Lossy functions do not amplify well

K.Z. Pietrzak, A. Rosen, G. Segev, in:, Springer, 2012, pp. 458–475.

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Conference Paper | Published | English
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Series Title
LNCS
Abstract
We consider the problem of amplifying the "lossiness" of functions. We say that an oracle circuit C*: {0,1} m → {0,1}* amplifies relative lossiness from ℓ/n to L/m if for every function f:{0,1} n → {0,1} n it holds that 1 If f is injective then so is C f. 2 If f has image size of at most 2 n-ℓ, then C f has image size at most 2 m-L. The question is whether such C* exists for L/m ≫ ℓ/n. This problem arises naturally in the context of cryptographic "lossy functions," where the relative lossiness is the key parameter. We show that for every circuit C* that makes at most t queries to f, the relative lossiness of C f is at most L/m ≤ ℓ/n + O(log t)/n. In particular, no black-box method making a polynomial t = poly(n) number of queries can amplify relative lossiness by more than an O(logn)/n additive term. We show that this is tight by giving a simple construction (cascading with some randomization) that achieves such amplification.
Publishing Year
Date Published
2012-05-04
Acknowledgement
We would like to thank Oded Goldreich and Omer Rein- gold for discussions at an early stage of this project, and Scott Aaronson for clarifications regarding the collision problem.
Volume
7194
Page
458 - 475
Conference
TCC: Theory of Cryptography Conference
Conference Location
Taormina, Sicily, Italy
Conference Date
2012-03-19 – 2012-03-21
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Cite this

Pietrzak KZ, Rosen A, Segev G. Lossy functions do not amplify well. In: Vol 7194. Springer; 2012:458-475. doi:10.1007/978-3-642-28914-9_26
Pietrzak, K. Z., Rosen, A., & Segev, G. (2012). Lossy functions do not amplify well (Vol. 7194, pp. 458–475). Presented at the TCC: Theory of Cryptography Conference, Taormina, Sicily, Italy: Springer. https://doi.org/10.1007/978-3-642-28914-9_26
Pietrzak, Krzysztof Z, Alon Rosen, and Gil Segev. “Lossy Functions Do Not Amplify Well,” 7194:458–75. Springer, 2012. https://doi.org/10.1007/978-3-642-28914-9_26.
K. Z. Pietrzak, A. Rosen, and G. Segev, “Lossy functions do not amplify well,” presented at the TCC: Theory of Cryptography Conference, Taormina, Sicily, Italy, 2012, vol. 7194, pp. 458–475.
Pietrzak KZ, Rosen A, Segev G. 2012. Lossy functions do not amplify well. TCC: Theory of Cryptography Conference, LNCS, vol. 7194. 458–475.
Pietrzak, Krzysztof Z., et al. Lossy Functions Do Not Amplify Well. Vol. 7194, Springer, 2012, pp. 458–75, doi:10.1007/978-3-642-28914-9_26.

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