The notion of non-malleability in cryptography refers to the setting where the adversary is a man-in-the-middle (MIM) who takes part in two or more protocol executions and tries to use information obtained in one, to violate the security of another. Despite two decades of research, non-malleable commitments (NMCs) have remained too inefficient to be implemented in practice, without some sort of trusted setup. In this work, we give a fast implementation of NMC in the plain model, based on the DDH assumption being hard over elliptic curve groups. Our main theoretical result is a new NMC scheme which can be thought of as a "high dimensional" generalization of the one in the recent work of [GRRV14]. Central to our efficiency improvements is a method of constraining challenges sent by the receiver. This new approach enables us to obtain dramatically improved parameters over those suggested in [GRRV14]. In particular, our work opens the door to implementations based on Elliptic Curves.

The notion of a universally utility-maximizing privacy mechanism was introduced by Ghosh, Roughgarden, and Sundararajan [Proceedings of STOC, 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, simultaneously, while preserving privacy. They demonstrated, quite surprisingly, a case where such a universally utility-maximizing privacy mechanism exists, when the information consumers are Bayesian. This result was later extended by Gupte and Sundararajan [Proceedings of PODS, 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a single count query (i.e., the number of individuals satisfying a specific property in a given population). We show that such universally optimal mechanisms do not exist for some natural extensions of count queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further and give a characterization of those functions that admit universally optimal mechanisms, showing that a universally optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function by its privacy constraint graph whose edges correspond to values resulting by applying the query function to neighboring databases.

The notion of universally utility-maximizing privacy mechanism was recently introduced by Ghosh, Rough garden, and Sundararajan [STOC 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, simultaneously, while preserving Differential Privacy [Dwork, McSherry, Nissim, and Smith, TCC 2006]. Ghosh, Rough garden and Sundararajan have demonstrated, quite surprisingly, a case where such a universally-optimal differentially-private mechanisms exists, when the information consumers are Bayesian. This result was recently extended by Gupte and Sundararajan [PODS 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a single count query (i.e., the number of individuals satisfying a specific property in a given population), and the starting point of our work is a trial at extending these results to similar settings, such as sum queries with non-binary individual values, histograms, and two (or more) count queries. We show, however, that universally-optimal mechanisms do not exist for all these queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further, and give a characterization of those functions that admit universally-optimal mechanisms, showing that a universally-optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function f by its privacy constraint graph Gfwhose edges correspond to values resulting by applying f to neighboring databases.