**Non-Malleable Extractors and Symmetric Key Cryptography from Weak Secrets**

*Yevgeniy Dodis and Daniel Wichs*

**Abstract: **We study the question of basing symmetric key cryptography on weak secrets. In this setting, Alice and Bob share an $n$-bit secret $W$, which might not be uniformly random, but the adversary has at least $k$ bits of uncertainty about it (formalized using conditional min-entropy). Since standard symmetric-key primitives require uniformly random secret keys, we would like to construct an authenticated key agreement protocol in which Alice and Bob use $W$ to agree on a nearly uniform key $R$, by communicating over a public channel controlled by an active adversary Eve. We study this question in the information theoretic setting where the attacker is computationally unbounded. We show that single-round (i.e. one message) protocols do not work when $k < n/2$, and require poor parameters even when
$n/2< k << n$.

On the other hand, for arbitrary values of $k$, we design a communication efficient two-round (challenge-response) protocol extracting nearly $k$ random bits. This dramatically improves the only previously known protocol of Renner and Wolf~\cite{RennerW03}, which required $O(\lambda)$ rounds where $\lambda$ is the security parameter. Our solution takes a new approach by studying and constructing ``non-malleable'' seeded randomness extractors --- if an attacker sees a random seed $X$ and comes up with an arbitrarily related seed $X'$, then we bound the relationship between $R= \Ext(W;X)$ and $R' = \Ext(W;X')$.

We also extend our two-round key agreement protocol to the ``fuzzy'' setting, where Alice and Bob share ``close'' (but not equal) secrets $W_A$ and $W_B$, and to the Bounded Retrieval Model (BRM) where the size of the secret $W$ is huge.

**Category / Keywords: **secret-key cryptography / Information Theoretic Security, Key Agreement, Weak Secrets

**Publication Info: **Full version of STOC 2009 paper.

**Date: **received 28 Nov 2008, last revised 5 Apr 2009

**Contact author: **wichs at cs nyu edu

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **The previous title of this paper was "One-round Authenticated Key Agreement from Weak Secrets". The technical content of the paper has not changed, but we corrected our terminology. In our previous version, we referred to our two message (challenge-response) protocol as "one-round" of interaction. Although we were explicit and consistent about the meaning of "round", the anonymous STOC reviewers correctly pointed out that this terminology is non-standard and confusing and thus we have corrected it (and changed the title) for the final version. We now use "round" to refer to a single exchanged message.

**Version: **20090405:211039 (All versions of this report)

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