The learning with rounding (LWR) problem, introduced by Banerjee, Peikert and Rosen at EUROCRYPT ’12, is a variant of learning with errors (LWE), where one replaces random errors with deterministic rounding. The LWR problem was shown to be as hard as LWE for a setting of parameters where the modulus and modulus-to-error ratio are super-polynomial. In this work we resolve the main open problem and give a new reduction that works for a larger range of parameters, allowing for a polynomial modulus and modulus-to-error ratio. In particular, a smaller modulus gives us greater efficiency, and a smaller modulus-to-error ratio gives us greater security, which now follows from the worst-case hardness of GapSVP with polynomial (rather than super-polynomial) approximation factors. As a tool in the reduction, we show that there is a “lossy mode” for the LWR problem, in which LWR samples only reveal partial information about the secret. This property gives us several interesting new applications, including a proof that LWR remains secure with weakly random secrets of sufficient min-entropy, and very simple constructions of deterministic encryption, lossy trapdoor functions and reusable extractors. Our approach is inspired by a technique of Goldwasser et al. from ICS ’10, which implicitly showed the existence of a “lossy mode” for LWE. By refining this technique, we also improve on the parameters of that work to only requiring a polynomial (instead of super-polynomial) modulus and modulus-to-error ratio.
This work was partly funded by the European Research Council under ERC Starting Grant 259668-PSPC and ERC Advanced Grant 321310-PERCY. Parts of this work were done while the second authors was at IST Austria, and the last author was at IBM Research, T.J. Watson. A full version of this paper is available online .
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CRYPTO: International Cryptology Conference
Santa Barbara, CA, United States
2013-08-18 – 2013-08-22
Alwen JF, Krenn S, Pietrzak KZ, Wichs D. Learning with rounding, revisited: New reduction properties and applications. 2013;8042(1):57-74. doi:10.1007/978-3-642-40041-4_4
Alwen, J. F., Krenn, S., Pietrzak, K. Z., & Wichs, D. (2013). Learning with rounding, revisited: New reduction properties and applications. Presented at the CRYPTO: International Cryptology Conference, Santa Barbara, CA, United States: Springer. https://doi.org/10.1007/978-3-642-40041-4_4
Alwen, Joel F, Stephan Krenn, Krzysztof Z Pietrzak, and Daniel Wichs. “Learning with Rounding, Revisited: New Reduction Properties and Applications.” Lecture Notes in Computer Science. Springer, 2013. https://doi.org/10.1007/978-3-642-40041-4_4.
J. F. Alwen, S. Krenn, K. Z. Pietrzak, and D. Wichs, “Learning with rounding, revisited: New reduction properties and applications,” vol. 8042, no. 1. Springer, pp. 57–74, 2013.
Alwen JF, Krenn S, Pietrzak KZ, Wichs D. 2013. Learning with rounding, revisited: New reduction properties and applications. 8042(1), 57–74.
Alwen, Joel F., et al. Learning with Rounding, Revisited: New Reduction Properties and Applications. Vol. 8042, no. 1, Springer, 2013, pp. 57–74, doi:10.1007/978-3-642-40041-4_4.
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