Motivated by the classical problem of privacy amplification, Dodis and Wichs [in Proceedings of the 41st Annual ACM Symposium on Theory of Computing, 2009, pp. 601--610] introduced the notion of a nonmalleable extractor, significantly strengthening the notion of a strong extractor. A nonmalleable extractor is a function $\mathsf{nmExt}:\{0,1\}^n\times\{0,1\}^d\to\{0,1\}^m$ that takes two inputs---a weak source $W$ and a uniform (independent) seed $S$ ---and outputs a string $\mathsf{nmExt}(W,S)$ that is nearly uniform given the seed $S$ as well as the value $\mathsf{nmExt}(W,S')$ for any seed $S'\neq S$ that may be determined as an arbitrary function of $S$ . The first explicit construction of a nonmalleable extractor was recently provided by Dodis et al. [Privacy Amplification and Non-malleable Extractors via Character Sums, preprint, arXiv:1102.5415 [cs.CR], 2011]. Their extractor works for any weak source with min-entropy rate $1/2+\delta$ , where $\delta>0$ is an arbitrary constant and outputs up to a linear number of bits but suffers from two drawbacks. First, the length of its seed is linear in the length of the weak source (which leads to privacy amplification protocols with high communication complexity). Second, the construction is conditional: when outputting more than a logarithmic number of bits (as required for privacy amplification protocols), its efficiency relies on a longstanding conjecture on the distribution of prime numbers. In this paper we present an unconditional construction of a nonmalleable extractor with short seeds. For any integers $n$ and $d$ such that $2.01\cdot\log n\leq d\leq n$ , we present an explicit construction of a nonmalleable extractor $\mathsf{nmExt}\colon\{0,1\}^n\times\{0,1\}^d\to\{0,1\}^m$ , with $m=\Omega(d)$ and error exponentially small in $m$ . The extractor works for any weak source with min-entropy rate $1/2+\delta$ , where $\delta>0$ is an arbitrary constant. Moreover, our extractor in fact satisfies an even more general notion of nonmalleability: its output $\mathsf{nmExt}(W,S)$ is nearly uniform given the seed $S$ as well as the values $\mathsf{nmExt}(W,S_1),\dots,\mathsf{nmExt}(W,S_t)$ for several seeds $S_1,\dots,S_t$ that may be determined as an arbitrary function of $S$ , as long as $S\notin\{S_1,\dots,S_t\}$ . By instantiating the framework of Dodis and Wichs with our nonmalleable extractor, we obtain the first 2-round privacy amplification protocol for min-entropy rate $1/2+\delta$ with asymptotically optimal entropy loss and polylogarithmic communication complexity. This improves the previously known 2-round privacy amplification protocols: the protocol of Dodis and Wichs, whose entropy loss is not asymptotically optimal, and the protocol of Dodis et al., whose communication complexity is linear.Motivated by the classical problem of privacy amplification, Dodis and Wichs [in Proceedings of the 41st Annual ACM Symposium on Theory of Computing, 2009, pp. 601--610] introduced the notion of a nonmalleable extractor, significantly strengthening the notion of a strong extractor. A nonmalleable extractor is a function $\mathsf{nmExt}:\{0,1\}^n\times\{0,1\}^d\to\{0,1\}^m$ that takes two inputs---a weak source $W$ and a uniform (independent) seed $S$---and outputs a string $\mathsf{nmExt}(W,S)$ that is nearly uniform given the seed $S$ as well as the value $\mathsf{nmExt}(W,S')$ for any seed $S'\neq S$ that may be determined as an arbitrary function of $S$. The first explicit construction of a nonmalleable extractor was recently provided by Dodis et al. [Privacy Amplification and Non-malleable Extractors via Character Sums, preprint, arXiv:1102.5415 [cs.CR], 2011]. Their extractor works for any weak source with min-entropy rate $1/2+\delta$, where $\delta>0$ is an arbitrary constant and outputs up to a linear number of bits but suffers from two drawbacks. First, the length of its seed is linear in the length of the weak source (which leads to privacy amplification protocols with high communication complexity). Second, the construction is conditional: when outputting more than a logarithmic number of bits (as required for privacy amplification protocols), its efficiency relies on a longstanding conjecture on the distribution of prime numbers. In this paper we present an unconditional construction of a nonmalleable extractor with short seeds. For any integers $n$ and $d$ such that $2.01\cdot\log n\leq d\leq n$, we present an explicit construction of a nonmalleable extractor $\mathsf{nmExt}\colon\{0,1\}^n\times\{0,1\}^d\to\{0,1\}^m$, with $m=\Omega(d)$ and error exponentially small in $m$. The extractor works for any weak source with min-entropy rate $1/2+\delta$, where $\delta>0$ is an arbitrary constant. Moreover, our extractor in fact satisfies an even more general notion of nonmalleability: its output $\mathsf{nmExt}(W,S)$ is nearly uniform given the seed $S$ as well as the values $\mathsf{nmExt}(W,S_1),\dots,\mathsf{nmExt}(W,S_t)$ for several seeds $S_1,\dots,S_t$ that may be determined as an arbitrary function of $S$, as long as $S\notin\{S_1,\dots,S_t\}$. By instantiating the framework of Dodis and Wichs with our nonmalleable extractor, we obtain the first 2-round privacy amplification protocol for min-entropy rate $1/2+\delta$ with asymptotically optimal entropy loss and polylogarithmic communication complexity. This improves the previously known 2-round privacy amplification protocols: the protocol of Dodis and Wichs, whose entropy loss is not asymptotically optimal, and the protocol of Dodis et al., whose communication complexity is linear.

SIAM Journal on ComputingJan 1, 201415 citations

Privacy with Imperfect Randomness

Yevgeniy Dodis, Yanqing Yao

Lecture notes in computer scienceJan 1, 201511 citations

Privacy Amplification Against Active Quantum Adversaries

Cohen, Gil, Thomas Vidick

Privacy amplification is the task by which two cooperating parties transform a shared weak secret, about which an eavesdropper may have side information, into a uniformly random string uncorrelated from the eavesdropper. Privacy amplification against passive adversaries, where it is assumed that the communication is over a public but authenticated channel, can be achieved in the presence of classical as well as quantum side information by a single-message protocol based on strong extractors. In 2009 Dodis and Wichs devised a two-message protocol to achieve privacy amplification against active adversaries, where the public communication channel is no longer assumed to be authenticated, through the use of a strengthening of strong extractors called non-malleable extractors which they introduced. Dodis and Wichs only analyzed the case of classical side information. We consider the task of privacy amplification against active adversaries with quantum side information. Our main result is showing that the Dodis-Wichs protocol remains secure in this scenario provided its main building block, the non-malleable extractor, satisfies a notion of quantum-proof non-malleability which we introduce. We show that an adaptation of a recent construction of non-malleable extractors due to Chattopadhyay et al. is quantum proof, thereby providing the first protocol for privacy amplification that is secure against active quantum adversaries. Our protocol is quantitatively comparable to the near-optimal protocols known in the classical setting.Privacy amplification is the task by which two cooperating parties transform a shared weak secret, about which an eavesdropper may have side information, into a uniformly random string uncorrelated from the eavesdropper. Privacy amplification against passive adversaries, where it is assumed that the communication is over a public but authenticated channel, can be achieved in the presence of classical as well as quantum side information by a single-message protocol based on strong extractors. In 2009 Dodis and Wichs devised a two-message protocol to achieve privacy amplification against active adversaries, where the public communication channel is no longer assumed to be authenticated, through the use of a strengthening of strong extractors called non-malleable extractors which they introduced. Dodis and Wichs only analyzed the case of classical side information. We consider the task of privacy amplification against active adversaries with quantum side information. Our main result is showing that the Dodis-Wichs protocol remains secure in this scenario provided its main building block, the non-malleable extractor, satisfies a notion of quantum-proof non-malleability which we introduce. We show that an adaptation of a recent construction of non-malleable extractors due to Chattopadhyay et al. is quantum proof, thereby providing the first protocol for privacy amplification that is secure against active quantum adversaries. Our protocol is quantitatively comparable to the near-optimal protocols known in the classical setting.

arXiv:1608.06318Aug 22, 20164 citations

\mathsf{nmExt}:\{0,1\}^n\times\{0,1\}^d\to\{0,1\}^m

that takes two inputs---a weak source

W

and a uniform (independent) seed

S

---and outputs a string

\mathsf{nmExt}(W,S)

that is nearly uniform given the seed

S

as well as the value

\mathsf{nmExt}(W,S')

for any seed

S'\neq S

that may be determined as an arbitrary function of

S

. The first explicit construction of a nonmalleable extractor was recently provided by Dodis et al. [Privacy Amplification and Non-malleable Extractors via Character Sums, preprint, arXiv:1102.5415 [cs.CR], 2011]. Their extractor works for any weak source with min-entropy rate

1/2+\delta

, where

\delta>0

is an arbitrary constant and outputs up to a linear number of bits but suffers from two drawbacks. First, the length of its seed is linear in the length of the weak source (which leads to privacy amplification protocols with high communication complexity). Second, the construction is conditional: when outputting more than a logarithmic number of bits (as required for privacy amplification protocols), its efficiency relies on a longstanding conjecture on the distribution of prime numbers. In this paper we present an unconditional construction of a nonmalleable extractor with short seeds. For any integers

n

and

d

such that

2.01\cdot\log n\leq d\leq n

, we present an explicit construction of a nonmalleable extractor

\mathsf{nmExt}\colon\{0,1\}^n\times\{0,1\}^d\to\{0,1\}^m

, with

m=\Omega(d)

and error exponentially small in

m

. The extractor works for any weak source with min-entropy rate

1/2+\delta

, where

\delta>0

is an arbitrary constant. Moreover, our extractor in fact satisfies an even more general notion of nonmalleability: its output

\mathsf{nmExt}(W,S)

is nearly uniform given the seed

S

as well as the values

\mathsf{nmExt}(W,S_1),\dots,\mathsf{nmExt}(W,S_t)

for several seeds

S_1,\dots,S_t

that may be determined as an arbitrary function of

S

, as long as

S\notin\{S_1,\dots,S_t\}

. By instantiating the framework of Dodis and Wichs with our nonmalleable extractor, we obtain the first 2-round privacy amplification protocol for min-entropy rate

1/2+\delta

with asymptotically optimal entropy loss and polylogarithmic communication complexity. This improves the previously known 2-round privacy amplification protocols: the protocol of Dodis and Wichs, whose entropy loss is not asymptotically optimal, and the protocol of Dodis et al., whose communication complexity is linear.