ECCC-Report TR17-038https://eccc.weizmann.ac.il/report/2017/038Comments and Revisions published for TR17-038en-usWed, 08 Mar 2017 12:28:26 +0200
Revision 1
| Conditional Disclosure of Secrets: Amplification, Closure, Amortization, Lower-bounds, and Separations |
Benny Applebaum,
Barak Arkis,
Pavel Raykov,
Prashant Nalini Vasudevan
https://eccc.weizmann.ac.il/report/2017/038#revision1In the \emph{conditional disclosure of secrets} problem (Gertner et al., J. Comput. Syst. Sci., 2000) Alice and Bob, who hold inputs $x$ and $y$ respectively, wish to release a common secret $s$ to Carol (who knows both $x$ and $y$) if only if the input $(x,y)$ satisfies some predefined predicate $f$. Alice and Bob are allowed to send a single message to Carol which may depend on their inputs and some joint randomness and the goal is to minimize the communication complexity while providing information-theoretic security.
Following Gay, Kerenidis, and Wee (Crypto 2015), we study the communication complexity of CDS protocols and derive the following positive and negative results.
1. *Closure* A CDS for $f$ can be turned into a CDS for its complement $\bar{f}$ with only a minor blow-up in complexity. More generally, for a (possibly non-monotone) predicate $h$, we obtain a CDS for $h(f_1,\ldots,f_m)$ whose cost is essentially linear in the formula size of $h$ and polynomial in the CDS complexity of $f_i$.
2. *Amplification* It is possible to reduce the privacy and correctness error of a CDS from constant to $2^{-k}$ with a multiplicative overhead of $O(k)$. Moreover, this overhead can be amortized over $k$-bit secrets.
3. *Amortization* Every predicate $f$ over $n$-bit inputs admits a CDS for multi-bit secrets whose amortized communication complexity per secret bit grows linearly with the input length $n$ for sufficiently long secrets. In contrast, the best known upper-bound for single-bit secrets is exponential in $n$.
4. *Lower-bounds* There exists a (non-explicit) predicate $f$ over $n$-bit inputs for which any perfect (single-bit) CDS requires communication of at least $\Omega(n)$. This is an exponential improvement over the previously known $\Omega(\log n)$ lower-bound.
5. *Separations* There exists an (explicit) predicate whose CDS complexity is exponentially smaller than its randomized communication complexity. This matches a lower-bound of Gay et. al., and, combined with another result of theirs, yields an exponential separation between the communication complexity of linear CDS and non-linear CDS. This is the first provable gap between the communication complexity of linear CDS (which captures most known protocols) and non-linear CDS.
Wed, 08 Mar 2017 12:28:26 +0200https://eccc.weizmann.ac.il/report/2017/038#revision1
Paper TR17-038
| Conditional Disclosure of Secrets: Amplification, Closure, Amortization, Lower-bounds, and Separations |
Benny Applebaum,
Barak Arkis,
Pavel Raykov,
Prashant Nalini Vasudevan
https://eccc.weizmann.ac.il/report/2017/038In the \emph{conditional disclosure of secrets} problem (Gertner et al., J. Comput. Syst. Sci., 2000) Alice and Bob, who hold inputs $x$ and $y$ respectively, wish to release a common secret $s$ to Carol (who knows both $x$ and $y$) if only if the input $(x,y)$ satisfies some predefined predicate $f$. Alice and Bob are allowed to send a single message to Carol which may depend on their inputs and some joint randomness and the goal is to minimize the communication complexity while providing information-theoretic security.
Following Gay, Kerenidis, and Wee (Crypto 2015), we study the communication complexity of CDS protocols and derive the following positive and negative results.
1. *Closure* A CDS for $f$ can be turned into a CDS for its complement $\bar{f}$ with only a minor blow-up in complexity. More generally, for a (possibly non-monotone) predicate $h$, we obtain a CDS for $h(f_1,\ldots,f_m)$ whose cost is essentially linear in the formula size of $h$ and polynomial in the CDS complexity of $f_i$.
2. *Amplification* It is possible to reduce the privacy and correctness error of a CDS from constant to $2^{-k}$ with a multiplicative overhead of $O(k)$. Moreover, this overhead can be amortized over $k$-bit secrets.
3. *Amortization* Every predicate $f$ over $n$-bit inputs admits a CDS for multi-bit secrets whose amortized communication complexity per secret bit grows linearly with the input length $n$ for sufficiently long secrets. In contrast, the best known upper-bound for single-bit secrets is exponential in $n$.
4. *Lower-bounds* There exists a (non-explicit) predicate $f$ over $n$-bit inputs for which any perfect (single-bit) CDS requires communication of at least $\Omega(n)$. This is an exponential improvement over the previously known $\Omega(\log n)$ lower-bound.
5. *Separations* There exists an (explicit) predicate whose CDS complexity is exponentially smaller than its randomized communication complexity. This matches a lower-bound of Gay et. al., and, combined with another result of theirs, yields an exponential separation between the communication complexity of linear CDS and non-linear CDS. This is the first provable gap between the communication complexity of linear CDS (which captures most known protocols) and non-linear CDS.
Our results solve several open problems posed by Gay et al., and have applications to secret-sharing schemes for forbidden-graph access structures. Tue, 28 Feb 2017 02:41:37 +0200https://eccc.weizmann.ac.il/report/2017/038