Paper 2010/544

Semantic Security Under Related-Key Attacks and Applications

Benny Applebaum, Danny Harnik, and Yuval Ishai

Abstract

In a related-key attack (RKA) an adversary attempts to break a cryptographic primitive by invoking the primitive with several secret keys which satisfy some known, or even chosen, relation. We initiate a formal study of RKA security for \emph{randomized encryption} schemes. We begin by providing general definitions for semantic security under passive and active RKAs. We then focus on RKAs in which the keys satisfy known linear relations over some Abelian group. We construct simple and efficient schemes which resist such RKAs even when the adversary can choose the linear relation adaptively during the attack. More concretely, we present two approaches for constructing RKA-secure encryption schemes. The first is based on standard randomized encryption schemes which additionally satisfy a natural ``key-homomorphism'' property. We instantiate this approach under number-theoretic or lattice-based assumptions such as the Decisional Diffie-Hellman (DDH) assumption and the Learning Noisy Linear Equations assumption. Our second approach is based on RKA-secure pseudorandom generators. This approach can yield either {\em deterministic,} {\em one-time use} schemes with optimal ciphertext size or randomized unlimited use schemes. We instantiate this approach by constructing a simple RKA-secure pseurodandom generator under a variant of the DDH assumption. Finally, we present several applications of RKA-secure encryption by showing that previous protocols which made a specialized use of random oracles in the form of \emph{operation respecting synthesizers} (Naor and Pinkas, Crypto 1999) or \emph{correlation-robust hash functions} (Ishai et. al., Crypto 2003) can be instantiated with RKA-secure encryption schemes. This includes the Naor-Pinkas protocol for oblivious transfer (OT) with adaptive queries, the IKNP protocol for batch-OT, the optimized garbled circuit construction of Kolesnikov and Schneider (ICALP 2008), and other results in the area of secure computation. Hence, by plugging in our constructions we get instances of these protocols that are provably secure in the standard model under standard assumptions.