A counterexample to the chain rule for conditional HILL entropy
published
yes
Stephan
Krenn
author 329FCCF0-F248-11E8-B48F-1D18A9856A87
Krzysztof Z
Pietrzak
author 3E04A7AA-F248-11E8-B48F-1D18A9856A87
Akshay
Wadia
author
Daniel
Wichs
author
KrPi
department
Provable Security for Physical Cryptography
project
Most entropy notions H(.) like Shannon or min-entropy satisfy a chain rule stating that for random variables X,Z, and A we have H(X|Z,A)≥H(X|Z)−|A|. That is, by conditioning on A the entropy of X can decrease by at most the bitlength |A| of A. Such chain rules are known to hold for some computational entropy notions like Yao’s and unpredictability-entropy. For HILL entropy, the computational analogue of min-entropy, the chain rule is of special interest and has found many applications, including leakage-resilient cryptography, deterministic encryption, and memory delegation. These applications rely on restricted special cases of the chain rule. Whether the chain rule for conditional HILL entropy holds in general was an open problem for which we give a strong negative answer: we construct joint distributions (X,Z,A), where A is a distribution over a single bit, such that the HILL entropy H HILL (X|Z) is large but H HILL (X|Z,A) is basically zero.
Our counterexample just makes the minimal assumption that NP⊈P/poly. Under the stronger assumption that injective one-way function exist, we can make all the distributions efficiently samplable.
Finally, we show that some more sophisticated cryptographic objects like lossy functions can be used to sample a distribution constituting a counterexample to the chain rule making only a single invocation to the underlying object.
/download/1479/5012/IST-2017-766-v1+1_678.pdf
application/pdfno
cc_by
Springer2016
eng
Computational Complexity10.1007/s00037-015-0120-9
253567 - 605
/record/2940
S. Krenn, K. Z. Pietrzak, A. Wadia, and D. Wichs, “A counterexample to the chain rule for conditional HILL entropy,” <i>Computational Complexity</i>, vol. 25, no. 3, pp. 567–605, 2016.
Krenn, Stephan, et al. “A Counterexample to the Chain Rule for Conditional HILL Entropy.” <i>Computational Complexity</i>, vol. 25, no. 3, Springer, 2016, pp. 567–605, doi:<a href="https://doi.org/10.1007/s00037-015-0120-9">10.1007/s00037-015-0120-9</a>.
Krenn, Stephan, Krzysztof Z Pietrzak, Akshay Wadia, and Daniel Wichs. “A Counterexample to the Chain Rule for Conditional HILL Entropy.” <i>Computational Complexity</i> 25, no. 3 (2016): 567–605. <a href="https://doi.org/10.1007/s00037-015-0120-9">https://doi.org/10.1007/s00037-015-0120-9</a>.
Krenn S, Pietrzak KZ, Wadia A, Wichs D. A counterexample to the chain rule for conditional HILL entropy. <i>Computational Complexity</i>. 2016;25(3):567-605. doi:<a href="https://doi.org/10.1007/s00037-015-0120-9">10.1007/s00037-015-0120-9</a>
S. Krenn, K.Z. Pietrzak, A. Wadia, D. Wichs, Computational Complexity 25 (2016) 567–605.
Krenn, S., Pietrzak, K. Z., Wadia, A., & Wichs, D. (2016). A counterexample to the chain rule for conditional HILL entropy. <i>Computational Complexity</i>, <i>25</i>(3), 567–605. <a href="https://doi.org/10.1007/s00037-015-0120-9">https://doi.org/10.1007/s00037-015-0120-9</a>
Krenn S, Pietrzak KZ, Wadia A, Wichs D. 2016. A counterexample to the chain rule for conditional HILL entropy. Computational Complexity. 25(3), 567–605.
14792018-12-11T11:52:16Z2019-11-14T08:42:27Z