We introduce the notion of pseudorandom obfuscation (PRO), a way to obfuscate (keyed) pseudorandom functions $f_K$ in an average-case sense. We introduce several variants of pseudorandom obfuscation and show constructions and applications. For some of our applications that can be achieved using full-fledged indistinguishability obfuscation (iO), we show constructions using lattice-based assumptions alone; the other applications we enable using PRO are simply not known even assuming iO. We...

Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory. Is there a...

This work describes a digital signature scheme constructed from a zero-knowledge proof of knowledge of a pre-image of the Rescue Prime Optimized (RPO) permutation. The proof of knowledge is constructed with the DEEP-ALI interactive oracle proof combined with the Ben-Sasson--Chiesa--Spooner (BCS) transformation in the random oracle model. The EUF-CMA security of the resulting signature scheme is established from the UC-friendly security properties of the BCS transformation and the pre-image...

This paper presents a novel reduction from the average-case hardness of the Module Inhomogeneous Short Integer Solution (M-ISIS) problem to the worst-case hardness of the Closest Vector Problem (CVP) by defining and leveraging “perfect” lattices for cryptographic purposes. Perfect lattices, previously only theoretical constructs, are characterized by their highly regular structure, optimal density, and a central void, which we term the “Origin Cell.” The simplest Origin Cell is a...

Cryptography often considers the strongest yet plausible attacks in the real world. Preprocessing (a.k.a. non-uniform attack) plays an important role in both theory and practice: an efficient online attacker can take advantage of advice prepared by a time-consuming preprocessing stage. Salting is a heuristic strategy to counter preprocessing attacks by feeding a small amount of randomness to the cryptographic primitive. We present general and tight characterizations of preprocessing...

In an Instance-Hiding Interactive Proof (IHIP) [Beaver et al. CRYPTO 90], an efficient verifier with a _private_ input x interacts with an unbounded prover to determine whether x is contained in a language L. In addition to completeness and soundness, the instance-hiding property requires that the prover should not learn anything about x in the course of the interaction. Such proof systems capture natural privacy properties, and may be seen as a generalization of the influential concept of...

In this work, we study the worst-case to average-case hardness of the Learning with Errors problem (LWE) under an alternative measure of hardness $−$ the maximum success probability achievable by a probabilistic polynomial-time (PPT) algorithm. Previous works by Regev (STOC 2005), Peikert (STOC 2009), and Brakerski, Peikert, Langlois, Regev, Stehle (STOC 2013) give worst-case to average-case reductions from lattice problems to LWE, specifically from the approximate decision variant of the...

While classic result in the 1980s establish that one-way functions (OWFs) imply the existence of pseudorandom generators (PRGs) which in turn imply pseudorandom functions (PRFs), the constructions (most notably the one from OWFs to PRGs) is complicated and inefficient. Consequently, researchers have developed alternative \emph{direct} constructions of PRFs from various different concrete hardness assumptions. In this work, we continue this thread of work and demonstrate the first direct...

Over the past few decades, we have seen a proliferation of advanced cryptographic primitives with lossy or homomorphic properties built from various assumptions such as Quadratic Residuosity, Decisional Diffie-Hellman, and Learning with Errors. These primitives imply hard problems in the complexity class $\mathcal{SZK}$ (statistical zero-knowledge); as a consequence, they can only be based on assumptions that are broken in $\mathcal{BPP}^{\mathcal{SZK}}$. This poses a barrier for building...

The Learning with Errors (LWE) problem has been widely utilized as a foundation for numerous cryptographic tools over the years. In this study, we focus on an algebraic variant of the LWE problem called Group ring LWE (GR-LWE). We select group rings (or their direct summands) that underlie specific families of finite groups constructed by taking the semi-direct product of two cyclic groups. Unlike the Ring-LWE problem described in \cite{lyubashevsky2010ideal}, the multiplication operation in...

We prove a tight parallel repetition theorem for $3$-message computationally-secure quantum interactive protocols between an efficient challenger and an efficient adversary. We also prove under plausible assumptions that the security of $4$-message computationally secure protocols does not generally decrease under parallel repetition. These mirror the classical results of Bellare, Impagliazzo, and Naor [BIN97]. Finally, we prove that all quantum argument systems can be generically compiled...

The presumed hardness of the Shortest Vector Problem for ideal lattices (Ideal-SVP) has been a fruitful assumption to understand other assumptions on algebraic lattices and as a security foundation of cryptosystems. Gentry [CRYPTO'10] proved that Ideal-SVP enjoys a worst-case to average-case reduction, where the average-case distribution is the uniform distribution over the set of inverses of prime ideals of small algebraic norm (below $d^{O(d)}$ for cyclotomic fields, here $d$ refers to...

Whether one-way functions (OWF) exist is arguably the most important problem in Cryptography, and beyond. While lots of candidate constructions of one-way functions are known, and recently also problems whose average-case hardness characterize the existence of OWFs have been demonstrated, the question of whether there exists some \emph{worst-case hard problem} that characterizes the existence of one-way functions has remained open since their introduction in 1976. In this work, we...

The low-degree method postulates that no efficient algorithm outperforms low-degree polynomials in certain hypothesis-testing tasks. It has been used to understand computational indistinguishability in high-dimensional statistics. We explore the use of the low-degree method in the context of cryptography. To this end, we apply it in the design and analysis of a new public-key encryption scheme whose security is based on Goldreich's pseudorandom generator. The scheme is a combination of...

A famous reduction by Regev shows that random instances of the Learning With Errors (LWE) problem are asymptotically at least as hard as a worst-case lattice problem. As such, by assuming that standard lattice problems are hard to solve, the asymptotic security of cryptosystems based on the LWE problem is guaranteed. However, it has not been clear to which extent, if any, this reduction provides support for the security of present concrete parametrizations. In this work we therefore use...

In the average-case $k$-SUM problem, given $r$ integers chosen uniformly at random from $\{0,\ldots,M-1\}$, the objective is to find a "solution" set of $k$ numbers that sum to $0$ modulo $M$. In the dense regime of $M \leq r^k$, where solutions exist with high probability, the complexity of these problems is well understood. Much less is known in the sparse regime of $M\gg r^k$, where solutions are unlikely to exist. In this work, we initiate the study of the sparse regime for...

Symmetry of Information (SoI) is a fundamental property of Kolmogorov complexity that relates the complexity of a pair of strings and their conditional complexities. Understanding if this property holds in the time-bounded setting is a longstanding open problem. In the nineties, Longpré and Mocas (1993) and Longpré and Watanabe (1995) established that if SoI holds for time-bounded Kolmogorov complexity then cryptographic one-way functions do not exist, and asked if a converse holds. We...

Recent code-based cryptosystems rely, among other things, on the hardness of the decisional decoding problem. If the search version is well understood, both from practical and theoretical standpoints, the decision version has been less studied in the literature, and little is known about its relationships with the search version, especially for structured variants. On the other hand, in the world of Euclidean lattices, the situation is rather different, and many reductions exist, both for...

In this work, we consider the worst and average case hardness of the decoding problems that are the basis for code-based cryptography. By a decoding problem, we consider inputs of the form $(\mathbf{G}, \mathbf{m} \mathbf{G} + \mathbf{t})$ for a matrix $\mathbf{G}$ that generates a code and a noise vector $\mathbf{t}$, and the algorithm's goal is to recover $\mathbf{m}$. We consider a natural strategy for creating a reduction to an average-case problem: from our input we simulate a...

Typical lattice-based cryptosystems are commonly believed to resist multi-target attacks. For example, the New Hope proposal stated that it avoids "all-for-the-price-of-one attacks". An ACM CCS 2021 paper from Duman–Hövelmanns–Kiltz–Lyubashevsky–Seiler stated that "we can show that Adv_{PKE}^{IND-CPA} ≈ Adv_{PKE}^{(n,q_C)-IND-CPA} for "lattice-based schemes" such as Kyber, i.e. that one-out-of-many-target IND-CPA is as difficult to break as single-target IND-CPA, assuming "the hardness of...

As a quantum analogue of one-way function, the notion of one-way quantum state generator is recently proposed by Morimae and Yamakawa (CRYPTO'22), which is proved to be implied by the pseudorandom state and can be used to devise the one-time secure digital signature. Due to Kretschmer's result (TQC'20), it's believed that pseudorandom state generator requires less than post-quantum secure one-way function. Unfortunately, it remains to be unknown how to achieve the one-way quantum...

It has been half a century since the first several NP-complete problems were discovered by Cook, Karp and Levin in the early 1970s. Till today, thousands of NP-complete problems have been found. Most of them are of combinatorial flavor. We discover new possibilities in purer mathematics and introduce more structures to the theory of computation. We propose a family of abstract problems related to the subset product problem. To describe hardness of abstract problems, we propose a new hardness...

Can we hope to provide provable security against model extraction attacks? As a step towards a theoretical study of this question, we unify and abstract a wide range of "observational" model extraction defenses (OMEDs) --- roughly, those that attempt to detect model extraction by analyzing the distribution over the adversary's queries. To accompany the abstract OMED, we define the notion of complete OMEDs --- when benign clients can freely interact with the model --- and sound OMEDs --- when...

The finite field isomorphism (FFI) problem was introduced in PKC'18, as an alternative to average-case lattice problems (like LWE, SIS, or NTRU). As an application, the same paper used the FFI problem to construct a fully homomorphic encryption scheme. In this work, we prove that the decision variant of the FFI problem can be solved in polynomial time for any field characteristics $q= \Omega(\beta n^2)$, where $q,\beta,n$ parametrize the FFI problem. Then we use our result from the FFI...

$\newcommand{\Z}{\mathbb{Z}} \newcommand{\basis}{B}$We study the computational problem of finding a shortest non-zero vector in a rotation of $\Z^n$, which we call $\Z$SVP. It has been a long-standing open problem to determine if a polynomial-time algorithm for $\Z$SVP exists, and there is by now a beautiful line of work showing how to solve it efficiently in certain very special cases. However, despite all of this work, the fastest known algorithm that is proven to solve $\Z$SVP is still...

We show polynomial-time quantum algorithms for the following problems: (*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a constant. (*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions, including bounded uniform distributions and Laplace...

A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity, ${\rm K}^t$, is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures: * We show, perhaps surprisingly, that the $\rm KT$ complexity is bounded-error average-case hard if and only if there exist one-way functions in *constant parallel time* (i.e., ${\sf NC}^0$). This...

We show equivalence between the existence of one-way functions and the existence of a \emph{sparse} language that is hard-on-average w.r.t. some efficiently samplable ``high-entropy'' distribution. In more detail, the following are equivalent: - The existence of a $S(\cdot)$-sparse language $L$ that is hard-on-average with respect to some samplable distribution with Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq 4\log n$; - The existence of a $S(\cdot)$-sparse...

The 25 year-old NTRU problem is an important computational assumption in public-key cryptography. However, from a reduction perspective, its relative hardness compared to other problems on Euclidean lattices is not well-understood. Its decision version reduces to the search Ring-LWE problem, but this only provides a hardness upper bound. We provide two answers to the long-standing open problem of providing reduction-based evidence of the hardness of the NTRU problem. First, we reduce the...

The dual attack has long been considered a relevant attack on lattice-based cryptographic schemes relying on the hardness of learning with errors (LWE) and its structured variants. As solving LWE corresponds to finding a nearest point on a lattice, one may naturally wonder how efficient this dual approach is for solving more general closest vector problems, such as the classical closest vector problem (CVP), the variants bounded distance decoding (BDD) and approximate CVP, and preprocessing...

Liu and Pass (FOCS'20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity---namely, Levin's notion of Kolmogorov Complexity---whose hardness is closely related to the problem of whether $\EXP \neq \BPP$. In more detail, let $Kt(x)$ denote the Levin-Kolmogorov...

Let $\mktp[s]$ be the set of strings $x$ such that $K^t(x) \leq s(|x|)$, where $K^t(x)$ denotes the $t$-bounded Kolmogorov complexity of the truthtable described by $x$. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every $\varepsilon>0$, polynomial $t(n) \geq (1+\varepsilon)n$, and every ``nice'' class $\F$ of super-polynomial functions, the following are equivalent: - the existence of some function $T \in \F$ such that $T$-hard one-way functions...

We present the first natural $\NP$-complete problem whose average-case hardness w.r.t. the uniform distribution over instances is \emph{equivalent} to the existence of one-way functions (OWFs). The problem, which originated in the 1960s, is the \emph{Conditional Time-Bounded Kolmogorov Complexity Problem}: let $K^t(x \mid z)$ be the length of the shortest ``program'' that, given the ``auxiliary input'' $z$, outputs the string $x$ within time $t(|x|)$, and let $\mcktp[\zeta]$ be the set of...

A secret-sharing scheme allows to distribute a secret $s$ among $n$ parties such that only some predefined ``authorized'' sets of parties can reconstruct the secret, and all other ``unauthorized'' sets learn nothing about $s$. The collection of authorized/unauthorized sets can be captured by a monotone function $f:\{0,1\}^n\rightarrow \{0,1\}$. In this paper, we focus on monotone functions that all their min-terms are sets of size $a$, and on their duals -- monotone functions whose max-terms...

Several recent proposals of efficient public-key encryption are based on variants of the polynomial learning with errors problem ($\mathsf{PLWE}^f$) in which the underlying polynomial ring $\mathbb{Z}_q[x]/f$ \ is replaced with the (related) modular integer ring $\mathbb{Z}_{f(q)}$; the corresponding problem is known as Integer Polynomial Learning with Errors ($\mathsf{I-PLWE}^f$). Cryptosystems based on $\mathsf{I-PLWE}^f$ and its variants can exploit optimised big-integer arithmetic to...

Constructing one-way functions from average-case hardness is a long-standing open problem. A positive result would exclude Pessiland (Impagliazzo ’95) and establish a highly desirable win-win situation: either (symmetric) cryptography exists unconditionally, or all NP problems can be solved efficiently on the average. Motivated by the lack of progress on this seemingly very hard question, we initiate the investigation of weaker yet meaningful candidate win-win results of the following type:...

The complexity class TFNP consists of all NP search problems that are total in the sense that a solution is guaranteed to exist for all instances. Over the years, this class has proved to illuminate surprising connections among several diverse subfields of mathematics like combinatorics, computational topology, and algorithmic game theory. More recently, we are starting to better understand its interplay with cryptography. We know that certain cryptographic primitives (e.g. one-way...

We construct a succinct non-interactive publicly-verifiable delegation scheme for any log-space uniform circuit under the sub-exponential Learning With Errors ($\mathsf{LWE}$) assumption. For a circuit $C:\{0,1\}^N\rightarrow\{0,1\}$ of size $S$ and depth $D$, the prover runs in time $\mathsf{poly}(S)$, the communication complexity is $D \cdot \mathsf{polylog} (S)$, and the verifier runs in time $(D+N) \cdot \mathsf{polylog} (S)$. To obtain this result, we introduce a new cryptographic...

Learning parity with noise (LPN) is a notorious (average-case) hard problem that has been well studied in learning theory, coding theory and cryptography since the early 90's. It further inspires the Learning with Errors (LWE) problem [Regev, STOC 2005], which has become one of the central building blocks for post-quantum cryptography and advanced cryptographic primitives. Unlike LWE whose hardness can be reducible from worst-case lattice problems, no corresponding worst-case hardness...

We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial $t(n)\geq (1+\varepsilon)n, \varepsilon>0$, the following are equivalent: - One-way functions exists (which in turn is equivalent to the existence of secure private-key encryption schemes, digital signatures, pseudorandom generators, pseudorandom functions, commitment schemes, and more); - $t$-time bounded Kolmogorov Complexity, $K^t$, is mildly hard-on-average (i.e., there exists a...

Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving the former. The subject of this work is ``lossy'' reductions, where the reduction loses some information about the input instance. We show that such reductions, when they exist, have interesting and powerful consequences for lifting hardness into ``useful'' hardness, namely cryptography. Our...

Fixing a number field, the space of all ideal lattices, up to isometry, is naturally an Abelian group, called the *Arakelov class group*. This fact, well known to number theorists, has so far not been explicitly used in the literature on lattice-based cryptography. Remarkably, the Arakelov class group is a combination of two groups that have already led to significant cryptanalytic advances: the class group and the unit torus. In the present article, we show that the Arakelov class group...

At CRYPTO 2017, Rosca et al. introduce a new variant of the Learning With Errors (LWE) problem, called the Middle-Product LWE (MP-LWE). The hardness of this new assumption is based on the hardness of the Polynomial LWE (P-LWE) problem parameterized by a set of polynomials, making it more secure against the possible weakness of a single defining polynomial. As a cryptographic application, they also provide an encryption scheme based on the MP-LWE problem. In this paper, we propose a...

Consider the following two fundamental open problems in complexity theory: (a) Does a hard-on-average language in NP imply the existence of one-way functions?, or (b) Does a hard-on-average language in NP imply a hard-on-average problem in TFNP (i.e., the class of total NP search problem)? Our main result is that the answer to (at least) one of these questions is yes. Both one-way functions and problems in TFNP can be interpreted as promise-true distributional NP search problems---namely,...

Cryptography is largely based on unproven assumptions, which, while believable, might fail. Notably if $P = NP$, or if we live in Pessiland, then all current cryptographic assumptions will be broken. A compelling question is if any interesting cryptography might exist in Pessiland. A natural approach to tackle this question is to base cryptography on an assumption from fine-grained complexity. Ball, Rosen, Sabin, and Vasudevan [BRSV'17] attempted this, starting from popular hardness...

There have been many successes in constructing explicit non-malleable codes for various classes of tampering functions in recent years, and strong existential results are also known. In this work we ask the following question: "When can we rule out the existence of a non-malleable code for a tampering class $\mathcal{F}$?" We show that non-malleable codes are impossible to construct for three different tampering classes: 1. Functions that change $d/2$ symbols, where $d$ is the distance of...

The hardness of finding short vectors in ideals of cyclotomic number fields (hereafter, Ideal-SVP) can serve as a worst-case assumption for numerous efficient cryptosystems, via the average-case problems Ring-SIS and Ring-LWE. For a while, it could be assumed the Ideal-SVP problem was as hard as the analog problem for general lattices (SVP), even when considering quantum algorithms. But in the last few years, a series of works has lead to a quantum algorithm for Ideal-SVP that outperforms...

Distributional collision resistance is a relaxation of collision resistance that only requires that it is hard to sample a collision (x,y) where x is uniformly random and y is uniformly random conditioned on colliding with x. The notion lies between one-wayness and collision resistance, but its exact power is still not well-understood. On one hand, distributional collision resistant hash functions cannot be built from one-way functions in a black-box way, which may suggest that they are...

Hard learning problems are central topics in recent cryptographic research. Many cryptographic primitives relate their security to difficult problems in lattices, such as the shortest vector problem. Such schemes include the possibility of decryption errors with some very small probability. In this paper we propose and discuss a generic attack for secret key recovery based on generating decryption errors. In a standard PKC setting, the model first consists of a precomputation phase...

It has been shown that, for appropriate parameters, solving the $\mathsf{SIS}$ problem in the average case is at least as hard as approximating certain lattice problems in the worst case %on any $n$-dimensional lattice to within polynomial factor $\beta\cdot\widetilde{O}(\sqrt n)$, where typically $\beta=O(\sqrt{n\log n})$ such that random $\mathsf{SIS}$ instances admit a solution. In this work, we show that $\beta=O(1)$, i.e., $\beta$ is essentially upper-bounded by a constant. This...

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from...

We give Proofs of Work (PoWs) whose hardness is based on well-studied worst-case assumptions from fine-grained complexity theory. This extends the work of (Ball et al., STOC '17), that presents PoWs that are based on the Orthogonal Vectors, 3SUM, and All-Pairs Shortest Path problems. These, however, were presented as a `proof of concept' of provably secure PoWs and did not fully meet the requirements of a conventional PoW: namely, it was not shown that multiple proofs could not be generated...

Collision resistant hashing is a fundamental concept that is the basis for many of the important cryptographic primitives and protocols. Collision resistant hashing is a family of compressing functions such that no efficient adversary can find any collision given a random function in the family. In this work we study a relaxation of collision resistance called distributional collision resistance, introduced by Dubrov and Ishai (STOC '06). This relaxation of collision resistance only...

Our main result in this work is the extension of the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. We prove that our generalization for Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the addition of a new parameter, the...

We present a worst case decoding problem whose hardness reduces to that of solving the Learning Parity with Noise (LPN) problem, in some parameter regime. Prior to this work, no worst case hardness result was known for LPN (as opposed to syntactically similar problems such as Learning with Errors). The caveat is that this worst case problem is only mildly hard and in particular admits a quasi-polynomial time algorithm, whereas the LPN variant used in the reduction requires extremely high...

Homomorphic secret sharing (HSS) is the secret sharing analogue of homomorphic encryption. An HSS scheme supports a local evaluation of functions on shares of one or more secret inputs, such that the resulting shares of the output are short. Some applications require the stronger notion of additive HSS, where the shares of the output add up to the output over a finite Abelian group. While strong feasibility results for HSS are known under specific cryptographic assumptions, many natural...

We show a general framework for constructing non-malleable codes against tampering families with average-case hardness bounds. Our framework adapts ideas from the Naor-Yung double encryption paradigm such that to protect against tampering in a class F, it suffices to have average-case hard distributions for the class, and underlying primitives (encryption and non-interactive, simulatable proof systems) satisfying certain properties with respect to the class. We instantiate our scheme in a...

Collision resistant hash functions are functions that shrink their input, but for which it is computationally infeasible to find a collision, namely two strings that hash to the same value (although collisions are abundant). In this work we study multi-collision resistant hash functions (MCRH) a natural relaxation of collision resistant hash functions in which it is difficult to find a t-way collision (i.e., t strings that hash to the same value) although finding (t-1)-way collisions could...

We give a polynomial-time quantum reduction from worst-case (ideal) lattice problems directly to the decision version of (Ring-)LWE. This extends to decision all the worst-case hardness results that were previously known for the search version, for the same or even better parameters and with no algebraic restrictions on the modulus or number field. Indeed, our reduction is the first that works for decision Ring-LWE with any number field and any modulus.

We give Proofs of Work (PoWs) whose hardness is based on a wide array of computational problems, including Orthogonal Vectors, 3SUM, All-Pairs Shortest Path, and any problem that reduces to them (this includes deciding any graph property that is statable in first-order logic). This results in PoWs whose completion does not waste energy but instead is useful for the solution of computational problems of practical interest. The PoWs that we propose are based on delegating the evaluation of...

We present functions that can be computed in some fixed polynomial time but are hard on average for any algorithm that runs in slightly smaller time, assuming widely-conjectured worst-case hardness for problems from the study of fine-grained complexity. Unconditional constructions of such functions are known from before (Goldmann et al., IPL '94), but these have been canonical functions that have not found further use, while our functions are closely related to well-studied problems and have...

The possibility of basing cryptography on the minimal assumption NP$\nsubseteq$BPP is at the very heart of complexity-theoretic cryptography. The closest we have gotten so far is lattice-based cryptography whose average-case security is based on the worst-case hardness of approximate shortest vector problems on integer lattices. The state-of-the-art is the construction of a one-way function (and collision-resistant hash function) based on the hardness of the $\tilde{O}(n)$-approximate...

Many practical lattice-based schemes are built upon the Ring-SIS or Ring-LWE problems, which are problems that are based on the presumed difficulty of finding low-weight solutions to linear equations over polynomial rings $Z_q[x]/\langle f(x) \rangle$. Our belief in the asymptotic computational hardness of these problems rests in part on the fact that there are reduction showing that solving them is as hard as finding short vectors in all lattices that correspond to ideals of the...

\emph{Statistical Zero-knowledge proofs} (Goldwasser, Micali and Rackoff, SICOMP 1989) allow a computationally-unbounded server to convince a computationally-limited client that an input $x$ is in a language $\Pi$ without revealing any additional information about $x$ that the client cannot compute by herself. \emph{Randomized encoding} (RE) of functions (Ishai and Kushilevitz, FOCS 2000) allows a computationally-limited client to publish a single (randomized) message, $\enc(x)$, from which...

Dodis, Kalai and Lovett (STOC 2009) initiated the study of the Learning Parity with Noise (LPN) problem with (static) exponentially hard-to-invert auxiliary input. In particular, they showed that under a new assumption (called Learning Subspace with Noise) the above is quasi-polynomially hard in the high (polynomially close to uniform) noise regime. Inspired by the ``sampling from subspace'' technique by Yu (eprint 2009 / 467) and Goldwasser et al. (ITCS 2010), we show that standard LPN can...

We consider the question of whether PPAD hardness can be based on standard cryptographic assumptions, such as the existence of one-way functions or public-key encryption. This question is particularly well-motivated in light of new devastating attacks on obfuscation candidates and their underlying building blocks, which are currently the only known source for PPAD hardness. Central in the study of obfuscation-based PPAD hardness is the sink-of-verifiable-line (SVL) problem, an intermediate...

Lattice-based cryptography has been received significant attention in the past decade. It has attractive properties such as being a major post-quantum cryptography candidate, enjoying worst-case to average-case security reductions, and being supported by efficient implementations.In recent years, lattice-based schemes have achieved enough maturity to become interesting also for the industry. Additionally, authenticated encryption (AE) is another important topic in the community of...

Hardcore lemmas are results in complexity theory which state that average-case hardness must have a very hard ``kernel'', that is a subset of instances where the given problem is extremely hard. They find important applications in hardness amplification. In this paper we revisit the following two fundamental results: \begin{enumerate}[(a)] \item The hardcore lemma for unpredictability, due to Impagliazzo (FOCS '95). It states that if a boolean function $f$ is ``moderately'' hard to predict...

In lattice cryptography, worst-case to average-case reductions rely on two problems: Ajtai's SIS and Regev's LWE, which both refer to a very small class of random lattices related to the group $G=\mZ_q^n$. We generalize worst-case to average-case reductions to all integer lattices of sufficiently large determinant, by allowing $G$ to be any (sufficiently large) finite abelian group. In particular, we obtain a partition of the set of full-rank integer lattices of large volume such that...

In this paper, we show two new constructions of chosen ciphertext secure (CCA secure) public key encryption (PKE) from general assumptions. The key ingredient in our constructions is an obfuscator for point functions with multi-bit output (MBPF obfuscators, for short), that satisfies some (average-case) indistinguishability-based security, which we call AIND security, in the presence of hard-to-invert auxiliary input. Specifically, our first construction is based on a chosen plaintext secure...

This paper introduces a new obfuscation called obfuscation of encrypted blind signature. Informally, Alice is Signer and Bob is User. Bob needs Alice to sign a message, but he does not want Alice to know what the message is. Furthermore, Bob doesn't want anyone to know the interactive process. So we present a secure obfuscator for encrypted blind signature which makes the process of encrypted blind signature unintelligible for any third party, while still keeps the original encrypted blind...

In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence to no longer work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in...

An evasive circuit family is a collection of circuits C such that for every input x, a random circuit from C outputs 0 on x with overwhelming probability. We provide a combination of definitional, constructive, and impossibility results regarding obfuscation for evasive functions: - The (average case variants of the) notions of virtual black box obfuscation (Barak et al, CRYPTO '01) and virtual gray box obfuscation (Bitansky and Canetti, CRYPTO '10) coincide for evasive function families....

Most lattice-based cryptographic schemes are built upon the assumed hardness of the Short Integer Solution (SIS) and Learning With Errors (LWE) problems. Their efficiencies can be drastically improved by switching the hardness assumptions to the more compact Ring-SIS and Ring-LWE problems. However, this change of hardness assumptions comes along with a possible security weakening: SIS and LWE are known to be at least as hard as standard (worst-case) problems on euclidean lattices, whereas...

Cryptosystems based on the hardness of lattice problems have recently acquired much importance due to their average-case to worst-case equivalence, their conjectured resistance to quantum cryptanalysis, their ease of implementation and increasing practicality, and, lately, their promising potential as a platform for constructing advanced functionalities. In this work, we construct “Fuzzy” Identity Based Encryption from the hardness of the standard Learning With Errors (LWE) problem. We give...

We propose a linearly homomorphic signature scheme that authenticates vector subspaces of a given ambient space. Our system has several novel properties not found in previous proposals: - It is the first such scheme that authenticates vectors defined over *binary fields*; previous proposals could only authenticate vectors with large or growing coefficients. - It is the first such scheme based on the problem of *finding short vectors in integer lattices*, and thus enjoys the...

This paper revisits the construction of Universally 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)...

We revisit the problem of generating a `hard' random lattice together with a basis of relatively short vectors. This problem has gained in importance lately due to new cryptographic schemes that use such a procedure to generate public/secret key pairs. In these applications, a shorter basis corresponds to milder underlying complexity assumptions and smaller key sizes. The contributions of this work are twofold. First, we simplify and modularize an approach originally due to Ajtai (ICALP...

The SWIFFT compression functions, proposed by Lyubashevsky et al. at FSE 2008, are very efficient instantiations of generalized compact knapsacks for a specific set of parameters. They have the property that, asymptotically, finding collisions for a randomly chosen compression function implies being able to solve computationally hard ideal lattice problems in the worst-case. We present three results. First, we present new average-case problems, which may be used for all lattice schemes...

In MyCrypt 2005, Augot, Finiasz, and Sendrier proposed FSB, a family of cryptographic hash functions. The security claim of the FSB hashes is based on a coding theory problem with hard average-case complexity. In the ECRYPT 2007 Hash Function Workshop, new versions with essentially the same compression function but radically different security parameters and an additional final transformation were presented. We show that hardness of average-case complexity of the underlying problem is...

We demonstrate an \emph{average-case} problem which is as hard as finding $\gamma(n)$-approximate shortest vectors in certain $n$-dimensional lattices in the \emph{worst case}, where $\gamma(n) = O(\sqrt{\log n})$. The previously best known factor for any class of lattices was $\gamma(n) = \tilde{O}(n)$. To obtain our results, we focus on families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to \emph{ideals} in the ring of integers of...

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is ``secure'' against any polynomial-time quantum adversary. Our problem QSCDff is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational...

We study pseudorandom generator (PRG) constructions G^f : {0,1}^l -> {0,1}^{l+s} from one-way functions f : {0,1}^n \to {0,1}^m. We consider PRG constructions of the form G^f(x) = C(f(q_{1}) ... f(q_{poly(n)})) where C is a polynomial-size constant depth circuit (i.e. AC^0) and C and the q's are generated from x arbitrarily. We show that every black-box PRG construction of this form must have stretch s bounded as s < l ( log^{O(1)} n)/ m + O(1) = o(l). This holds even if the PRG construction...

Is it harder to solve many puzzles than it is to solve just one? This question has different answers, depending on how you define puzzles. For the case of inverting one-way functions it was shown by Yao that solving many independent instances simultaneously is indeed harder than solving a single instance (cf. the transformation from weak to strong one-way functions). The known proofs of that result, however, use in an essential way the fact that for one-way functions, verifying candidate...

We investigate the average case complexity of a generalization of the compact knapsack problem to arbitrary rings: given $m$ (random) ring elements a_1,...,a_m in R and a (random) target value b in R, find coefficients x_1,...,x_m in S (where S is an appropriately chosen subset of R) such that a_1*x_1 + ... + a_m*x_m = b. We consider compact versions of the generalized knapsack where the set S is large and the number of weights m is small. Most variants of this problem considered in the past...