We undertake a general study of hash functions secure under correlated inputs, meaning that security should be maintained when the adversary sees hash values of many related high-entropy inputs. Such a property is satisfied by a random oracle, and its importance is illustrated by study of the "avalanche effect," a well-known heuristic in cryptographic hash function design. One can interpret "security" in different ways: e.g., asking for one-wayness or that the hash values look uniformly and independently random; the latter case can be seen as a generalization of correlation-robustness introduced by Ishai et al. (CRYPTO 2003). We give specific applications of these notions to password-based login and efficient search on encrypted data. Our main construction achieves them (without random oracles) for inputs related by polynomials over the input space (namely Zp), based on corresponding variants of the q-Diffie Hellman Inversion assumption. Additionally, we show relations between correlated-input secure hash functions and cryptographic primitives secure under related-key attacks. Using our techniques, we are also able to obtain a host of new results for such related-key attack secure cryptographic primitives.

Constructing round-optimal blind signatures in the standard model has been a long standing open problem. In particular, Fischlin and Schröder recently ruled out a large class of three-move blind signatures in the standard model (Eurocrypt'10). In particular, their result shows that finding security proofs for the well-known blind signature schemes by Chaum, and by Pointcheval and Stern in the standard model via black-box reductions is hard. In this work we propose the first roundoptimal, i.e., two-move, blind signature scheme in the standard model (i.e., without assuming random oracles or the existence of a common reference string). Our scheme relies on the Decisional Diffie Hellman assumption and the existence of sub-exponentially hard 1-to-1 one way functions. This scheme is also secure in the concurrent setting.

Abstract. Constrained pseudorandom functions have recently been introduced independently by Boneh and Waters (Asiacrypt'13), Kiayias et al. (CCS'13), and Boyle et al. (PKC'14). In a standard pseudorandom function (PRF) a key K is used to evaluate the PRF on all inputs in the domain. Constrained PRFs additionally offer the functionality to delegate "constrained" keys KS which allow to evaluate the PRF only on a subset S of the domain. The three above-mentioned papers all show that the classical GGM construction (J.ACM'86) of a PRF from a pseudorandom generator (PRG) directly yields a constrained PRF where one can compute constrained keys to evaluate the PRF on all inputs with a given prefix. This constrained PRF has already found many interesting applications. Unfortunately, the existing security proofs only show selective security (by a reduction to the security of the underlying PRG). To achieve full security, one has to use complexity leveraging, which loses an exponential factor 2 N in security, where N is the input length. The first contribution of this paper is a new reduction that only loses a quasipolynomial factor q log N , where q is the number of adversarial queries. For this we develop a new proof technique which constructs a distinguisher by interleaving simple guessing steps and hybrid arguments a small number of times. This approach might be of interest also in other contexts where currently the only technique to achieve full security is complexity leveraging. Our second contribution is concerned with another constrained PRF, due to Boneh and Waters, which allows for constrained keys for the more general class of bit-fixing functions. Their security proof also suffers from a 2 N loss, which we show is inherent. We construct a meta-reduction which shows that any "simple" reduction of full security from a non-interactive hardness assumption must incur an exponential security loss.

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In [6,7], Dwork et al. posed the fundamental question of existence of commitment schemes that are secure against selective opening attacks (SOA, for short). In [2] Bellare, Hofheinz, and Yilek, and Hofheinz in [13] answered it affirmatively by presenting a scheme which is based solely on the non-black-box use of a one-way permutation needing a super-constant number of rounds. This result however opened other challenging questions about achieving a better round complexity and obtaining fully black-box schemes using underlying primitives and code of the adversary in a black-box manner. Recently, in TCC 2011, Xiao ([23]) investigated on how to achieve (nearly) optimal SOA-secure commitment schemes where optimality is in the sense of both the round complexity and the black-box use of cryptographic primitives. The work of Xiao focuses on a simulation-based security notion of SOA. Moreover, the various results in [23] focus only on either parallel or concurrent SOA. In this work we first point out various issues in the claims of [23] that actually reopen several of the questions left open in [2,13]. Then, we provide new lower bounds and concrete constructions that produce a very different state-of-the-art compared to the one claimed in [23].