International Association
for Cryptologic Research

Consider two-party secure function evaluation against an honest-but-curious adversary in the information-theoretic plain model. We study the round complexity of securely realizing a given secure function evaluation functionality. Chor-Kushilevitz-Beaver (1989) proved that the round complexity of securely evaluating a deterministic function depends solely on the cardinality of its domain and range. A folklore conjecture asserts that this phenomenon extends to functions with randomized output. Our work falsifies this folklore conjecture -- revealing intricate subtleties even for this elementary security notion. For every $r$, there is a function $f_r$ with binary inputs and five output alphabets that has round complexity $r$. Previously, such a construction was known using $(r+1)$ output symbols. This counter-example is optimal -- we prove that any securely realizable function with binary inputs and four output alphabets has round complexity at most four. We work in the geometric framework Basu-Khorasgani-Maji-Nguyen (FOCS--2022) introduced to investigate randomized functions' round complexity. Our work establishes a connection between secure computation and the lamination hull (geometric object originally motivated by applications in hydrodynamics). Our counterexample constructions are related to the ``tartan square'' construction in the lamination hull literature.

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 {\em 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 \BSC and \BEC, respectively. We show the impossibility of interconversion between \BSC and \BEC samples. Next, we prove that a \snis of a $\BEC(\eps')$ sample (a \BEC with noise characteristic $\eps'$) from $\BEC(\eps)$ is feasible if and only if $(1-\eps') = (1-\eps)^k$, for some $k\in\NN$. In this context, we prove that all \snis constructions must be linear. Furthermore, if $(1-\eps') = (1-\eps)^k$, then the rate of simulating multiple independent $\BEC(\eps')$ samples is at most $1/k$, which is also achievable using (block) linear constructions. Finally, we show that a \snis of a $\BSC(\eps')$ sample from $\BSC(\eps)$ samples is feasible if and only if $(1-2\eps')=(1-2\eps)^k$, for some $k\in\NN$. Interestingly, there are linear as well as non-linear \snis constructions. When $(1-2\eps')=(1-2\eps)^k$, we prove that the rate of a {\em 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 {\em 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 {\em homogeneous Boolean functions} and {\em distance-invariant codes}, which may be of independent interest.

{\em Secure non-interactive simulation} (SNIS), introduced in {EUROCRYPT} 2022, is the information-theoretic analog of {\em pseudo-correlation generators}. SNIS allows parties, starting with samples of a source correlated private randomness, to non-interactively and securely transform them into samples from a different correlated private randomness. Determining the feasibility, rate, and capacity of SNIS is natural and essential for the efficiency of secure computation. This work initiates the study of SNIS, where the target distribution $(U,V)$ is a random sample from the {\em binary symmetric or erasure channels}; however, the source distribution can be arbitrary. In this context, our work presents: \begin{enumerate} \item The characterization of all sources that facilitate such SNIS, \item An upper and lower bound on their maximum achievable rate, and \item Exemplar SNIS instances where non-linear reductions achieve optimal efficiency; however, any linear reduction is insecure. \end{enumerate} These results collectively yield the fascinating instances of {\em computer-assisted search} for secure computation protocols that identify ingenious protocols that are more efficient than all known constructions. Our work generalizes the algebraization of the simulation-based definition of SNIS as an approximate eigenvector problem. The following foundational and general technical contributions of ours are the underpinnings of the results mentioned above. \begin{enumerate} \item Characterization of Markov and adjoint Markov operators' effect on the Fourier spectrum of reduction functions. \item A new concentration phenomenon in the Fourier spectrum of reduction functions. \item A powerful statistical-to-perfect lemma with broad consequences for feasibility and rate characterization of SNIS. \end{enumerate} Our technical analysis relies on Fourier analysis over large alphabets with arbitrary measure, the orthogonal Efron-Stein decomposition, and junta theorems of Kindler-Safra and Friedgut. Our work establishes a fascinating connection between the rate of SNIS and the maximal correlation, a prominent information-theoretic property. Our technical approach motivates the new problem of ``security-preserving dimension reduction'' in harmonic analysis, which may be of independent and broader interest.

Consider the representative task of designing a distributed coin-tossing protocol for n processors such that the probability of heads is $$X_0\in [0,1]$$. This protocol should be robust to an adversary who can reset one processor to change the distribution of the final outcome. For $$X_0=1/2$$, in the information-theoretic setting, no adversary can deviate the probability of the outcome of the well-known Blum’s “majority protocol” by more than $$\frac{1}{\sqrt{2\pi n}}$$, i.e., it is $$\frac{1}{\sqrt{2\pi n}}$$ insecure.In this paper, we study discrete-time martingales $$(X_0,X_1,\dotsc ,X_n)$$ such that $$X_i\in [0,1]$$, for all $$i\in \{0,\dotsc ,n\}$$, and $$X_n\in {\{0,1\}} $$. These martingales are commonplace in modeling stochastic processes like coin-tossing protocols in the information-theoretic setting mentioned above. In particular, for any $$X_0\in [0,1]$$, we construct martingales that yield $$\frac{1}{2}\sqrt{\frac{X_0(1-X_0)}{n}}$$ insecure coin-tossing protocols. For $$X_0=1/2$$, our protocol requires only 40% of the processors to achieve the same security as the majority protocol.The technical heart of our paper is a new inductive technique that uses geometric transformations to precisely account for the large gaps in these martingales. For any $$X_0\in [0,1]$$, we show that there exists a stopping time $$\tau $$ such that The inductive technique simultaneously constructs martingales that demonstrate the optimality of our bound, i.e., a martingale where the gap corresponding to any stopping time is small. In particular, we construct optimal martingales such that any stopping time $$\tau $$ has Our lower-bound holds for all $$X_0\in [0,1]$$; while the previous bound of Cleve and Impagliazzo (1993) exists only for positive constant $$X_0$$. Conceptually, our approach only employs elementary techniques to analyze these martingales and entirely circumvents the complex probabilistic tools inherent to the approaches of Cleve and Impagliazzo (1993) and Beimel, Haitner, Makriyannis, and Omri (2018).By appropriately restricting the set of possible stopping-times, we present representative applications to constructing distributed coin-tossing/dice-rolling protocols, discrete control processes, fail-stop attacking coin-tossing/dice-rolling protocols, and black-box separations.