### On the Streaming Indistinguishability of a Random Permutation and a Random Function

Itai Dinur

##### Abstract

An adversary with $S$ bits of memory obtains a stream of $Q$ elements that are uniformly drawn from the set $\{1,2,\ldots,N\}$, either with or without replacement. This corresponds to sampling $Q$ elements using either a random function or a random permutation. The adversary's goal is to distinguish between these two cases. This problem was first considered by Jaeger and Tessaro (EUROCRYPT 2019), which proved that the adversary's advantage is upper bounded by $\sqrt{Q \cdot S/N}$. Jaeger and Tessaro used this bound as a streaming switching lemma which allowed proving that known time-memory tradeoff attacks on several modes of operation (such as counter-mode) are optimal up to a factor of $O(\log N)$ if $Q \cdot S \approx N$. However, the bound's proof assumed an unproven combinatorial conjecture. Moreover, if $Q \cdot S \ll N$ there is a gap between the upper bound of $\sqrt{Q \cdot S/N}$ and the $Q \cdot S/N$ advantage obtained by known attacks. In this paper, we prove a tight upper bound (up to poly-logarithmic factors) of $O(\log Q \cdot Q \cdot S/N)$ on the adversary's advantage in the streaming distinguishing problem. The proof does not require a conjecture and is based on a hybrid argument that gives rise to a reduction from the unique-disjointness communication complexity problem to streaming.

Note: Minor changes

##### Metadata
Available format(s)
Category
Secret-key cryptography
Publication info
A minor revision of an IACR publication in Eurocrypt 2020
Keywords
Streaming algorithmtime-memory tradeoffcommunication complexityprovable securityswitching lemmamode of operation.
Contact author(s)
dinuri @ cs bgu ac il
History
2020-07-18: last of 2 revisions
2019-04-22: received
See all versions
Short URL
https://ia.cr/2019/413
License

CC BY

BibTeX

@misc{cryptoeprint:2019/413,
author = {Itai Dinur},
title = {On the Streaming Indistinguishability of a Random Permutation and a Random Function},
howpublished = {Cryptology ePrint Archive, Paper 2019/413},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/413}},
url = {https://eprint.iacr.org/2019/413}
}

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