We consider (almost) k-wise independent hash functions, whose evaluations on any k inputs are (almost) uniformly random, for very large values of k. Such hash functions need to have a large key that grows linearly with k. However, it may be possible to evaluate them in sub-linear time by only reading a small subset of t ≪ k locations during each evaluation; we call such hash functions t-local. Such hash functions have applications to nearly optimal bounded-use information-theoretic cryptography. Local hash functions were previously studied in several works starting with Siegel (FOCS’89, SICOMP’04). For a hash function with n-bit input and output size, we get the following new results:

We study error-detection and error-correction codes for computationally bounded adversarial channels. We consider seeded codes where the polynomial-time encoding and decoding procedures share a public random seed, but are otherwise deterministic. An adversarial channel gets this seed and can perform arbitrary polynomial-time computation to adaptively select both the message to be encoded and a bounded number of errors to be added to the resulting codeword. The goal is to detect or correct such errors with overwhelming probability, while achieving better trade-offs between rate and error tolerance than those possible for computationally unbounded channels. For large alphabets, prior work (ITCS ‘25) achieves essentially optimal parameters under minimal cryptographic assumptions. However, for the binary alphabet, prior works (TCC ‘20, EUROCRYPT ‘25) either only achieved a weaker notion of selective security under the learning with errors (LWE) assumption, or relied on non-standard cryptographic assumptions to get the full notion of adaptive security. In this work, we construct binary codes that achieve the full notion of adaptive security assuming trapdoor hashing, which can in turn be instantiated under a variety of standard cryptographic assumptions such as LWE, or Decisional DiffieHellman (DDH), or Quadratic Residuosity (QR), or Decisional Composite Residuosity (DCR). For error detection, our codes get essentially optimal rate $R \approx 1$ and relative error tolerance $p \approx \frac{1}{2}$. For error correction, they can uniquely correct $p\lt1 / 4$ fraction of errors with a rate R matching that of the best known list-decodable codes for this error tolerance. As a central technical tool of potentially independent interest, we construct multi-input correlation intractable hashing for “shifted output relations” under the standard cryptographic assumptions above.

The random oracle model (ROM) allows us to opti-mistically reason about security properties of cryptographic hash functions, and has been hugely influential in designing practical cryptosystems. But it is overly optimistic against non-uniform adversaries, and often suggests security properties and security levels unachievable by any real hash function. To reconcile with this discrepancy, Unruh [CRYPTO '07] proposed the auxiliary-input random oracle model (AI-ROM), where a non-uniform attacker additionally gets a bounded amount of advice about the random oracle. Proving security in the AI-ROM is often much more difficult, but a series of works starting with Unruh provided useful technical tools to do so. Although these tools lead to good results in the information-theoretic setting, they are unsatisfactory in the computational setting, where the random oracle is used alongside other computational hardness assumptions. At the most basic level, we did not even know whether it is possible to efficiently simulate random oracle queries given auxiliary input, which has remained as an explicit open problem since the work of Unruh. In this work, we resolve the above open problem and show how to efficiently simulate auxiliary-input random oracles. Moreover, the simulation has low concrete overhead, leading to small losses in exact security. We use it to prove the security of a broad class of computational schemes in the AI-ROM, including the first non-interactive zero-knowledge (NIZK) scheme in the AI-ROM. As a tool of independent interest, we develop a new notion of ultra-secure pseudorandom functions with fast RAM evaluation, which can achieve$2^{\lambda}$security while having sublinear$\mathrm{o}(\lambda)$evaluation time.

We show how to generically improve the succinctness of non-interactive publicly verifiable batch argument (BARG) systems. In particular, we show (under a mild additional assumption) how to convert a BARG that generates proofs of length poly (m)· k1−є, where m is the length of a single instance and k is the number of instances being batched, into one that generates proofs of length poly (m, logk), which is the gold standard for succinctness of BARGs. By prior work, such BARGs imply the existence of SNARGs for deterministic time T computation with succinctness poly(logT).

A (single server) private information retrieval (PIR) allows a client to read data from a public database held on a remote server, without revealing to the server which locations she is reading. In a doubly efficient PIR (DEPIR), the database is first preprocessed, but the server can subsequently answer any client’s query in time that is sub-linear in the database size. Prior work gave a plausible candidate for a public-key variant of DEPIR, where a trusted party is needed to securely preprocess the database and generate a corresponding public key for the clients; security relied on a new non-standard code-based assumption and a heuristic use of ideal obfuscation. In this work we construct the stronger unkeyed notion of DEPIR, where the preprocessing is a deterministic procedure that the server can execute on its own. Moreover, we prove security under just the standard ring learning-with-errors (RingLWE) assumption. For a database of size N and any constant ε>0, the preprocessing run-time and size is O(N1+ε), while the run-time and communication-complexity of each PIR query is polylog(N). We also show how to update the preprocessed database in time O(Nε). Our approach is to first construct a standard PIR where the server’s computation consists of evaluating a multivariate polynomial; we then convert it to a DEPIR by preprocessing the polynomial to allow for fast evaluation, using the techniques of Kedlaya and Umans (STOC ’08).

We give new instantiations of the Fiat-Shamir transform using explicit, efficiently computable hash functions. We improve over prior work by reducing the security of these protocols to qualitatively simpler and weaker computational hardness assumptions. As a consequence of our framework, we obtain the following concrete results.

We introduce a new cryptographic primitive called laconic function evaluation (LFE). Using LFE, Alice can compress a large circuit f into a small digest. Bob can encrypt some data x under this digest in a way that enables Alice to recover f(x) without learning anything else about Bob's data. For the scheme to be laconic, we require that the size of the digest, the run-time of the encryption algorithm and the size of the ciphertext should all be small, much smaller than the circuit-size of f. We construct an LFE scheme for general circuits under the learning with errors (LWE) assumption, where the above parameters only grow polynomially with the depth but not the size of the circuit. We then use LFE to construct secure 2-party and multi-party computation (2PC, MPC) protocols with novel properties: We construct a 2-round 2PC protocol between Alice and Bob with respective inputs xA, xBin which Alice learns the output f(xA, xB) in the second round. This is the first such protocol which is “Bob-optimized”, meaning that Alice does all the work while Bob's computation and the total communication of the protocol are smaller than the size of the circuit f or even Alice's input xA. In contrast, prior solutions based on fully homomorphic encryption are “Alice-optimized”. . We construct an MPC protocol, which allows N parties to securely evaluate a function f(x1, ..., xN) over their respective inputs, where the total amount of computation performed by the parties during the protocol execution is smaller than that of evaluating the function itself! Each party has to individually pre-process the circuit f before the protocol starts and post-process the protocol transcript to recover the output after the protocol ends, and the cost of these steps is larger than the circuit size. However, this gives the first MPC where the computation performed by each party during the actual protocol execution, from the time the first protocol message is sent until the last protocol message is received, is smaller than the circuit size.

We construct a delegation scheme for verifying non-deterministic computations, with complexity proportional only to the non-deterministic space of the computation. Specifically, letting n denote the input length, we construct a delegation scheme for any language verifiable in non-deterministic time and space (T(n), S(n)) with communication complexity poly(S(n)), verifier runtime n.polylog(T(n))+poly(S(n)), and prover runtime poly(T(n)).

We introduce the notion of “non-malleable codes” which relaxes the notion of error correction and error detection. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely unrelated value. In contrast to error correction and error detection, non-malleability can be achieved for very rich classes of modifications. We construct an efficient code that is non-malleable with respect to modifications that affect each bit of the codeword arbitrarily (i.e., leave it untouched, flip it, or set it to either 0 or 1), but independently of the value of the other bits of the codeword. Using the probabilistic method, we also show a very strong and general statement: there exists a non-malleable code for every “small enough” family F of functions via which codewords can be modified. Although this probabilistic method argument does not directly yield efficient constructions, it gives us efficient non-malleable codes in the random-oracle model for very general classes of tampering functions—e.g., functions where every bit in the tampered codeword can depend arbitrarily on any 99% of the bits in the original codeword. As an application of non-malleable codes, we show that they provide an elegant algorithmic solution to the task of protecting functionalities implemented in hardware (e.g., signature cards) against “tampering attacks.” In such attacks, the secret state of a physical system is tampered, in the hopes that future interaction with the modified system will reveal some secret information. This problem was previously studied in the work of Gennaro et al. in 2004 under the name “algorithmic tamper proof security” (ATP). We show that non-malleable codes can be used to achieve important improvements over the prior work. In particular, we show that any functionality can be made secure against a large class of tampering attacks, simply by encoding the secret state with a non-malleable code while it is stored in memory.

A watermarking scheme for programs embeds some information called a mark into a program while preserving its functionality. No adversary can remove the mark without damaging the functionality of the program. In this work, we study the problem of watermarking various cryptographic programs such as pseudorandom function (PRF) evaluation, decryption, and signing. For example, given a PRF $F$, we create a marked program $\widetilde{C}$ that evaluates $F(\cdot)$. An adversary that gets $\widetilde{C}$ cannot come up with any program $C^*$ in which the mark is removed but which still evaluates the PRF correctly on even a small fraction of the inputs. The work of Barak et al. [ CRYPTO 2001, Springer, Berlin, 2001, pp. 1--18; J. ACM, 59 (2012), 6] shows that, assuming indistinguishability obfuscation (iO), such watermarking is impossible if the marked program $\widetilde{C}$ evaluates the original program with perfect correctness. In this work we show that, assuming iO, such watermarking is possible if the marked program $\widetilde{C}$ is allowed to err with even a negligible probability, which would be undetectable to the user. We also significantly extend the impossibility results to our relaxed setting. Our watermarking schemes are public key, meaning that we use a secret marking key to embed marks in programs, and a public detection key that allows anyone to detect marks in programs. Our schemes are secure against chosen program attacks where the adversary is given oracle access to the marking functionality. We emphasize that our security notion of watermark nonremovability considers arbitrary adversarial strategies to modify the marked program, in contrast to the prior works [R. Nishimaki in EUROCRYPT 2013, Springer, Berlin, pp. 111--125].

We show how to obfuscate a large and expressive class of programs, which we call compute-and-compare programs, under the learning-with-errors (LWE) assumption. Each such program CC[f, y] is parametrized by an arbitrary polynomial-time computable function f along with a target value y and we define CC[f,y](x) to output 1 if f(x) = y and 0 otherwise. In other words, the program performs an arbitrary computation f and then compares its output against a target y. Our obfuscator satisfies distributional virtual-blackbox security, which guarantees that the obfuscated program does not reveal any partial information about the function f or the target value y, as long as they are chosen from some distribution where y has sufficient pseudo-entropy given f. We also extend our result to multi-bit compute-and-compare programs MBCC[f, y, z](x) which output a message z if f(x) = y. Compute-and-compare programs are powerful enough to capture many interesting obfuscation tasks as special cases. This includes obfuscating conjunctions, and therefore we improve on the prior work of Brakerski et al. (ITCS '16) which constructed a conjunction obfuscator under a non-standard “entropic” ring-LWE assumption, while here we obfuscate a significantly broader class of programs under standard LWE. We show that our obfuscator has several interesting applications. For example, we can take any encryption scheme and publish an obfuscated plaintext equality tester that allows users to check whether a ciphertext decrypts to some target value y; as long as y has sufficient pseudo-entropy this will not harm semantic security. We can also use our obfuscator to generically upgrade attribute-based encryption to predicate encryption with one-sided attribute-hiding security, and to upgrade witness encryption to indistinguishability obfuscation which is secure for all “null circuits”. Furthermore, we show that our obfuscator gives new circular-security counterexamples for public-key bit encryption and for unbounded length key cycles. Our result uses the graph-induced multi-linear maps of Gentry, Gorbunov and Halevi (TCC '15), but only in a carefully restricted manner which is provably secure under LWE. Our technique is inspired by ideas introduced in a recent work of Goyal, Koppula and Waters (EUROCRYPT '17) in a seemingly unrelated context.

We show how to securely obfuscate conjunctions, which are functions f(x1,...,xn) = ∧i∈I yi where I ⊆ [n] and each literal yi is either just xi or ¬ xi e.g., f(xi,...,x_n) = xi ⊆ ¬ x3 ⊆ ¬ x7 ... ⊆ x{n-1. Whereas prior work of Brakerski and Rothblum (CRYPTO 2013) showed how to achieve this using a non-standard object called cryptographic multilinear maps, our scheme is based on an "entropic" variant of the Ring Learning with Errors (Ring LWE) assumption. As our core tool, we prove that hardness assumptions on the recent multilinear map construction of Gentry, Gorbunov and Halevi (TCC 2015) can be established based on entropic Ring LWE. We view this as a first step towards proving the security of additional mutlilinear map based constructions, and in particular program obfuscators, under standard assumptions.

A watermarking scheme for programs embeds some information called a mark into a program while preserving its functionality. No adversary can remove the mark without damaging the functionality of the program. In this work, we study the problem of watermarking various cryptographic programs such as pseudorandom function (PRF) evaluation, decryption, and signing. For example, given a PRF key K, we create a marked program C that evaluates the PRF F(K,). An adversary that gets C cannot come up with any program C* in which the mark is removed but which still evaluates the PRF correctly on even a small fraction of the inputs. The work of Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan, and Yang (CRYPTO'01 and Journal of ACM 59(2)) shows that, assuming indistinguishability obfuscation (iO), such watermarking is impossible if the marked program C evaluates the original program with perfect correctness. In this work we show that, assuming iO, such watermarking is possible if the marked program C is allowed to err with even a negligible probability, which would be undetectable to the user. Our watermarking schemes are public key, namely we use a secret marking key to embed marks in programs, and a public detection key that allows anyone to detect marks in programs. Our schemes are secure against chosen program attacks, that is even if the adversary is given oracle access to the marking functionality. We emphasize that our security notion of watermark non-removability considers arbitrary adversarial strategies to modify the marked program, in contrast to the prior works (Nishimaki, EUROCRYPT '13).

Non-malleable codes, defined by Dziembowski, Pietrzak, and Wichs (ICS '10), provide roughly the following guarantee: if a codeword c encoding some message x is tampered to c'= f (c) such that c' ≠ c, then the tampered message x' contained in c' reveals no information about x. The nonmalleable codes have applications to immunizing cryptosystems against tampering attacks and related-key attacks. One cannot have an efficient non-malleable code that protects against all efficient tampering functions f . However, in this paper we show “the next best thing”: for any polynomial bound s given a-priori, there is an efficient non-malleable code that protects against all tampering functions f computable by a circuit of size s. More generally, for any family of tampering functions F of size |F| ≤ 2s, there is an efficient non-malleable code that protects against all f ∈ F. The rate of our codes, defined as the ratio of message to codeword size, approaches 1. Our results are information-theoretic and our main proof technique relies on a careful probabilistic method argument using limited independence. As a result, we get an efficiently samplable family of efficient codes, such that a random member of the family is non-malleable with overwhelming probability. Alternatively, we can view the result as providing an efficient non-malleable code in the “common reference string” model. We also introduce a new notion of non-malleable key derivation, which uses randomness x to derive a secret key y = h(x) in such a way that, even if x is tampered to a different value x'= f (x), the derived key y' = h(x') does not reveal any information about y. Our results for non-malleable key derivation are analogous to those for non-malleable codes. As a useful tool in our analysis, we rely on the notion of “leakage-resilient storage” of Davi, Dziembowski, and Venturi (SCN '10), and, as a result of independent interest, we also significantly improve on the parameters of such schemes.

We study the communication complexity of secure function evaluation (SFE). Consider a setting where Alice has a short input χA, Bob has an input χB and we want Bob to learn some function y = f(χA, χB) with large output size. For example, Alice has a small secret decryption key, Bob has a large encrypted database and we want Bob to learn the decrypted data without learning anything else about Alice's key. In a trivial insecure protocol, Alice can just send her short input χA to Bob. However, all known SFE protocols have communication complexity that scales with size of the output y, which can potentially be much larger. Is such 'output-size dependence' inherent in SFE'

In a homomorphic signature scheme, a user Alice signs some large dataset x using her secret signing key and uploads the signed data to an untrusted remote server. The server can then run some computation y=f(x) over the signed data and homomorphically derive a short signature σf,y certifying that y is the correct output of the computation f. Anybody can verify the tuple (f, y, σf,y) using Alice's public verification key and become convinced of this fact without having to retrieve the entire underlying data. In this work, we construct the first leveled fully homomorphic signature} schemes that can evaluate arbitrary {circuits} over signed data. Only the maximal {depth} d of the circuits needs to be fixed a-priori at setup, and the size of the evaluated signature grows polynomially in d, but is otherwise independent of the circuit size or the data size. Our solution is based on the (sub-exponential) hardness of the small integer solution (SIS) problem in standard lattices and satisfies full (adaptive) security. In the standard model, we get a scheme with large public parameters whose size exceeds the total size of a dataset. In the random-oracle model, we get a scheme with short public parameters. In both cases, the schemes can be used to sign many different datasets. The complexity of verifying a signature for a computation f is at least as large as that of computing f, but can be amortized when verifying the same computation over many different datasets. Furthermore, the signatures can be made context-hiding so as not to reveal anything about the data beyond the outcome of the computation.

We construct the first schemes that allow a client to privately outsource arbitrary program executions to a remote server while ensuring that: (I) the client's work is small and essentially independent of the complexity of the computation being outsourced, and (II) the server's work is only proportional to the run-time of the computation on a random access machine (RAM), rather than its potentially much larger circuit size. Furthermore, our solutions are non-interactive and have the structure of reusable garbled RAM programs, addressing an open question of Lu and Ostrovsky (Eurocrypt 2013). We also construct schemes for an augmented variant of the above scenario, where the client can initially outsource a large private and persistent database to the server, and later outsource arbitrary program executions with read/write access to this database. Our solutions are built from non-reusable garbled RAM in conjunction with new types of reusable garbled circuits that are more efficient than prior solutions but only satisfy weaker security. For the basic setting without a persistent database, we can instantiate the required type of reusable garbled circuits from indistinguishability obfuscation or from functional encryption for circuits as a black-box. For the more complex setting with a persistent database, we can instantiate the required type of reusable garbled circuits using stronger notions of obfuscation. Our basic solution also requires the client to perform a one-time pre-processing step to garble a program at the cost of its RAM run-time, and we can avoid this cost using stronger notions of obfuscation. It remains an open problem to instantiate these new types of reusable garbled circuits under weaker assumptions, possibly avoiding obfuscation altogether. We show several simple extensions of our results and techniques to achieve: efficiency proportional to the input-specific RAM run-time, verifiability of outsourced RAM computation, functional encryption for RAMs, and a candidate obfuscation for RAMs.

A pseudo-random number generator (PRNG) is a deterministic algorithm that produces numbers whose distribution is indistinguishable from uniform. A formal security model for PRNGs with input was proposed in 2005 by Barak and Halevi (BH). This model involves an internal state that is refreshed with a (potentially biased) external random source, and a cryptographic function that outputs random numbers from the continually internal state. In this work we extend the BH model to also include a new security property capturing how it should accumulate the entropy of the input data into the internal state after state compromise. This property states that a good PRNG should be able to eventually recover from compromise even if the entropy is injected into the system at a very slow pace, and expresses the real-life expected behavior of existing PRNG designs. Unfortunately, we show that neither the model nor the specific PRNG construction proposed by BH meet this new property, despite meeting a weaker robustness notion introduced by BH. From a practical side, we give a precise assessment of the Linux PRNGs, /dev/random and /dev/urandom. In particular, we show attacks proving that these PRNGs are not robust according to our definition, due to vulnerabilities in their entropy estimator and their internal mixing function. Finally, we propose a simple PRNG construction that is provably robust in our new and stronger adversarial model and we show that it is more efficient than the Linux PRNGs. We therefore recommend to use this construction whenever a PRNG with input is used for cryptography.

There has been much recent progress in constructing cryptosystems that maintain their security without requiring uniform randomness and perfect secrecy. These schemes are motivated by a diverse set of problems such as providing resilience to side-channel leakage, using weak physical sources of randomness as secret keys, and allowing deterministic encryption for high-entropy messages. Nevertheless, despite this progress, some basic and seemingly achievable security properties have eluded our reach. For example, we are unable to prove the security of basic tools for manipulating weak/leaky random sources, such as as pseudo-entropy generators and seed-dependent computational condensers. We also do not know how to prove leakage-resilient security of any cryptosystem with a uniquely determined secret key. In the context of deterministic encryption we do not have a standard-model constructions achieving the strongest notion of security proposed by Bellare, Boldyreva and O'Neill (CRYPTO '07), that would allow us to encrypt arbitrarily correlated messages of sufficiently large individual entropy.

We consider the question of how to store a value secretly on devices that continually leak information about their internal state to an external attacker. If the secret value is stored on a single device from which it is efficiently retrievable, and the attacker can leak even a single predicate of the internal state of that device, then she may learn some information about the secret value itself. Therefore, we consider a setting where the secret value is shared between multiple devices (or multiple components of a single device), each of which continually leaks arbitrary adaptively chosen predicates its individual state. Since leakage is continual, each device must also continually update its state so that an attacker cannot just leak it entirely one bit at a time. In our model, the devices update their state individually and asynchronously, without any communication between them. The update process is necessarily randomized, and its randomness can leak as well. As our main result, we construct a sharing scheme for two devices, where a constant fraction of the internal state of each device can leak in between and during updates. Our scheme has the structure of a public-key encryption, where one share is a secret key and the other is a ciphertext. As a contribution of independent interest, we also get public-key encryption in the continual leakage model, introduced by Brakerski et al. and Dodis et al. (FOCS '10). This scheme tolerates continual leakage on the secret key and the updates, and simplifies the recent construction of Lewko, Lewko and Waters (STOC '11). For our main result, we show how to update the ciphertexts of the encryption scheme so that the message remains hidden even if an attacker interleaves leakage on secret key and ciphertext shares. The security of our scheme is based on the linear assumption in prime-order bilinear groups. We also provide an extension to general access structures realizable by linear secret sharing schemes across many devices. The main advantage of this extension is that the state of some devices can be compromised entirely, while that of the all remaining devices is susceptible to continual leakage. Lastly, we show impossibility of information theoretic sharing schemes in our model, where continually leaky devices update their state individually.

An argument system for NP is succinct, if its communication complexity is polylogarithmic the instance and witness sizes. The seminal works of Kilian '92 and Micali '94 show that such arguments can be constructed under standard cryptographic hardness assumptions with four rounds of interaction, and that they be made non-interactive in the random-oracle model. However, we currently do not have any construction of succinct non-interactive arguments (SNARGs) in the standard model with a proof of security under any simple cryptographic assumption.

We say that a cryptographic scheme is Continuous Leakage-Resilient (CLR), if it allows users to refresh their secret keys, using only fresh local randomness, such that: 1. The scheme remains functional after any number of key refreshes, although the public key never changes. Thus, the “outside world'' is neither affected by these key refreshes, nor needs to know about their frequency. 2. The scheme remains secure even if the adversary can continuously leak arbitrary information about the current secret-key, as long as the amount of leaked information is bounded in between any two successive key refreshes. There is no bound on the total amount of information that can be leaked during the lifetime of the system. In this work, we construct a variety of practical CLR schemes, including CLR one-way relations, CLR signatures, CLR identification schemes, and CLR authenticated key agreement protocols. For each of the above, we give general constructions, and then show how to instantiate them efficiently using a well established assumption on bilinear groups, called the K-Linear assumption (for any constant K greater than or equal to 1). Our constructions are highly modular, and we develop many interesting techniques and building-blocks along the way, including: leakage-indistinguishable re-randomizable relations, homomorphic NIZKs, and leakage-of-cipher text non-malleable encryption schemes.

We study the question of basing symmetric key cryptography on weak secrets. In this setting, Alice and Bob share an n-bit secret W, which might not be uniformly random, but the adversary has at least k bits of uncertainty about it (formalized using conditional min-entropy). Since standard symmetric-key primitives require uniformly random secret keys, we would like to construct an authenticated key agreement protocol in which Alice and Bob use W to agree on a nearly uniform key R, by communicating over a public channel controlled by an active adversary Eve. We study this question in the information theoretic setting where the attacker is computationally unbounded. We show that single-round (i.e. one message) protocols do not work when k ≤ n/2, and require poor parameters even when n/2<k<