Negation-Limited Formulas

We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC\u2715), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas

Differentially Oblivious Turing Machines

Oblivious RAM (ORAM) is a machinery that protects any RAM from leaking information about its secret input by observing only the access pattern. It is known that every ORAM must incur a logarithmic overhead compared to the non-oblivious RAM. In fact, even the seemingly weaker notion of differential obliviousness, which intuitively "protects" a single access by guaranteeing that the observed access pattern for every two "neighboring" logical access sequences satisfy (?,?)-differential privacy, is subject to a logarithmic lower bound. In this work, we show that any Turing machine computation can be generically compiled into a differentially oblivious one with only doubly logarithmic overhead. More precisely, given a Turing machine that makes N transitions, the compiled Turing machine makes O(N ? log log N) transitions in total and the physical head movements sequence satisfies (?,?)-differential privacy (for a constant ? and a negligible ?). We additionally show that ?(log log N) overhead is necessary in a natural range of parameters (and in the balls and bins model). As a corollary, we show that there exist natural data structures such as stack and queues (supporting online operations) on N elements for which there is a differentially oblivious implementation on a Turing machine incurring amortized O(log log N) overhead per operation, while it is known that any oblivious implementation must consume ?(log N) operations unconditionally even on a RAM. Therefore, we obtain the first unconditional separation between obliviousness and differential obliviousness in the most natural setting of parameters where ? is a constant and ? is negligible. Before this work, such a separation was only known in the balls and bins model. Note that the lower bound applies in the RAM model while our upper bound is in the Turing machine model, making our separation stronger

Communication with Contextual Uncertainty

We introduce a simple model illustrating the role of context in communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information $x$ and Bob gets $y$, where $(x,y)$ is drawn from a known distribution, and Bob wishes to compute some function $g(x,y)$ (with high probability over $(x,y)$). In our variant, Alice does not know $g$, but only knows some function $f$ which is an approximation of $g$. Thus, the function being computed forms the context for the communication, and knowing it imperfectly models (mild) uncertainty in this context. A naive solution would be for Alice and Bob to first agree on some common function $h$ that is close to both $f$ and $g$ and then use a protocol for $h$ to compute $h(x,y)$. We show that any such agreement leads to a large overhead in communication ruling out such a universal solution. In contrast, we show that if $g$ has a one-way communication protocol with complexity $k$ in the standard setting, then it has a communication protocol with complexity $O(k \cdot (1+I))$ in the uncertain setting, where $I$ denotes the mutual information between $x$ and $y$. In the particular case where the input distribution is a product distribution, the protocol in the uncertain setting only incurs a constant factor blow-up in communication and error. Furthermore, we show that the dependence on the mutual information $I$ is required. Namely, we construct a class of functions along with a non-product distribution over $(x,y)$ for which the communication complexity is a single bit in the standard setting but at least $\Omega(\sqrt{n})$ bits in the uncertain setting.Comment: 20 pages + 1 title pag

Leakage Resilient One-Way Functions: The Auxiliary-Input Setting

Most cryptographic schemes are designed in a model where perfect secrecy of the secret key is assumed. In most physical implementations, however, some form of information leakage is inherent and unavoidable. To deal with this, a flurry of works showed how to construct basic cryptographic primitives that are resilient to various forms of leakage. Dodis et al. (FOCS \u2710) formalized and constructed leakage resilient one-way functions. These are one-way functions $f$ such that given a random image $f(x)$ and leakage $g(x)$ it is still hard to invert $f(x)$. Based on any one-way function, Dodis et al. constructed such a one-way function that is leakage resilient assuming that an attacker can leak any lossy function g of the input. In this work we consider the problem of constructing leakage resilient one-way functions that are secure with respect to arbitrary computationally hiding leakage (a.k.a auxiliary-input). We consider both types of leakage --- selective and adaptive --- and prove various possibility and impossibility results. On the negative side, we show that if the leakage is an adaptively-chosen arbitrary one-way function, then it is impossible to construct leakage resilient one-way functions. The latter is proved both in the random oracle model (without any further assumptions) and in the standard model based on a strong vector-variant of DDH. On the positive side, we observe that when the leakage is chosen ahead of time, there are leakage resilient one-way functions based on a variety of assumption

A Lower Bound for Adaptively-Secure Collective Coin-Flipping Protocols

In 1985, Ben-Or and Linial (Advances in Computing Research \u2789) introduced the collective coin-flipping problem, where n parties communicate via a single broadcast channel and wish to generate a common random bit in the presence of adaptive Byzantine corruptions. In this model, the adversary can decide to corrupt a party in the course of the protocol as a function of the messages seen so far. They showed that the majority protocol, in which each player sends a random bit and the output is the majority value, tolerates O(sqrt n) adaptive corruptions. They conjectured that this is optimal for such adversaries. We prove that the majority protocol is optimal (up to a poly-logarithmic factor) among all protocols in which each party sends a single, possibly long, message. Previously, such a lower bound was known for protocols in which parties are allowed to send only a single bit (Lichtenstein, Linial, and Saks, Combinatorica \u2789), or for symmetric protocols (Goldwasser, Kalai, and Park, ICALP \u2715)

From Minicrypt to Obfustopia via Private-Key Functional Encryption

Private-key functional encryption enables fine-grained access to symmetrically-encrypted data. Although private-key functional encryption (supporting an unbounded number of keys and ciphertexts) seems significantly weaker than its public-key variant, its known realizations all rely on public-key functional encryption. At the same time, however, up until recently it was not known to imply any public-key primitive, demonstrating our poor understanding of this extremely-useful primitive. Recently, Bitansky et al. [TCC \u2716B] showed that sub-exponentially-secure private-key function encryption bridges from nearly-exponential security in Minicrypt to slightly super-polynomial security in Cryptomania, and from sub-exponential security in Cryptomania to Obfustopia. Specifically, given any sub-exponentially-secure private-key functional encryption scheme and a nearly-exponentially-secure one-way function, they constructed a public-key encryption scheme with slightly super-polynomial security. Assuming, in addition, a sub-exponentially-secure public-key encryption scheme, they then constructed an indistinguishability obfuscator. We settle the problem of positioning private-key functional encryption within the hierarchy of cryptographic primitives by placing it in Obfustopia. First, given any quasi-polynomially-secure private-key functional encryption scheme, we construct an indistinguishability obfuscator for circuits with inputs of poly-logarithmic length. Then, we observe that such an obfuscator can be used to instantiate many natural applications of indistinguishability obfuscation. Specifically, relying on sub-exponentially-secure one-way functions, we show that quasi-polynomially-secure private-key functional encryption implies not just public-key encryption but leads all the way to public-key functional encryption for circuits with inputs of poly-logarithmic length. Moreover, relying on sub-exponentially-secure injective one-way functions, we show that quasi-polynomially-secure private-key functional encryption implies a hard-on-average distribution over instances of a PPAD-complete problem. Underlying our constructions is a new transformation from single-input functional encryption to multi-input functional encryption in the private-key setting. The previously known such transformation [Brakerski et al., EUROCRYPT \u2716] required a sub-exponentially-secure single-input scheme, and obtained a scheme supporting only a slightly super-constant number of inputs. Our transformation both relaxes the underlying assumption and supports more inputs: Given any quasi-polynomially-secure single-input scheme, we obtain a scheme supporting a poly-logarithmic number of inputs

Another Step Towards Realizing Random Oracles: Non-Malleable Point Obfuscation

The random oracle paradigm allows us to analyze the security of protocols and constructions in an idealized model, where all parties have access to a truly random function. This is one of the most popular and well-studied models in cryptography. However, being such a strong idealized model, it is known to be susceptible to various weaknesses when implemented naively in ``real-life\u27\u27, as shown by Canetti, Goldreich and Halevi (J. ACM 2004). As a counter-measure, one could try to identify and implement only one or few of the properties a random oracle possesses that are needed for a specific setting. Such a systematic study was initiated by Canetti (CRYPTO 1997), who showed how to implement the property that the output of the function does not reveal anything regarding the input by constructing a point function obfucator. This property turned out to suffice in many follow-up works and applications. In this work, we tackle another natural property of random oracles and implement it in the standard model. The property we focus on is non-malleability, where it is required that the output on an input cannot be used to generate an output on any related point. We construct a point obfuscator that is both hiding (a la Canetti) and is non-malleable for a non-trivial class of mauling functions. Our construction does not use heavy cryptographic machinery (such as zero-knowledge proofs) and is comparable to that of Canetti in terms of time complexity and obfuscation size. The security of our construction relies on variants of the DDH and power-DDH assumptions. On the technical side, we introduce a new technique for proving security of a construction based on a DDH-like assumption. We call this technique ``double-exponentiation\u27\u27 and believe it will be useful in the future

Cutting-Edge Cryptography Through the Lens of Secret Sharing

Secret sharing is a mechanism by which a trusted dealer holding a secret splits the secret into many shares and distributes the shares to a collection of parties. Associated with the sharing is a monotone access structure, that specifies which parties are qualified and which are not: any qualified subset of parties can (efficiently) reconstruct the secret, but no unqualified subset can learn anything about the secret. In the most general form of secret sharing, the access structure can be any monotone NP language. In this work, we consider two very natural extensions of secret sharing. In the first, which we call distributed} secret sharing, there is no trusted dealer at all, and instead the role of the dealer is distributed amongst the parties themselves. Distributed secret sharing can be thought of as combining the features of multiparty non-interactive key exchange and standard secret sharing, and may be useful in settings where the secret is so sensitive that no one individual dealer can be trusted with the secret. Our second notion is called functional secret sharing, which incorporates some of the features of functional encryption into secret sharing by providing more fine-grained access to the secret. Qualified subsets of parties do not learn the secret, but instead learn some function applied to the secret, with each set of parties potentially learning a different function. Our main result is that both of the extensions above are equivalent to several recent cutting-edge primitives. In particular, general-purpose distributed secret sharing is equivalent to witness PRFs, and general-purpose functional secret sharing is equivalent to indistinguishability obfuscation. Thus, our work shows that it is possible to view some of the recent developments in cryptography through a secret sharing lens, yielding new insights about both these cutting-edge primitives and secret sharing