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**2008/454**

**Title:**

Complexity of Multiparty Computation Problems: The Case of 2-Party Symmetric Secure Function Evaluation

**Authors:**
Hemanta K. Maji, Manoj Prabhakaran, Mike Rosulek

**Abstract:**

In symmetric secure function evaluation (SSFE), Alice has an input x, Bob has an input y, and both parties wish to securely compute f(x,y). We classify these functions f according to their ``cryptographic complexities,‘’ and show that the landscape of complexity among these functions is surprisingly rich. We give combinatorial characterizations of the SSFE functions f that have passive-secure protocols, and those which are protocols secure in the standalone setting. With respect to universally composable security (for unbounded parties), we show that there is an infinite hierarchy of increasing complexity for SSFE functions, That is, we describe a family of SSFE functions f_1, f_2, \ldots such that there exists a UC-secure protocol for f_i in the f_j-hybrid world if and only if i \le j. Our main technical tool for deriving complexity separations is a powerful protocol simulation theorem which states that, even in the strict setting of UC security, the canonical protocol for f is as secure as any other protocol for f, as long as f satisfies a certain combinatorial characterization. We can then show intuitively clear impossibility results by establishing the combinatorial properties of f and then describing attacks against the very simple canonical protocols, which by extension are also feasible attacks against {\em any} protocol for the same functionality.

**ePrint:**
https://eprint.iacr.org/2008/454

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