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Pseudorandom Correlation Functions from Variable-Density LPN, Revisited

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

Pseudorandom correlation functions (PCF), introduced in the work of (Boyle et al., FOCS 2020), allow two parties to locally generate, from short correlated keys, a near-unbounded amount of pseudorandom samples from a target correlation. PCF is an extremely appealing primitive in secure computation, where they allow to confine all preprocessing phases of all future computations two parties could want to execute to a single short interaction with low communication and computation, followed solely by offline computations. Beyond introducing the notion, Boyle et al. gave a candidate construction, using a new variable-density variant of the learning parity with noise (LPN) assumption. Then, to provide support for this new assumption, the authors showed that it provably resists a large class of linear attacks, which captures in particular all known attacks on LPN.

In this work, we revisit the analysis of the VDLPN assumption. We make two key contributions:

  • First, we observe that the analysis of Boyle et al. is purely asymptotic: they do not lead to any concrete and efficient PCF instantiation within the bounds that offer security guarantees. To improve this state of affairs, we combine a new variant of a VDLPN assumption with an entirely new, much tighter security analysis, which we further tighten using extensive computer simulations to optimize parameters. This way, we manage to obtain for the first time a set of provable usable parameters (under a simple combinatorial conjecture which is easy to verify experimentally), leading to a concretely efficient PCF resisting all linear tests.

  • Second, we identify a flaw in the security analysis of Boyle et al., which invalidates their proof that VDLPN resists linear attacks. Using several new non-trivial arguments, we repair the proof and fully demonstrate that VDLPN resists linear attack; our new analysis is more involved than the original (flawed) analysis.

Our parameters set leads to PCFs with keys around 3 MB allowing \(\sim 500\) evaluations per second on one core of a standard laptop for 110 bits of security; these numbers can be improved to 350 kB keys and \(\sim 3950\) evaluations/s using a more aggressive all-prefix variant. All numbers are quite tight: only within a factor 3 of the best bounds one could heuristically hope for.

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Notes

  1. 1.

    E.g. A laptop equipped with an Intel i5 2540M processor can compute an RSA decryption in 1.4ms of amortized time.

  2. 2.

    The choice of 80 bits of security is more conservative than it appears: it means that an adversary will have to compute \(2^{80}\) inner products with a length-\(2^{30}\) vector to detect a \(2^{-80}\) bias in the output. In terms of bit-security, this corresponds to at least 110 bits of security.

  3. 3.

    E.g. if Z is 0 with probability 1/2, and 10 else, then \(\mathbb {E} [Z] = 5 \in [4,5]\), but \(\mathbb {E} [|Z-5|] = 5 > 5-4\).

  4. 4.

    The improvement comes from a better estimation of the \(\beta \) parameter on one hand, but also from the better inherent quality of the new proof. In fact, we can consider the same calculation as before, but with \(l= 2^7\) and \(\beta = 0.44\). This is a non-computer-optimized, provable value of \(\beta \) using \(i^* = 7\). With this value, we get \(w \approx 13 000\), which is already a big gain from the previous method: this is already several orders of magnitudes better than the previous method, though still not practical.

References

  1. Al Jabri, A.A.: A statistical decoding algorithm for general linear block codes (2001)

    Google Scholar 

  2. Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. Second Series 19(3), 357–367 (1967)

    MathSciNet  MATH  Google Scholar 

  3. Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Orrù, M.: Homomorphic secret sharing: Optimizations and applications. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D., (eds.) ACM CCS 2017, pp. 2105–2122. ACM Press (October/November 2017)

    Google Scholar 

  4. Boyle, E., et al.: Efficient two-round OT extension and silent non-interactive secure computation. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J., (eds.) ACM CCS 2019, pp. 291–308. ACM Press (November 2019)

    Google Scholar 

  5. Boyle, E., et al.: Efficient pseudorandom correlation generators: silent ot extension and more. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 489–518. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_16

    Chapter  Google Scholar 

  6. Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Scholl, P.: Correlated pseudorandom functions from variable-density LPN. In: 61st FOCS, pages 1069–1080. IEEE Computer Society Press (November 2020)

    Google Scholar 

  7. Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Scholl, P.: Efficient pseudorandom correlation generators from ring-LPN. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 387–416. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_14

    Chapter  Google Scholar 

  8. Boyle, E., Couteau, G., Gilboa, N., Ishai, Y.: Compressing vector OLE. In: Lie, D., Mannan, M., Backes, M., Wang, X., (eds.) ACM CCS 2018, pp. 896–912. ACM Press (October 2018)

    Google Scholar 

  9. Boyle, E., Goldwasser, S., Ivan, I.: Functional signatures and pseudorandom functions. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 501–519. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54631-0_29

    Chapter  Google Scholar 

  10. Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 337–367. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_12

    Chapter  Google Scholar 

  11. Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing: Improvements and extensions. In: Weippl, E.R., Katzenbeisser, S., Kruegel, C., Myers, A.C., Halevi, S., (eds.) ACM CCS 2016, pp. 1292–1303. ACM Press (October 2016)

    Google Scholar 

  12. Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in 2n/20: How 1 + 0 improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_31

    Chapter  MATH  Google Scholar 

  13. Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. In: 32nd ACM STOC, pp. 435–440. ACM Press (May 2000)

    Google Scholar 

  14. Bernstein, D.J., Lange, T., Peters, C.: Smaller decoding exponents: ball-collision decoding. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 743–760. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_42

    Chapter  Google Scholar 

  15. Bellare, M., Micciancio, D.: A new paradigm for collision-free hashing: incrementality at reduced cost. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 163–192. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-69053-0_13

    Chapter  Google Scholar 

  16. Both, L., May, A.: Decoding linear codes with high error rate and its impact for LPN security. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 25–46. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_2

    Chapter  Google Scholar 

  17. Bogos, S., Tramer, F., Vaudenay, S.: On solving lpn using bkw and variants (2016)

    Google Scholar 

  18. Bogos, S., Vaudenay, S.: Optimization of \(\sf LPN\) solving algorithms. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 703–728. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_26

    Chapter  Google Scholar 

  19. Boneh, D., Waters, B.: constrained pseudorandom functions and their applications. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8270, pp. 280–300. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42045-0_15

    Chapter  Google Scholar 

  20. Couteau, G., Rindal, P., Raghuraman, S.: Silver: silent VOLE and oblivious transfer from hardness of decoding structured LDPC Codes. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12827, pp. 502–534. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84252-9_17

    Chapter  Google Scholar 

  21. Debris-Alazard, T., Tillich, J.-P.: Statistical decoding (2017)

    Google Scholar 

  22. Dodis, Y., Halevi, S., Rothblum, R.D., Wichs, D.: Spooky encryption and its applications. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 93–122. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_4

    Chapter  Google Scholar 

  23. Damgård, I., Pastro, V., Smart, N., Zakarias, S.: Multiparty computation from somewhat homomorphic encryption. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 643–662. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_38

    Chapter  Google Scholar 

  24. Esser, A., Kübler, R., May, A.: LPN decoded. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 486–514. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_17

    Chapter  Google Scholar 

  25. Fossorier, M.P.C., Kobara, K., Imai, H.: Modeling bit flipping decoding based on nonorthogonal check sums with application to iterative decoding attack of mceliece cryptosystem (2006)

    Google Scholar 

  26. Finiasz, M., Sendrier, N.: Security bounds for the design of code-based cryptosystems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 88–105. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_6

    Chapter  Google Scholar 

  27. Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gilboa, N., Ishai, Y.: Distributed point functions and their applications. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 640–658. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_35

    Chapter  Google Scholar 

  29. Guo, Q., Johansson, T., Löndahl, C.: Solving LPN using covering codes. J. Cryptol. 33(1), 1–33 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kirchner, P.: Improved generalized birthday attack. Cryptology ePrint Archive, Report 2011/377 (2011). https://eprint.iacr.org/2011/377

  31. Kamath, A., Motwani, R., Palem, K.V., Spirakis, P.G.: Tail bounds for occupancy and the satisfiability threshold conjecture. In: 35th FOCS, pp. 592–603. IEEE Computer Society Press (November 1994)

    Google Scholar 

  32. Kiayias, A., Papadopoulos, S., Triandopoulos, N., Zacharias, T.: Delegatable pseudorandom functions and applications. In: Sadeghi, A.-R., Gligor, V.D., Yung, M., (eds.) ACM CCS 2013, pp. 669–684. ACM Press (November 2013)

    Google Scholar 

  33. Levieil, É., Fouque, P.-A.: An improved LPN algorithm. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 348–359. Springer, Heidelberg (2006). https://doi.org/10.1007/11832072_24

    Chapter  Google Scholar 

  34. Lyubashevsky, V.: The parity problem in the presence of noise, decoding random linear codes, and the subset sum problem (2005)

    Google Scholar 

  35. McDiarmid. C.: On the method of bounded differences. In: Simons, J., (ed.)"Survey in Combinatorics," London Mathematical Society Lecture Notes, vol. 141 (1989)

    Google Scholar 

  36. May, A., Meurer, A., Thomae, E.: Decoding random linear codes in \(\tilde{\cal{O}}(2^{0.054n})\). In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 107–124. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_6

    Chapter  MATH  Google Scholar 

  37. May, A., Ozerov, I.: On computing nearest neighbors with applications to decoding of binary linear codes. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 203–228. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_9

    Chapter  Google Scholar 

  38. Münch, J.-P., Schneider, T., Yalame, H.: Vasa: Vector aes instructions for security applications. In: Annual Computer Security Applications Conference, pp. 131–145 (2021)

    Google Scholar 

  39. Orlandi, C., Scholl, P., Yakoubov, S.: The rise of paillier: homomorphic secret sharing and public-key silent OT. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12696, pp. 678–708. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_24

    Chapter  Google Scholar 

  40. Overbeck, R.: Statistical decoding revisited. In: Batten, L.M., Safavi-Naini, R. (eds.) ACISP 2006. LNCS, vol. 4058, pp. 283–294. Springer, Heidelberg (2006). https://doi.org/10.1007/11780656_24

    Chapter  Google Scholar 

  41. Prange, E.: The use of information sets in decoding cyclic codes (1962)

    Google Scholar 

  42. Saarinen, M.-J.O.: Linearization attacks against syndrome based hashes. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 1–9. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-77026-8_1

    Chapter  Google Scholar 

  43. Schoppmann, P., Gascón, A., Reichert, L., Raykova, M.: Distributed vector-OLE: Improved constructions and implementation. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J., (eds.) ACM CCS 2019, pp. 1055–1072. ACM Press (November 2019)

    Google Scholar 

  44. Shpilka, A.: Constructions of low-degree and error-correcting \(\varepsilon \)-biased generators (2009)

    Google Scholar 

  45. Stern, J.: A method for finding codewords of small weight (1988)

    Google Scholar 

  46. Wagner, D.: A generalized birthday problem. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 288–303. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  47. Zichron, L.: Locally computable arithmetic pseudorandom generators (2017)

    Google Scholar 

  48. Zhang, B., Jiao, L., Wang, M.: Faster algorithms for solving LPN. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. Part I, volume 9665 of LNCS, pp. 168–195. Springer, Heidelberg (2016)

    Chapter  Google Scholar 

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Acknowledgements

The first author acknowledges the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-20-CE39-0001 (project SCENE), and of the France 2030 ANR Project ANR-22-PECY-003 SecureCompute. The second author was supported by the DIM RFSI grant LICENCED.

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Authors and Affiliations

  1. CNRS, IRIF, Université de Paris, Paris, France

    Geoffroy Couteau

  2. IRIF, INRIA, Université de Paris, Paris, France

    Clément Ducros

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  1. Geoffroy Couteau
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Correspondence to Geoffroy Couteau .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Couteau, G., Ducros, C. (2023). Pseudorandom Correlation Functions from Variable-Density LPN, Revisited. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_8

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