACM STOC, Victoria, BC, Canada, 17–20 May 2008, pp. 187–196. ACM Press (2008)
Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. SIAM J. Comput. 40(6), 1803–1844 (2011)
Reed, I.S., Solomon, G.: Polynomial codes over certain finite fields. J. Soc. Ind. Appl. Math. 8(2), 300–304 (1960)
Welch, L.R., Berlekamp, E.R.: Error correction for algebraic block codes, December 30 1986. US Patent 4,633,470
Wee, H.: Dual Projective hashing and its applications—lossy trapdoor functions and more. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 246–262. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_16
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Döttling, N., Garg, S., Hajiabadi, M., Liu, K., Malavolta, G. (2019). Rate-1 Trapdoor Functions from the Diffie-Hellman Problem. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11923. Springer, Cham. https://doi.org/10.1007/978-3-030-34618-8_20
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