Abstract
Correlated pairs of random variables are a central concept in information-theoretically secure cryptography. Secure reductions between different correlations have been studied, and completeness results are known. Further, the complexity of such reductions is intimately connected with circuit complexity and efficiency of locally decodable codes. As such, making progress on these complexity questions faces strong barriers. Motivated by this, in this work, we study a restricted form of secure reductions—namely, Secure Non-Interactive Reductions (SNIR)—which is still closely related to the original problem, and establish several fundamental results and relevant techniques for it.
We uncover striking connections between SNIR and linear algebraic properties of correlations. Specifically, we define the spectrum of a correlation, and show that a target correlation has a SNIR to a source correlation only if the spectrum of the latter contains the entire spectrum of the former. We also establish a “mirroring lemma” that shows an unexpected symmetry between the two parties in a SNIR, when viewed through the lens of spectral analysis. We also use cryptographic insights and elementary linear algebraic analysis to fully characterize the role of common randomness as well as local randomness in SNIRs. We employ these results to resolve several fundamental questions about SNIRs, and to define future directions.
V. Narayanan—Supported by the Department of Atomic Energy, India project RTI4001, ERC Project NTSC (742754) and ISF Grants 1709/14 & 2774/20.
M. Prabhakaran—Supported by a Ramanujan Fellowship and Joint Indo-Israel Project DST/INT/ISR/P-16/2017 of Dept. of Science and Technology, India.
V. M. Prabhakaran—Supported by the Department of Atomic Energy, India project RTI4001, and Science & Engineering Research Board, India project MTR/2020/000308.
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Notes
- 1.
The choice of the exact correlation is not crucial and can be replaced by any finite complete functionality, without altering the complexity beyond constant factors [25].
- 2.
There does exist an alternate model of Zero Communication Reductions which allows non-interactive function computation conditioned on a predicate [27]. But here we consider the standard model of secure 2-party computation, except for the restriction of being non-interactive.
- 3.
Here we use the convention that symbols with 0 probability are omitted from the alphabets \(\mathcal {X}\) and \(\mathcal {Y}\), so that all the marginal probabilities \(\boldsymbol{a} _x\) and \(\boldsymbol{b} _y\) are strictly positive.
- 4.
There is a caveat: if the correlation involves common information – i.e., if it is over pairs that can be written as ((x, c), (y, c)) where c indicates a piece of information available to both parties – then, the distribution of the common random variable itself is not captured by the correlation operator. But as we shall see, this component of the distribution can indeed be ignored when studying the feasibility of \(\textsf {SNIR}\) s.
- 5.
Recall that we use the convention that a distribution matrix does not have an all-0 row or column, and hence the diagonal matrices \(\boldsymbol{\Delta }_{M^\intercal }\) and \(\boldsymbol{\Delta }_{M}\) have strictly positive entries in their diagonals.
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Agarwal, P., Narayanan, V., Pathak, S., Prabhakaran, M., Prabhakaran, V.M., Rehan, M.A. (2022). Secure Non-interactive Reduction and Spectral Analysis of Correlations. 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_28
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