Unifying computational entropies via Kullback-Leibler divergence

We introduce hardness in relative entropy, a new notion of hardness for search problems which on the one hand is satisfied by all one-way functions and on the other hand implies both next-block pseudoentropy and inaccessible entropy, two forms of computational entropy used in recent constructions of pseudorandom generators and statistically hiding commitment schemes, respectively. Thus, hardness in relative entropy unifies the latter two notions of computational entropy and sheds light on the apparent "duality" between them. Additionally, it yields a more modular and illuminating proof that one-way functions imply next-block inaccessible entropy, similar in structure to the proof that one-way functions imply next-block pseudoentropy (Vadhan and Zheng, STOC '12)

Universal One-Way Hash Functions via Inaccessible Entropy

This paper revisits the construction of Universal One-Way Hash Functions (UOWHFs) from any one-way function due to Rompel (STOC 1990). We give a simpler construction of UOWHFs, which also obtains better efficiency and security. The construction exploits a strong connection to the recently introduced notion of inaccessible entropy (Haitner et al. STOC 2009). With this perspective, we observe that a small tweak of any one-way function f is already a weak form of a UOWHF: Consider F (x, i) that outputs the i-bit long prefix of f(x). If F were a UOWHF then given a random x and i it would be hard to come up with x ′ ̸ = x such that F (x, i) = F (x ′ , i). While this may not be the case, we show (rather easily) that it is hard to sample x ′ with almost full entropy among all the possible such values of x ′. The rest of our construction simply amplifies and exploits this basic property. With this and other recent works, we have that the constructions of three fundamental cryptographic primitives (Pseudorandom Generators, Statistically Hiding Commitments and UOWHFs) out of one-way functions are to a large extent unified. In particular, all three constructions rely on and manipulate computational notions of entropy in similar ways. Pseudorandom Generators rely on the well-established notion of pseudoentropy, whereas Statistically Hiding Commitments and UOWHFs rely on the newer notion of inaccessible entropy

On the (Im-)Possibility of Extending Coin Toss

We consider the task of extending a given coin toss. By this, we mean the two-party task of using a single instance of a given coin toss protocol in order to interactively generate more random coins. A bit more formally, our goal is to generate n common random coins from a single use of an ideal functionality that gives m < n common random coins to both parties. In the framework of universal composability, we show the impossibility of securely extending a coin toss for statistical and perfect security. On the other hand, for computational security, the existence of a protocol for coin toss extension depends on the number m of random coins that can be obtained “for free.” For the case of stand-alone security, i.e., a simulation-based security definition without an environment, we present a protocol for statistically secure coin toss extension. Our protocol works for superlogarithmic m, which is optimal as we show the impossibility of statistically secure coin toss extension for smaller m. Combining our results with already known results, we obtain a (nearly) complete characterization under which circumstances coin toss extension is possible

On Tightly Secure Primitives in the Multi-Instance Setting

We initiate the study of general tight reductions in cryptography. There already exist a variety of works that offer tight reductions for a number of cryptographic tasks, ranging from encryption and signature schemes to proof systems. However, our work is the first to provide a universal definition of a tight reduction (for arbitrary primitives), along with several observations and results concerning primitives for which tight reductions have not been known. Technically, we start from the general notion of reductions due to Reingold, Trevisan, and Vadhan (TCC 2004), and equip it with a quantification of the respective reduction loss, and a canonical multi-instance extension to primitives. We then revisit several standard reductions whose tight security has not yet been considered. For instance, we revisit a generic construction of signature schemes from one-way functions, and show how to tighten the corresponding reduction by assuming collision-resistance from the used one-way function. We also obtain tightly secure pseudorandom generators (by using suitable rerandomisable hard-core predicates), and tightly secure lossy trapdoor functions

Non-Adaptive Universal One-Way Hash Functions from Arbitrary One-Way Functions

In this work we give the first non-adaptive construction of universal one-way hash functions (UOWHFs) from arbitrary one-way functions. Our construction uses $O(n^9)$ calls to the one-way function, has a key of length $O(n^{10})$, and can be implemented in NC1 assuming the underlying one-way function is in NC1. Prior to this work, the best UOWHF construction used O(n13) adaptive calls and a key of size O(n5) (Haitner, Holenstein, Reingold, Vadhan and Wee [Eurocrypt ’10]). By the result of Applebaum, Ishai and Kushilevitz [FOCS ’04], the above implies the existence of UOWHFs in NC0, given the existence of one-way functions in NC1. We also show that the PRG construction of Haitner, Reingold and Vadhan (HRV, [STOC ’10]), with small modifications, yields a relaxed notion of UOWHFs , which is a function family which can be (inefficiently) converted to UOWHF by changing the functions on a negligible fraction of the inputs. In order to analyze this construction, we introduce the notion of next-bit unreachable entropy, which replaces the next-bit pseudoentropy notion used by HRV

A Uniform Min-Max Theorem and Characterizations of Computational Randomness

This thesis develops several tools and techniques using ideas from information theory, optimization, and online learning, and applies them to a number of highly related fundamental problems in complexity theory, pseudorandomness theory, and cryptography.Engineering and Applied Science

Advances in Functional Encryption

Functional encryption is a novel paradigm for public-key encryption that enables both fine-grained access control and selective computation on encrypted data, as is necessary to protect big, complex data in the cloud. In this thesis, I provide a brief introduction to functional encryption, and an overview of my contributions to the area