From Impossibility to Completeness for Deterministic SFE Jonathan Katz ∗ Aggelos Kiayias † Ranjit Kumaresan ∗ abhi shelat ‡ Hong-Sheng Zhou ∗§ September 16, 2011 Abstract Cryptographic impossibility is an extensive phenomenon in the setting of composable secure function evaluation (SFE). Most functionalities turn out to be impossible to realize if compos- ability is required in the plain model. This is in sharp contrast to stand-alone security that is known to be feasible for all SFE tasks. In the light of this, it is important to understand the basic requirements in terms of setup assumptions that are needed to realize functionalities when general composition is expected to preserve security properties. Given that setup assumptions (which themselves can be casted as functionalities) have to be impossible in the standard model, the fundamental question that arises is which of the known impossible tasks are also complete i.e., they are sufficient to imply any other functionality. In this work, using as a starting point the extensive impossibility results for unbounded deterministic two-party functionalities given by Canetti, Kushilevitz and Lindell (Eurocrypt 2003, J Cryptology 2006) for universal com- posability (UC), we prove that a large class impossible SFE tasks are in fact complete; this significantly extends the class of functionalities that are known to imply all cryptographic tasks in the UC setting (specifically to include unbounded functionalities as opposed to just finite functionalities that was known before). We achieve this result by constructing UC puzzles, a complete primitive recently introduced by Lin, Pass, Venkitasubramaniam (STOC 2009). ∗ Department of Computer Science, University of Maryland, College Park, MD 20742, USA. Email: {jkatz, ranjit, hszhou}@cs.umd.edu. This work is supported in part by NSF award #1111599 and DARPA. † Dept. of Informatics & Telecommunications, University of Athens. Email: aggelos@kiayias.com ‡ Dept. of Computer Science, University of Virginia. Email: shelat@cs.virginia.edu § Supported by an NSF CI postdoctoral fellowship.