Function Private Functional Encryption and Property Preserving Encryption : New Definitions and Positive Results.

This work furthers the exploration of meaningful definitions for security of Functional Encryption. We propose new simulation based definitions for function privacy in addition to data privacy and study their achievability. In addition, we improve efficiency/ underlying assumptions/ security achieved by existing inner product Functional Encryption and Property Preserving Encryption schemes, in both the private and public key setting. Our results can be summarized as follows: • We present a new simulation based definition, which we call Relax-AD-SIM, that lies between simulation based (SIM) and indistinguishability based (IND) definitions for data privacy, and implies the function privacy definition of [BRS13a]. Our definition relaxes the requirements on the simulator to bypass impossibility of SIM in the standard model. We show that the inner product FE scheme of [KSW08] enjoys Relax-AD-SIM security for function hiding and the inner product FE scheme of [LOS10] enjoys Relax-AD-SIM security for data hiding. • We study whether known impossibilities for achieving strong SIM based security imply actual real world attacks. For this, we present a new UC-style SIM based definition of security that captures both data and function hiding, both public key and symmetric key settings and represents the “dream” security of FE. While known impossibilities rule out its achievability in the standard model, we show, surprisingly, that it can be achieved in the generic group model for Inner Product FE ([KSW08]). This provides evidence that FE implementations may enjoy extremely strong security against a large class of real world attacks, namely generic attacks. It also implies a program obfuscator for the inner product functionality in the generic group model, which is related to the hyperplane-membership obfuscator of [CRV10]. • We provide several improvements to known constructions of Inner Product FE. In the private key setting, the construction by Shen et al. was based on non-standard assumptions, used composite order groups, and only achieved selective security. We give the first construction of a symmetric key inner product FE which is built using prime order groups, and is fully secure under the standard DLIN assumption. Our scheme is more efficient in the size of key and ciphertext than [SSW09], when the latter is converted to prime-order groups. We also port the public key inner product scheme of [KSW08] to prime order groups. • We give the first standard model construction of a property preserving encryption (PPE) scheme [PR12] for inner-products. Our scheme is secure under the DLIN assumption and satisfies the strongest definition of security – Left-or-Right security. Note that previously known constructions were only known to be secure in the generic group model. ∗UIUC. Email: sagrawl2@illinois.edu. †I.I.T, Delhi. Email: shweta.a@gmail.com. ‡UIUC. Email: bsaikrishna7393@gmail.com. §UCLA Email: abishekk@cs.ucla.edu. ¶UIUC. Email: manojmp@gmail.com. ‖UCLA Email: sahai@cs.ucla.edu.