Abstract
A natural solution to increase the efficiency of secure computation will be to non-interactively and securely transform diverse inexpensive-to-generate correlated randomness, like, joint samples from noise sources, into correlations useful for secure computation protocols. Motivated by this general application for secure computation, our work introduces the notion of secure non-interactive simulation (SNIS). Parties receive samples of correlated randomness, and they, without any interaction, securely convert them into samples from another correlated randomness.
Our work presents a simulation-based security definition for SNIS and initiates the study of the feasibility and efficiency of SNIS. We also study SNIS among fundamental correlated randomnesses like random samples from the binary symmetric and binary erasure channels, represented by \(\mathsf {BSS}\) and \(\mathsf {BES}\), respectively. We show the impossibility of interconversion between \(\mathsf {BSS}\) and \(\mathsf {BES}\) samples.
Next, we prove that a SNIS of a \(\mathsf {BES} (\varepsilon ')\) sample (a \(\mathsf {BES}\) with noise characteristic \(\varepsilon '\)) from \(\mathsf {BES} (\varepsilon )\) is feasible if and only if \((1-\varepsilon ') = (1-\varepsilon )^k\), for some \(k\in \mathbb {N}\). In this context, we prove that all SNIS constructions must be linear. Furthermore, if \((1-\varepsilon ') = (1-\varepsilon )^k\), then the rate of simulating multiple independent \(\mathsf {BES} (\varepsilon ')\) samples is at most 1/k, which is also achievable using (block) linear constructions.
Finally, we show that a SNIS of a \(\mathsf {BSS} (\varepsilon ')\) sample from \(\mathsf {BSS} (\varepsilon )\) samples is feasible if and only if \((1-2\varepsilon ')=(1-2\varepsilon )^k\), for some \(k\in \mathbb {N}\). Interestingly, there are linear as well as non-linear SNIS constructions. When \((1-2\varepsilon ')=(1-2\varepsilon )^k\), we prove that the rate of a perfectly secure SNIS is at most 1/k, which is achievable using linear and non-linear constructions.
Our technical approach algebraizes the definition of SNIS and proceeds via Fourier analysis. Our work develops general analysis methodologies for Boolean functions, explicitly incorporating cryptographic security constraints. Our work also proves strong forms of statistical-to-perfect security transformations: one can error-correct a statistically secure SNIS to make it perfectly secure. We show a connection of our research with homogeneous Boolean functions and distance-invariant codes, which may be of independent interest.
H. Amini Khorasgani, H.K. Maji, H.H. Nguyen—The research effort is supported in part by an NSF CRII Award CNS–1566499, NSF SMALL Awards CNS–1618822 and CNS–2055605, the IARPA HECTOR project, MITRE Innovation Program Academic Cybersecurity Research Awards (2019–2020, 2020–2021), a Ross-Lynn Research Scholars Grant, a Purdue Research Foundation (PRF) Award, and The Center for Science of Information, an NSF Science and Technology Center, Cooperative Agreement CCF–0939370.
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Notes
- 1.
As is typical in this line of work in cryptography and information theory, the joint distributions (U, V) and (X, Y) assign probabilities to samples that are either 0 or at least a positive constant.
- 2.
The conditional distribution \((A|B=b)\) is \(\nu \)-close to being independent of b if there exists a distribution \(A^*\) such that \((A|B=b)\) is \(\nu \)-close to \(A^*\) in the statistical distance, for all \(b\in \mathrm {Supp}({B})\).
- 3.
Alice can perform a random walk on an appropriate expander graph using her samples to get one random bit that is statistically secure conditioned on Bob’s samples.
- 4.
A joint distribution (X, Y) is complete if there exists samples \(x_0,x_1\in \mathrm {Supp}({X})\) and \(y_0,y_1\in \mathrm {Supp}({Y})\) such that
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1.
\(\mathrm {Pr}\left[ {X=x_0,Y=y_0}\right] , \mathrm {Pr}\left[ {X=x_1,Y=y_0}\right] , \mathrm {Pr}\left[ {X=x_1,Y=y_1}\right] > 0\), and
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2.
\( \mathrm {Pr}\left[ {X=x_0,Y=y_0}\right] \cdot \mathrm {Pr}\left[ {X=x_1,Y=y_1}\right] \ne \mathrm {Pr}\left[ {X=x_0,Y=y_1}\right] \cdot \mathrm {Pr}\left[ {X=x_1,Y=y_0}\right] .\)
Multiple samples of a complete distributions can be used to (interactively) implement oblivious transfer [30], the atomic primitive for secure computation. The joint distribution \(\mathsf {BES} (\varepsilon )\), for \(\varepsilon \in (0,1)\), and \(\mathsf {BSS} (\varepsilon )\), for \(\varepsilon \in (0,1/2)\), are complete distributions. However, \(\mathsf {BSS} (0)=\mathsf {BES} (0)\), \(\mathsf {BES} (1)\), and \(\mathsf {BSS} (1/2)\) are not complete distributions.
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1.
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Amini Khorasgani, H., Maji, H.K., Nguyen, H.H. (2022). Secure Non-interactive Simulation: Feasibility and Rate. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13277. Springer, Cham. https://doi.org/10.1007/978-3-031-07082-2_27
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