Algebraic Lower Bounds for Computing on Encrypted Data

In cryptography, there has been tremendous success in building primitives out of homomorphic semantically-secure encryption schemes, using homomorphic properties in a blackbox way. A few notable examples of such primitives include items like private information retrieval schemes and collision-resistant hash functions (e.g. [14, 6, 13]). In this paper, we illustrate a general methodology for determining what types of protocols can be implemented in this way and which cannot. This is accomplished by analyzing the computational power of various algebraic structures which are preserved by existing cryptosystems. More precisely, we demonstrate lower bounds for algebraically generating generalized characteristic vectors tain algebraic structures, and subsequently we show how to directly apply this abstract algebraic results to put lower bounds on algebraic constructions of a number of cryptographic protocols, including PIR-writing and private keyword search protocols. We hope that this work...

• Type: Journal
• Year: 2007
• Where: ECCC
• Authors: Rafail Ostrovsky, William E. Skeith III
• Keywords: Algebraic Structures, ECCC 2007, Lower Bounds, Semantically-secure Encryption Schemes