International Association
for Cryptologic Research

We give a taxonomy of computational assumptions in the algebraic group model (AGM). We first analyze the Uber assumption family for bilinear groups defined by Boyen and then extend it in multiple ways to cover assumptions such as Gap Diffie-Hellman and the LRSW assumption. We show that in the AGM every member of these families reduces to the q-discrete logarithm (DL) problem, for some q that depends on the degrees of the polynomials defining the assumption. Using the meta-reduction technique, we then separate (q+1)-DL from q-DL, which thus yields a classification of all members of the extended Uber-assumption families. We finally show that there are strong assumptions, such as one-more DL, that provably fall outside our classification, as we prove that they cannot be reduced to q-DL even in the AGM.

In the implementation of post-quantum primitives, it is well known that all computations that handle secret information need to be implemented to run in constant time. Using the Fujisaki-Okamoto transformation or any of its different variants, a CPA-secure primitive can be converted into an IND-CCA secure KEM. In this paper we show that although the transformation does not handle secret information apart from calls to the CPA-secure primitive, it has to be implemented in constant time. Namely, if the ciphertext comparison step in the transformation is leaking side-channel information, we can launch a key-recovery attack. Several proposed schemes in round 2 of the NIST post-quantum standardization project are susceptible to the proposed attack and we develop and show the details of the attack on one of them, being FrodoKEM. It is implemented on the reference implementation of FrodoKEM, which is claimed to be secure against all timing attacks. In the experiments, the attack code is able to extract the secret key for all security levels using about $2^{30}$ decapsulation calls.

Today's most compact zero-knowledge arguments are based on the hardness of the discrete logarithm problem and related classical assumptions. If one is interested in quantum-safe solutions, then all of the known techniques stem from the PCP-based framework of Kilian (STOC 92) which can be instantiated based on the hardness of any collision-resistant hash function. Both approaches produce asymptotically logarithmic sized arguments but, by exploiting extra algebraic structure, the discrete logarithm arguments are a few orders of magnitude more compact in practice than the generic constructions.\\ In this work, we present the first (poly)-logarithmic \emph{post-quantum} zero-knowledge arguments that deviate from the PCP approach. At the core of succinct zero-knowledge proofs are succinct commitment schemes (in which the commitment and the opening proof are sub-linear in the message size), and we propose two such constructions based on the hardness of the (Ring)-Short Integer Solution (Ring-SIS) problem, each having certain trade-offs. For commitments to $N$ secret values, the communication complexity of our first scheme is $\tilde{O}(N^{1/c})$ for any positive integer $c$, and $O(\log^2 N)$ for the second. %Both of our protocols have somewhat large \emph{slack}, which in lattice constructions is the ratio of the norm of the extracted secrets to the norm of the secrets that the honest prover uses in the proof. The lower this factor, the smaller we can choose the practical parameters. For a fixed value of this factor, our $\tilde{O}(N^{1/c})$-argument actually achieves lower communication complexity. Both of these are a significant theoretical improvement over the previously best lattice construction by Bootle et al. (CRYPTO 2018) which gave $O(\sqrt{N})$-sized proofs.

At Crypto '99, Nguyen and Stern described a lattice based algorithm for solving the hidden subset sum problem, a variant of the classical subset sum problem where the n weights are also hidden. While the Nguyen-Stern algorithm works quite well in practice for moderate values of n, we argue that its complexity is actually exponential in n; namely in the final step one must recover a very short basis of a n-dimensional lattice, which takes exponential-time in n, as one must apply BKZ reduction with increasingly large block-sizes. In this paper, we describe a variant of the Nguyen-Stern algorithm that works in polynomial-time. The first step is the same orthogonal lattice attack with LLL as in the original algorithm. In the second step, instead of applying BKZ, we use a multivariate technique that recovers the short lattice vectors and finally the hidden secrets in polynomial time. Our algorithm works quite well in practice, as we can reach n=250 in a few hours on a single PC.

Hardness amplification is a central problem in the study of interactive protocols. While "natural" parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols (Bellare, Impagliazzo, and Naor [FOCS '97]) and public-coin protocols (Hastad, Pass, Wikstrom, and Pietrzak [TCC '10], Chung and Lu [TCC '10] and Chung and Pass [TCC '15]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikstrom [TCC '07]). The only known round-preserving approach that applies to all interactive arguments is Haitner's random-terminating transformation [SICOMP '13], who showed that the parallel repetition of the transformed protocol reduces the soundness error at a weak exponential rate: if the original m-round protocol has soundness error (1 − ε) then the n-parallel repetition of its random-terminating variant has soundness error (1 − ε)^{ε n/m^4} (omitting constant factors). Hastad et al. have generalized this result to the so-called partially simulatable interactive arguments. In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the n parallel repetition is (1 − ε)^{n/m}, only an m factor from the optimal rate of (1 − ε)^n achievable in public-coin and three-message arguments. The result generalizes to partially simulatable arguments. This is achieved by presenting a tight bound on a relaxed variant of the KL-divergence between the distribution induced by our reduction and its ideal variant, a result whose scope extends beyond parallel repetition proofs. We prove the tightness of the above bound for random-terminating arguments, by presenting a matching protocol.

Constrained pseudorandom functions (CPRFs) allow learning "constrained" PRF keys that can evaluate the PRF on a subset of the input space, or based on some predicate. First introduced by Boneh and Waters [AC’13], Kiayias et al. [CCS’13] and Boyle et al. [PKC’14], they have shown to be a useful cryptographic primitive with many applications. These applications often require CPRFs to be adaptively secure, which allows the adversary to learn PRF values and constrained keys in an arbitrary order. However, there is no known construction of adaptively secure CPRFs based on a standard assumption in the standard model for any non-trivial class of predicates. Moreover, even if we rely on strong tools such as indistinguishability obfuscation (IO), the state-of-the-art construction of adaptively secure CPRFs in the standard model only supports the limited class of NC1 predicates. In this work, we develop new adaptively secure CPRFs for various predicates from different types of assumptions in the standard model. Our results are summarized below. - We construct adaptively secure and O(1)-collusion-resistant CPRFs for t-conjunctive normal form (t-CNF) predicates from one-way functions (OWFs) where t is a constant. Here, O(1)-collusion-resistance means that we can allow the adversary to obtain a constant number of constrained keys. Note that t-CNF includes bit-fixing predicates as a special case. - We construct adaptively secure and single-key CPRFs for inner-product predicates from the learning with errors (LWE) assumption. Here, single-key means that we only allow the adversary to learn one constrained key. Note that inner-product predicates include t-CNF predicates for a constant t as a special case. Thus, this construction supports a more expressive class of predicates than that supported by the first construction though it loses the collusion-resistance and relies on a stronger assumption. - We construct adaptively secure and O(1)-collusion-resistant CPRFs for all circuits from the LWE assumption and indistinguishability obfuscation (IO). The first and second constructions are the first CPRFs for any non-trivial predicates to achieve adaptive security outside of the random oracle model or relying on strong cryptographic assumptions. Moreover, the first construction is also the first to achieve any notion of collusion-resistance in this setting. Besides, we prove that the first and second constructions satisfy weak 1-key privacy, which roughly means that a constrained key does not reveal the corresponding constraint. The third construction is an improvement over previous adaptively secure CPRFs for less expressive predicates based on IO in the standard model.

Protocols for secure Multi-Party Computation (MPC) can be classified according to the underlying communication model. Two prominent communication models considered in the literature are the synchronous and asynchronous models, which considerably differ in terms of the achievable security guarantees. Synchronous MPC protocols can achieve the optimal corruption threshold $n/2$ and allow every party to give input, but become completely insecure when synchrony assumptions are violated. On the other hand, asynchronous MPC protocols remain secure under arbitrary network conditions, but can tolerate only $n/3$ corruptions and parties with slow connections unavoidably cannot give input. A natural question is whether there exists a protocol for MPC that can tolerate up to $t_s < n/2$ corruptions under a synchronous network and $t_a < n/3$ corruptions even when the network is asynchronous. We answer this question by showing tight feasibility and impossibility results. More specifically, we show that such a protocol exists if and only if $t_a + 2t_s < n$ and the number of inputs taken into account under an asynchronous network is at most $n-t_s$.

S-boxes are the only source of non-linearity in many symmetric cryptographic primitives. While they are often defined as being functions operating on a small space, some recent designs propose the use of much larger ones (e.g., 32 bits). In this context, an S-box is then defined as a subfunction whose cryptographic properties can be estimated precisely. In this paper, we present a 64-bit ARX-based S-box called Alzette which can be evaluated in constant time using only 12 instructions on modern CPUs. Its parallel application can also leverage vector (SIMD) instructions. One iteration of Alzette has differential and linear properties comparable to those of the AES S-box, while two iterations are at least as secure as the AES super S-box. Since the state size is much larger than the typical 4 or 8 bits, the study of the relevant cryptographic properties of Alzette is not trivial. We further discuss how such wide S-boxes could be used to construct round functions of 64-, 128- and 256-bit (tweakable) block ciphers with good cryptographic properties that are guaranteed even in the related-tweak setting. We use these structures to design a very lightweight 64-bit block cipher (CRAX) which outerperforms SPECK-64/128 for short messages on micro-controllers, and a 256-bit tweakable block cipher (TRAX) which can be used to obtain strong security guarantees against powerful adversaries (nonce misuse, quantum attacks).

Security amplification is a fundamental problem in cryptography. In this work, we study security amplification for functional encryption. We show two main results: - For any constant epsilon in (0,1), we can amplify an epsilon-secure FE scheme for P/poly which is secure against all polynomial sized adversaries to a fully secure FE scheme for P/poly, unconditionally. - For any constant epsilon in (0,1), we can amplify an epsilon-secure FE scheme for P/poly which is secure against subexponential sized adversaries to a subexponentially secure FE scheme for P/poly, unconditionally. Furthermore, both of our amplification results preserve compactness of the underlying FE scheme. Previously, amplification results for FE were only known assuming subexponentially secure LWE. Along the way, we introduce a new form of homomorphic secret sharing called set homomorphic secret sharing that may be of independent interest. Additionally, we introduce a new technique, which allows one to argue security amplification of nested primitives, and prove a general theorem that can be used to analyze the security amplification of parallel repetitions.

We present a cryptographic construction for anonymous tokens with private metadata bit, called PMBTokens. This primitive enables an issuer to provide a user with a lightweight, single-use anonymous trust token that can embed a single private bit, which is accessible only to the party who holds the secret authority key and is private with respect to anyone else. Our construction generalizes and extends the functionality of Privacy Pass (PETS’18) with this private metadata bit capability. It provides unforgeability, unlinkability, and privacy for the metadata bit properties based on the DDH and CTDH assumptions in the random oracle model. Both Privacy Pass and PMBTokens rely on non-interactive zero-knowledge proofs (NIZKs). We present new techniques to remove the need for NIZKs, while still achieving unlinkability. We implement our constructions and we report their efficiency costs.

We study the discrete logarithm problem at the boundary case between small and medium characteristic finite fields, which is precisely the area where finite fields used in pairing-based cryptosystems live. In order to evaluate the security of pairing-based protocols, we thoroughly analyze the complexity of all the algorithms that coexist at this boundary case: the Quasi-Polynomial algorithms, the Number Field Sieve and its many variants, and the Function Field Sieve. We adapt the latter to the particular case where the extension degree is composite, and show how to lower the complexity by working in a shifted function field. All this study finally allows us to give precise values for the characteristic asymptotically achieving the highest security level for pairings. Surprisingly enough, there exist special characteristics that are as secure as general ones.

Since Keccak was selected as the SHA-3 standard, more and more permutation-based primitives have been proposed. Different from block ciphers, there is no round key in the underlying permutation for permutation-based primitives. Therefore, there is a higher risk for a differential characteristic of the underlying permutation to become incompatible when considering the dependency of difference transitions over different rounds. However, in most of the MILP or SAT based models to search for differential characteristics, only the difference transitions are involved and are treated as independent in different rounds, which may cause that an invalid one is found for the underlying permutation. To overcome this obstacle, we are motivated to design a model which automatically avoids the inconsistency in the search for differential characteristics. Our technique is to involve both the difference transitions and value transitions in the constructed model. Such an idea is inspired by the algorithm to find SHA-2 characteristics as proposed by Mendel et al. in ASIACRYPT 2011, where the differential characteristic and the conforming message pair are simultaneously searched. As a first attempt, our new technique will be applied to the Gimli permutation, which was proposed in CHES 2017. As a result, we reveal that some existing differential characteristics of reduced Gimli are indeed incompatible, one of which is found in the Gimli document. In addition, since only the permutation is analyzed in the Gimli document, we are lead to carry out a comprehensive study, covering the proposed hash scheme and the authenticated encryption (AE) scheme specified for Gimli, which has become a second round candidate of the NIST lightweight cryptography standardization process. For the hash scheme, a semi-free-start (SFS) collision attack can reach up to 8 rounds starting from an intermediate round. For the AE scheme, a state recovery attack is demonstrated to achieve up to 9 rounds. It should be emphasized that our analysis does not threaten the security of Gimli.

We study the concrete security of high-performance implementations of half-gates garbling, which all rely on (hardware-accelerated) AES. We find that current instantiations using k-bit wire labels can be completely broken—in the sense that the circuit evaluator learns all the inputs of the circuit garbler—in time O(2k/C), where C is the total number of (non-free) gates that are garbled, possibly across multiple independent executions. The attack can be applied to existing circuit-garbling libraries using k = 80 when C ≈ $10^9$, and would require 267 machine-months and cost about $3500 to implement on the Google Cloud Platform. Since the attack can be entirely parallelized, the attack could be carried out in about a month using ≈ 250 machines. With this as our motivation, we seek a way to instantiate the hash function in the half-gates scheme so as to achieve better concrete security. We present a construction based on AES that achieves optimal security in the single-instance setting (when only a single circuit is garbled). We also show how to modify the half-gates scheme so that its concrete security does not degrade in the multi-instance setting. Our modified scheme is as efficient as prior work in networks with up to 2 Gbps bandwidth.

In the context of secure computation, protocols with security against covert adversaries ensure that any misbehavior by malicious parties will be detected by the honest parties with some constant probability. As such, these protocols provide better security guarantees than passively secure protocols and, moreover, are easier to construct than protocols with full security against active adversaries. Protocols that, upon detecting a cheating attempt, allow the honest parties to compute a certificate that enables third parties to verify whether an accused party misbehaved or not are called publicly verifiable. In this work, we present the first generic compilers for constructing two-party protocols with covert security and public verifiability from protocols with passive security. We present two separate compilers, which are both fully blackbox in the underlying protocols they use. Both of them only incur a constant multiplicative factor in terms of bandwidth overhead and a constant additive factor in terms of round complexity on top of the passively secure protocols they use. The first compiler applies to all two-party protocols that have no private inputs. This class of protocols covers the important class of preprocessing protocols that are used to setup correlated randomness among parties. We use our compiler to obtain the first secret-sharing based two-party protocol with covert security and public verifiability. Notably, the produced protocol achieves public verifiability essentially for free when compared with the best known previous solutions based on secret-sharing that did not provide public verifiability Our second compiler constructs protocols with covert security and public verifiability for arbitrary functionalities from passively secure protocols. It uses our first compiler to perform a setup phase, which is independent of the parties' inputs as well as the protocol they would like to execute. Finally, we show how to extend our techniques to obtain multiparty computation protocols with covert security and public verifiability against arbitrary constant fractions of corruptions.

A two-party fair coin-tossing protocol guarantees output delivery to the honest party even when the other party aborts during the protocol execution. Cleve (STOC--1986) demonstrated that a computationally bounded fail-stop adversary could alter the output distribution of the honest party by (roughly) $1/r$ (in the statistical distance) in an $r$-message coin-tossing protocol. An optimal fair coin-tossing protocol ensures that no adversary can alter the output distribution beyond $1/r$. In a seminal result, Moran, Naor, and Segev (TCC--2009) constructed the first optimal fair coin-tossing protocol using (unfair) oblivious transfer protocols. Whether the existence of oblivious transfer protocols is a necessary hardness of computation assumption for optimal fair coin-tossing remains among the most fundamental open problems in theoretical cryptography. The results of Impagliazzo and Luby (FOCS–1989) and Cleve and Impagliazzo (1993) prove that optimal fair coin-tossing implies the necessity of one-way functions' existence; a significantly weaker hardness of computation assumption compared to the existence of secure oblivious transfer protocols. However, the sufficiency of the existence of one-way functions is not known. Towards this research endeavor, our work proves a black-box separation of optimal fair coin-tossing from the existence of one-way functions. That is, the black-box use of one-way functions cannot enable optimal fair coin-tossing. Following the standard Impagliazzo and Rudich (STOC--1989) approach of proving black-box separations, our work considers any $r$-message fair coin-tossing protocol in the random oracle model where the parties have unbounded computational power. We demonstrate a fail-stop attack strategy for one of the parties to alter the honest party's output distribution by $1/\sqrt r$ by making polynomially-many additional queries to the random oracle. As a consequence, our result proves that the $r$-message coin-tossing protocol of Blum (COMPCON--1982) and Cleve (STOC--1986), which uses one-way functions in a black-box manner, is the best possible protocol because an adversary cannot change the honest party's output distribution by more than $1/\sqrt r$. Several previous works, for example, Dachman--Soled, Lindell, Mahmoody, and Malkin (TCC--2011), Haitner, Omri, and Zarosim (TCC--2013), and Dachman--Soled, Mahmoody, and Malkin (TCC--2014), made partial progress on proving this black-box separation assuming some restrictions on the coin-tossing protocol. Our work diverges significantly from these previous approaches to prove this black-box separation in its full generality. The starting point is the recently introduced potential-based inductive proof techniques for demonstrating large gaps in martingales in the information-theoretic plain model. Our technical contribution lies in identifying a global invariant of communication protocols in the random oracle model that enables the extension of this technique to the random oracle model.

In this paper, we use genus theory to analyze the hardness of the decisional Diffie--Hellman problem (DDH) for ideal class groups of imaginary quadratic orders, acting on sets of elliptic curves through isogenies; such actions are used in the Couveignes--Rostovtsev--Stolbunov protocol and in CSIDH. Concretely, genus theory equips every imaginary quadratic order $\mathcal{O}$ with a set of assigned characters $\chi : \text{cl}(\mathcal{O}) \to \{ \pm 1 \}$, and for each such character and every secret ideal class $[\mathfrak{a}]$ connecting two public elliptic curves $E$ and $E' = [\mathfrak{a}] \star E$, we show how to compute $\chi([\mathfrak{a}])$ given only $E$ and $E'$, i.e., without knowledge of $[\mathfrak{a}]$. In practice, this breaks DDH as soon as the class number is even, which is true for a density $1$ subset of all imaginary quadratic orders. For instance, our attack works very efficiently for all supersingular elliptic curves over $\mathbb{F}_p$ with $p \equiv 1 \bmod 4$. Our method relies on computing Tate pairings and walking down isogeny volcanoes.

We provide a construction of chosen ciphertext secure public-key encryption from (injective) trapdoor functions. Our construction is black box and assumes no special properties (e.g. ``lossy'', ``correlated product secure'') of the trapdoor function.

A software watermarking scheme can embed a message into a program without significantly changing its functionality. Moreover, any attempt to remove the embedded message in a marked program will substantially change the functionality of the program. Prior constructions of watermarking schemes focus on watermarking cryptographic functions, such as pseudorandom function (PRF), public key encryption, etc. A natural security requirement for watermarking schemes is collusion resistance, where the adversary’s goal is to remove the embedded messages given multiple marked versions of the same program. Currently, this strong security guarantee has been achieved by watermarking schemes for public key cryptographic primitives from standard assumptions (Goyal et al., CRYPTO 2019) and by watermarking schemes for PRFs from indistinguishability obfuscation (Yang et al., ASIACRYPT 2019). However, no collusion resistant watermarking scheme for PRF from standard assumption is known. In this work, we solve this problem by presenting a generic construction that upgrades a watermarkable PRF without collusion resistance to a collusion resistant one. One appealing feature of our construction is that it can preserve the security properties of the original scheme. For example, if the original scheme has security with extraction queries, the new scheme is also secure with extraction queries. Besides, the new scheme can achieve unforgeability even if the original scheme does not provide this security property. Instantiating our construction with existing watermarking schemes for PRF, we obtain collusion resistant watermarkable PRFs from standard assumptions, offering various security properties.

We report on two new records: the factorization of RSA-240, a 795-bit number, and a discrete logarithm computation over a 795-bit prime field. Previous records were the factorization of RSA-768 in 2009 and a 768-bit discrete logarithm computation in 2016. Our two computations at the 795-bit level were done using the same hardware and software, and show that computing a discrete logarithm is not much harder than a factorization of the same size. Moreover, thanks to algorithmic variants and well-chosen parameters, our computations were significantly less expensive than anticipated based on previous records. The last page of this paper also reports on the factorization of RSA-250.

Sigma-Protocols provide a well-understood basis for secure algorithmics. Recently, Bulletproofs (Bootle et al., EUROCRYPT 2016, and Bünz et al., S&P 2018) have been proposed as a drop-in replacement in case of zero-knowledge (ZK) for arithmetic circuits, achieving logarithmic communication instead of linear. Its pivot is an ingenious, logarithmic-size proof of knowledge BP for certain quadratic relations. However, reducing ZK for general relations to it forces a somewhat cumbersome ``reinvention'' of cryptographic protocol theory. We take a rather different viewpoint and reconcile Bulletproofs with Sigma-Protocol Theory such that (a) simpler circuit ZK is developed within established theory, while (b) achieving exactly the same logarithmic communication. The natural key here is linearization. First, we repurpose BPs as a blackbox compression mechanism for standard Sigma-Protocols handling ZK proofs of general linear relations (on compactly committed secret vectors); our pivot. Second, we reduce the case of general nonlinear relations to blackbox applications of our pivot via a novel variation on arithmetic secret sharing based techniques for Sigma-Protocols (Cramer et al., ICITS 2012). Orthogonally, we enhance versatility by enabling scenarios not previously addressed, e.g., when a secret input is dispersed across several commitments. Standard implementation platforms leading to logarithmic communication follow from a Discrete-Log assumption or a generalized Strong-RSA assumption. Also, under a Knowledge-of-Exponent Assumption (KEA) communication drops to constant, as in ZK-SNARKS. All in all, our theory should more generally be useful for modular (``plug & play'') design of practical cryptographic protocols; this is further evidenced by our separate work (2020) on proofs of partial knowledge.

We report on the concrete cryptanalysis of LEDAcrypt, a 2nd Round candidate in NIST's Post-Quantum Cryptography standardization process and one of 17 encryption schemes that remain as candidates for near-term standardization. LEDAcrypt consists of a public-key encryption scheme built from the McEliece paradigm and a key-encapsulation mechanism (KEM) built from the Niederreiter paradigm, both using a quasi-cyclic low-density parity-check (QC-LDPC) code. In this work, we identify a large class of extremely weak keys and provide an algorithm to recover them. For example, we demonstrate how to recover $1$ in $2^{47.79}$ of LEDAcrypt's keys using only $2^{18.72}$ guesses at the 256-bit security level. This is a major, practical break of LEDAcrypt. Further, we demonstrate a continuum of progressively less weak keys (from extremely weak keys up to all keys) that can be recovered in substantially less work than previously known. This demonstrates that the imperfection of LEDAcrypt is fundamental to the system's design.

In 2017, Ward Beullens et al. submitted Lifted Unbalanced Oil and Vinegar (LUOV), a signature scheme based on the famous multivariate public-key cryptosystem (MPKC) called Unbalanced Oil and Vinegar (UOV), to NIST for the competition for post-quantum public-key scheme standardization. The defining feature of LUOV is that, though the public key P works in the extension field of degree r of F2, the coefficients of P come from F2. This is done to significantly reduce the size of P. The LUOV scheme is now in the second round of the NIST PQC standardization process. In this paper, we introduce a new attack on LUOV. It exploits the "lifted" structure of LUOV to reduce direct attacks on it to those over a subfield. We show that this reduces the complexity below the targeted security for the NIST postquantum standardization competition.

Spook is one of the 32 candidates that has made it to the second round of the NIST Lightweight Cryptography Standardization process, and is particularly interesting since it proposes differential side channel resistance. In this paper, we present practical distinguishers of the full 6-step version of the underlying permutations of Spook, namely Shadow-512 and Shadow-384, solving challenges proposed by the designers on the permutation. We also propose practical forgeries with 4-step Shadow for the S1P mode of operation in the nonce misuse scenario, which is allowed by the CIML2 security game considered by the authors. All the results presented in this paper have been implemented.

We argue that the machine learning problem of model extraction is actually a cryptanalytic problem in disguise, and should be studied as such. Given oracle access to a neural network, we introduce a differential attack that can efficiently steal the parameters of the remote model up to floating point precision. Our attack relies on the fact that ReLU neural networks are piecewise linear functions, and thus queries at the critical points reveal information about the model parameters. We evaluate our attack on multiple neural network models and extract models that are 2^20 times more precise and require 100x fewer queries than prior work. For example, we extract a 100,000 parameter neural network trained on the MNIST digit recognition task with 2^21.5 queries in under an hour, such that the extracted model agrees with the oracle on all inputs up to a worst-case error of 2^-25, or a model with 4,000 parameters in 2^18.5 queries with worst-case error of 2^-40.4. Code is available at https://github.com/google-research/cryptanalytic-model-extraction.

In this work, we show the hardness of finding a Nash equilibrium, a \PPAD-complete problem, based on the quasi-polynomial hardness of the decisional assumption on groups with bilinear maps introduced by Kalai, Paneth and Yang [STOC 2019]. Towards this goal, we construct an {\em unambiguous} and {\em updatable} delegation scheme under this assumption for deterministic computations running in super-polynomial time and polynomial space. This delegation scheme, which is of independent interest, is publicly verifiable and non-interactive in the common reference string (CRS) model. It is {\em unambiguous} meaning that it is hard to compute two different proofs for the same statement. It is {\em updatable} meaning that given a proof for the statement that a Turing machine $M$ reaches configuration $\conf_T$ in $T$ steps, one can {\em efficiently} generate a proof for the statement that $M$ reaches configuration $\conf_{T+1}$ in $T+1$ steps.

We introduce Dynamic Decentralized Functional Encryption (DDFE), a generalization of Functional Encryption which allows multiple users to join the system dynamically, without relying on a trusted third party or on expensive and interactive Multi-Party Computation protocols. This notion subsumes existing multi-user extensions of Functional Encryption, such as Multi-Input, Multi-Client, and Ad Hoc Multi-Input Functional Encryption. We define and construct schemes for various functionalities which serve as building-blocks for latter primitives and may be useful in their own right, such as a scheme for dynamically computing sums in any Abelian group. These constructions build upon simple primitives in a modular way, and have instantiations from well-studied assumptions, such as DDH or LWE. Our constructions culminate in an Inner-Product scheme for computing weighted sums on aggregated encrypted data, from standard assumptions in prime-order groups in the Random Oracle Model.

Recent years have seen a tremendous growth in the interest in se- cure multiparty computation (MPC) and its applications. While much progress has been made concerning its efficiency, many current, state-of-the-art protocols are vulnerable to Denial of Service attacks, where a cheating party may prevent the honest parties from learning the output of the computation, whilst remaining anonymous. The security model of identifiable abort aims to prevent these at- tacks, by allowing honest parties to agree upon the identity of a cheating party, who can then be excluded in the future. Several existing MPC protocols offer security with identifiable abort against a dishonest majority of corrupted parties. However, all of these protocols have a round complexity that scales linearly with the depth of the circuit (and are therefore unsuitable for use in high latency net- works) or use cryptographic primitives or techniques that have a high computa- tional overhead. In this work, we present the first efficient MPC protocols with identifiable abort in the dishonest majority setting, which run in a constant number of rounds and make only black-box use of cryptographic primitives. Our main construction is built from highly efficient primitives in a careful way to achieve identifiability at a low cost. In particular, we avoid the use of public-key operations outside of a setup phase, incurring a relatively low overhead on top of the fastest currently known constant-round MPC protocols based on garbled circuits. Our construction also avoids the use of adaptively secure primitives and heavy zero-knowledge machinery, which was inherent in previous works. In addition, we show how to upgrade our protocol to achieve public verifiability using a public bulletin board, allowing any external party to verify correctness of the computation or identify a cheating party.

Secure multiparty computation can often utilize a trusted source of correlated randomness to achieve better efficiency. A recent line of work, initiated by Boyle et al. (CCS 2018, Crypto 2019), showed how useful forms of correlated randomness can be generated using a cheap, one-time interaction, followed by only ``silent'' local computation. This is achieved via a \emph{pseudorandom correlation generator} (PCG), a deterministic function that stretches short correlated seeds into long instances of a target correlation. Previous works constructed concretely efficient PCGs for simple but useful correlations, including random oblivious transfer and vector-OLE, together with efficient protocols to distribute the PCG seed generation. Most of these constructions were based on variants of the Learning Parity with Noise (LPN) assumption. PCGs for other useful correlations had poor asymptotic and concrete efficiency. In this work, we design a new class of efficient PCGs based on different flavors of the {\em ring-LPN} assumption. Our new PCGs can generate OLE correlations, authenticated multiplication triples, matrix product correlations, and other types of useful correlations over large fields. These PCGs are more efficient by orders of magnitude than the previous constructions and can be used to improve the preprocessing phase of many existing MPC protocols.

Updatable encryption allows a client to outsource ciphertexts to some untrusted server and periodically rotate the encryption key. The server can update ciphertexts from an old key to a new key with the help of an update token, received from the client, which should not reveal anything about plaintexts to an adversary. We provide a new and highly efficient suite of updatable encryption schemes that we collectively call SHINE. In the variant designed for short messages, ciphertext generation consists of applying one permutation and one exponentiation (per message block), while updating ciphertexts requires just one exponentiation. Variants for longer messages provide much stronger security guarantees than prior work that has comparable efficiency. We present a new confidentiality notion for updatable encryption schemes that implies prior notions. We prove that SHINE is secure under our new confidentiality definition while also providing ciphertext integrity.

We introduce a framework generalizing lattice reduction algorithms to module lattices in order to \emph{practically} and \emph{efficiently} solve the $\gamma$-Hermite Module-SVP problem over arbitrary cyclotomic fields. The core idea is to exploit the structure of the subfields for designing a recursive strategy of reduction in the tower of fields we are working in. Besides, we demonstrate how to leverage the inherent symplectic geometry existing such fields to provide a significant speed-up of the reduction for rank two modules. As a byproduct, we also generalize to all cyclotomic fields and provide speedups for many previous number theoretical algorithms, in particular to the rounding in the so-called Log-unit lattice. Quantitatively, we show that a module of rank 2 over a cyclotomic field of degree $n$ can be heuristically reduced within approximation factor $2^{\tilde{O}(n)}$ in time $\tilde{O}(n^2B)$, where $B$ is the bitlength of the entries. For $B$ large enough, this complexity shrinks to $\tilde{O}(n^{\log_2 3}B)$. This last result is particularly striking as it goes below the estimate of $n^2B$ swaps given by the classical analysis of the LLL algorithm using the decrease of the \emph{potential} of the basis. Finally, all this framework is fully parallelizable, and we provide a full implementation. We apply it to break multilinear cryptographic candidates on concrete proposed parameters. We were able to reduce matrices of dimension 4096 with 6675-bit integers in 4 days, which is more than a million times faster than previous state-of-the-art implementations. Eventually, we demonstrate a quasicubic time for the Gentry-Szydlo algorithm which finds a generator given the relative norm and a basis of an ideal. This algorithm is important in cryptanalysis and requires efficient ideal multiplications and lattice reductions; as such we can practically use it in dimension 1024.

We give a lattice reduction algorithm that achieves root Hermite factor k^(1/(2k)) in time k^(k/8 + o(k)) and polynomial memory. This improves on the previously best known enumeration-based algorithms which achieve the same quality, but in time k^(k/(2e) + o(k)). A cost of k^(k/8 + o(k)) was previously mentioned as potentially achievable (Hanrot-Stehlé’10) or as a heuristic lower bound (Nguyen’10) for enumeration algorithms. We prove the complexity and quality of our algorithm under a heuristic assumption and provide empirical evidence from simulation and implementation experiments attesting to its performance for practical and cryptographic parameter sizes. Our work also suggests potential avenues for achieving costs below k^(k/8 + o(k)) for the same root Hermite factor, based on the geometry of SDBKZ-reduced bases.

The Fiat-Shamir transform is a methodology for compiling a (public-coin) interactive proof system for a language $L$ into a {\em non-interactive} argument system for $L$. Proving security of the Fiat-Shamir transform in the standard model, especially in the context of \emph{succinct} arguments, is largely an unsolved problem. The work of Canetti et al. (STOC 2019) proved the security of the Fiat-Shamir transform applied to the Goldwasser-Kalai-Rothblum (STOC 2008) succinct interactive proof system under a very strong ``optimal learning with errors'' assumption. Achieving a similar result under standard assumptions remains an important open question. In this work, we consider the problem of compiling a different succinct interactive proof system: Pietrzak's proof system (ITCS 2019) for the iterated squaring problem. We construct a hash function family (with evaluation time roughly $2^{\lambda^{\epsilon}}$) that guarantees the soundness of Fiat-Shamir for this protocol assuming the sub-exponential ($2^{-n^{1-\epsilon}}$)-hardness of the $n$-dimensional learning with errors problem. (The latter follows from the worst-case $2^{n^{1-\epsilon}}$ hardness of lattice problems.) More generally, we extend the ``bad-challenge function'' methodology of Canetti et al. for proving the soundness of Fiat-Shamir to a class of protocols whose bad-challenge functions are {\em not} efficiently computable. As a corollary (following Choudhuri et al., ePrint 2019 and Ephraim et al., EUROCRYPT 2020), we construct hard-on-average problems in the complexity class $\mathbf{CLS}\subset \mathbf{PPAD}$ under the $2^{\secp^\epsilon}$-hardness of the repeated squaring problem and the $2^{-n^{1-\epsilon}}$-hardness of the learning with errors problem. Under the additional assumption that the repeated squaring problem is ``inherently sequential'', we also obtain a Verifiable Delay Function (Boneh et al., EUROCRYPT 2018) in the standard model. Finally, we give additional PPAD-hardness and VDF instantiations demonstrating a broader tradeoff between the strength of the repeated squaring assumption and the strength of the lattice assumption.

Deniable encryption (Canetti \emph{et al.}, Crypto 1996) enhances secret communication over public channels, providing the additional guarantee that the secrecy of communication is protected even if the parties are later coerced (or willingly bribed) to expose their entire internal states: plaintexts, keys and randomness. To date, constructions of deniable encryption --- and more generally, interactive deniable communication --- only address restricted cases where only {\em one} party is compromised (Sahai and Waters, STOC 2014). The main question --- whether deniable communication is at all possible if {\em both} parties are coerced at once --- has remained open. We resolve this question in the affirmative, presenting a communication protocol that is {\em fully deniable} under coercion of both parties. Our scheme has three rounds, assumes subexponentially secure indistinguishability obfuscation and one-way functions, and uses a short global reference string that is generated once at system set-up and suffices for an unbounded number of encryptions and decryptions. Of independent interest, we introduce a new notion called \emph{off-the-record deniability}, which protects parties even when their claimed internal states are inconsistent (a case not covered by prior definitions). Our scheme satisfies both standard deniability and off-the-record deniability.

We present functional encryption schemes for attribute-weighted sums, where encryption takes as input N attribute-value pairs (x_i,z_i) where x_i is public and z_i is private; secret keys are associated with arithmetic branching programs f, and decryption returns the weighted sum \sum_{i=1}^N f(x_i) z_i while leaking no additional information about the z_i's. Our main construction achieves (1) compact public parameters and key sizes that are independent of N and the secret key can decrypt a ciphertext for any a-priori unbounded N; (2) short ciphertexts that grow with N and the size of z_i but not x_i; (3) simulation-based security against unbounded collusions; (4) relies on the standard k-linear assumption in prime-order bilinear groups.

Despite the fundamental importance of delay functions, repeated squaring in RSA groups (Rivest, Shamir and Wagner '96) is the only candidate offering both a useful structure and a realistic level of practicality. Somewhat unsatisfyingly, its sequentiality is provided directly by assumption (i.e., the function is assumed to be a delay function). We prove sharp thresholds on the sequentiality of all generic-ring delay functions relative to an RSA modulus based on the hardness of factoring in the standard model. In particular, we show that generically speeding-up repeated squaring (even with a preprocessing stage and any polynomial number parallel processors) is equivalent to factoring. More generally, based on the (essential) hardness of factoring, we prove that any generic-ring function is in fact a delay function, admitting a sharp sequentiality threshold that is determined by our notion of sequentiality depth. Moreover, we show that generic-ring functions admit not only sharp sequentiality thresholds, but also sharp pseudorandomness thresholds.

We study the communication complexity of unconditionally secure MPC with guaranteed output delivery over point-to-point channels for corruption threshold t < n/2, assuming the existence of a public broadcast channel. We ask the question: “is it possible to construct MPC in this setting s.t. the communication complexity per multiplication gate is linear in the number of parties?” While a number of works have focused on reducing the communication complexity in this setting, the answer to the above question has remained elusive until now. We also focus on the concrete communication complexity of evaluating each multiplication gate. We resolve the above question in the affirmative by providing an MPC with communication complexity O(Cn\phi) bits (ignoring fixed terms which are independent of the circuit) where \phi is the length of an element in the field, C is the size of the (arithmetic) circuit, n is the number of parties. This is the first construction where the asymptotic communication complexity matches the best-known semi-honest protocol. This represents a strict improvement over the previously best-known communication complexity of O(C(n\phi + \kappa) + D_Mn^2\kappa) bits, where \kappa is the security parameter and D_M is the multiplicative depth of the circuit. Furthermore, the concrete communication complexity per multiplication gate is 5.5 field elements per party in the best case and 7.5 field elements in the worst case when one or more corrupted parties have been identified. This also roughly matches the best-known semi-honest protocol, which requires 5.5 field elements per gate. The above also yields the first secure-with-abort MPC protocol with the same cost per multiplication gate as the best-known semi-honest protocol. Our main result is obtained by compiling the secure-with-abort MPC protocol into a fully secure one.

We provide a new definitional framework capturing the multi-user security of encryption schemes and pseudorandom functions in the face of adversaries that can adaptively compromise users' keys. We provide a sequence of results establishing the security of practical symmetric encryption schemes under adaptive compromise in the random oracle or ideal cipher model. The bulk of analysis complexity for adaptive compromise security is relegated to the analysis of lower-level primitives such as pseudorandom functions. We apply our framework to give proofs of security for the BurnBox system for privacy in the face of border searches and the in-use searchable symmetric encryption scheme due to Cash et al. In both cases, prior analyses had bugs that our framework helps avoid.

We present several improvements to the framework of differential-linear attacks with a special focus on ARX ciphers. As a demonstration of their impact, we apply them to Chaskey and ChaCha and we are able to significantly improve upon the best attacks published so far.

This work introduces novel techniques to improve the translation between arithmetic and binary data types in multi-party computation. To this end, we introduce a new approach to performing these conversions, using what we call \emph{extended doubly-authenticated bits} (edaBits), which correspond to shared integers in the arithmetic domain whose bit decomposition is shared in the binary domain. These can be used to considerably increase the efficiency of non-linear operations such as truncation, secure comparison and bit-decomposition. Our eDaBits are similar to the \emph{daBits} technique introduced by Rotaru et al.~(Indocrypt 2019). However, our main observations are that (1) applications that benefit from daBits can also benefit from edaBits in the same way, and (2) we can generate edaBits directly in a much more efficeint way than computing them directly from a set of DaBits. Technically, the second contribution is much more challenging, and involves a novel cut and choose technique that may be of independent interest, and requires taking advantage of natural tamper-resilient properties of binary circuits that occur in our construction to obtain the best level of efficiency. Finally, we show how our eDaBits can be applied to efficiently implement various non-linear protocols of interest, and we thoroughly analyze their correctness for both signed and unsigned integers. The results of this work can be applied to any corruption threshold, although they seem best suited to dishonest majority protocols such as SPDZ. We implement and benchmark our constructions, and experimentally verify that our technique yield a substantial increase in effiency. Our eDaBits save in communication by a factor that lies between $2$ and $170$ for secure comparisons with respect to a purely arithmetic approach, and between $2$ and $60$ with respect to using daBits. Improvements in throughput per second are more subdued but still as high as a factor of $47$. We also apply our novel machinery to the tasks of biometric matching and convolutional neural networks, obtaining a noticeable improvement as well.

An incompressible encoding can probabilistically encode some data $m$ into a codeword $c$, which is not much larger. Anyone can decode $c$ to recover the original data $m$. However, $c$ cannot be efficiently compressed, even if the original data $m$ is given to the decompression procedure for free. In other words, $c$ is a representation of $m$, yet is computationally incompressible even given $m$. An incompressible encoding is composable if many encodings cannot be simultaneously compressed into anything sufficiently smaller than their concatenation. A recent work of Damgard, Ganesh and Orlandi (CRYPTO '19) defined a variant of incompressible encodings and gave an applications to ``proofs of replicated storage''. They constructed incompressible encodings in an ideal permutation model over a structured domain, but it was left open if they can be constructed under standard assumptions, or even in the more basic random-oracle model. In this work, we give new constructions, negative results and applications of incompressible encodings: * We construct incompressible encodings in the common random string (CRS) model under the Decisional Composite Residuosity (DCR) or Learning with Errors (LWE) assumptions. However, the construction has several drawbacks: (1) it is not composable, (2) it only achieves selective security, and (3) the CRS is as long as the data $m$. * We leverage the above construction to also get a scheme in the random-oracle model, under the same assumptions, that avoids all of the above drawbacks. Furthermore, it is significantly more efficient than the prior ideal-model construction. * We give black-box separations, showing that incompressible encodings in the plain model cannot be proven secure under any standard hardness assumption, and incompressible encodings in the CRS model must inherently suffer from all of the drawbacks above. * We give a new application to ``big-key cryptography in the bounded-retrieval model'', where secret keys are made intentionally huge to make them hard to exfiltrate. Using incompressible encodings, we can get all the security benefits of a big key without wasting storage space, by having the key to encode useful data.

We initiate the study of indifferentiability for public key encryption and other public key primitives. Our main results are definitions and constructions of public key cryptosystems that are indifferentiable from ideal cryptosystems, in the random oracle model. Cryptosystems include: 1) Public key encryption; 2) Digital signatures; 3) Non-interactive key agreement. Our schemes are based on relatively standard public key assumptions. By being indifferentiable from an ideal object, our schemes automatically satisfy a wide range of security properties, including any property representable as a single-stage game, and can be composed to operate in higher-level protocols.

We consider interactive proofs for social graphs, where the verifier has only oracle access to the graph and can query for the $i^{th}$ neighbor of a vertex $v$, given $i$ and $v$. In this model, we construct a doubly-efficient public-coin two-message interactive protocol for estimating the size of the graph to within a multiplicative factor $\epsilon>0$. The verifier performs $\widetilde{O}(1/\epsilon^2 \cdot \tau \cdot \Delta)$ queries to the graph, where $\tau$ is the mixing time of the graph and $\Delta$ is the average degree of the graph. The prover runs in quasi-linear time in the number of nodes in the graph. Furthermore, we develop a framework for computing the average of essentially any (reasonable) function $f$ of vertices of the graph. Using this framework, we can estimate many health measures of social graphs such as the clustering coefficients and the average degree, where the verifier performs only a small number of queries to the graph. Using the Fiat-Shamir paradigm, we are able to transform the above protocols to a non-interactive argument in the random oracle model. The result is that any social media company (Facebook, Twitter, etc.) can publish, once and for all, a short proof for the size or health of their social network. This proof can be publicly verified by any single user using a small number of queries to the graph.

We show how to generalize lattice reduction algorithms to module lattices. Specifically, we reduce $\gamma$-approximate ModuleSVP over module lattices with rank $k \geq2$ to $\gamma'$-approximate ModuleSVP over module lattices with rank $2 \leq \beta \leq k$. To do so, we modify the celebrated slide-reduction algorithm of Gama and Nguyen to work with module filtrations, a high-dimensional generalization of the ($\Z$-)basis of a lattice. The particular value of $\gamma$ that we achieve depends on the underlying number field $K$, the order $R \subseteq \mathcal{O}_K$, and the embedding (as well as, of course, $k$ and $\beta$). However, for reasonable choices of these parameters, the approximation factor that we achieve is surprisingly close to the one achieved by ``plain'' lattice reduction algorithms, which require an arbitrary SVP oracle in the same dimension. In other words, we show that ModuleSVP oracles are nearly as useful as SVP oracles for solving approximate ModuleSVP in higher dimensions. Our result generalizes the recent independent result of Lee, Pellet-Mary, Stehl\'e, and Wallet, which works in the important special case when $\beta = 2$ and $R = \mathcal{O}_K$ is the ring of integers of $K$ under the canonical embedding. Indeed, at a high level our reduction can be thought of as a generalization of theirs in roughly the same way that block reduction generalizes LLL reduction.

We observe that all previously known lattice-based blind signatures schemes contain subtle flaws in their security proofs (e.g.,~Rückert, ASIACRYPT '08) or can be attacked (e.g., BLAZE by Alkadri et al., FC~'20). Motivated by this, we revisit the problem of constructing blind signatures from standard lattice assumptions. We propose a new three-round lattice-based blind signature scheme whose security can be proved, in the random oracle model, from the standard SIS assumption. Our starting point is a modified version of the insecure three-round BLAZE scheme, which itself is based Lyubashevsky's three-round identification scheme combined with a new aborting technique to reduce the correctness error. Our proof builds upon and extends the recent modular framework for blind signatures of Hauck, Kiltz, and Loss (EUROCRYPT~'19). It also introduces several new techniques to overcome the additional challenges posed by the correctness error which is inherent to all lattice-based constructions. While our construction is mostly of theoretical interest, we believe it to be an important stepping stone for future works in this area.

Can Alice and Bob agree on a uniformly random secret key without having any truly secret randomness to begin with? Here we consider a setting where Eve can get partial leakage on the internal state of both Alice and Bob individually before the protocol starts. They then run a protocol using their states without any additional randomness and need to agree on a shared key that looks uniform to Eve, even after observing the leakage and the protocol transcript. We focus on non-interactive (one round) key exchange (NIKE), where Alice and Bob send one message each without waiting for one another. We first consider this problem in the symmetric-key setting, where the states of Alice and Bob include a shared secret as well as individual uniform randomness. However, since Eve gets leakage on these states, Alice and Bob need to perform privacy amplification to derive a fresh secret key from them. Prior solutions require Alice and Bob to sample fresh uniform randomness during the protocol, while in our setting all of their randomness was already part of their individual states a priori and was therefore subject to leakage. We show an information-theoretic solution to this problem using a novel primitive that we call a two-seed extractor, which we in turn construct by drawing a connection to communication-complexity lower-bounds in the number-on-forehead (NOF) model. We then turn to studying this problem in the public-key setting, where the states of Alice and Bob consist of independent uniform randomness. Unfortunately, we give a black-box separation showing that leakage-resilient NIKE in this setting cannot be proven secure via a black-box reduction under any game-based assumption when the leakage is super-logarithmic. This includes virtually all assumptions used in cryptography, and even very strong assumptions such as indistinguishability obfuscation (iO). Nevertheless, we also provide positive results that get around the above separation: -We show that every key exchange protocol (e.g., Diffie-Hellman) is secure when the leakage amount is logarithmic, or potentially even greater if we assume sub-exponential security without leakage. -We notice that the black-box separation does not extend to schemes in the common reference string (CRS) model, or to schemes with preprocessing, where Alice and Bob can individually pre-process their random coins to derive their secret state prior to leakage. We give a solution in the CRS model with preprocessing using bilinear maps. We also give solutions in just the CRS model alone (without preprocessing) or just with preprocessing (without a CRS), using iO and lossy functions.

Encrypted multi-maps (EMMs) enable clients to outsource the storage of a multi-map to a potentially untrusted server while maintaining the ability to perform operations in a privacy-preserving manner. EMMs are an important primitive as they are an integral building block for many practical applications such as searchable encryption and encrypted databases. In this work, we formally examine the tradeoffs between privacy and efficiency for EMMs. Currently, all known dynamic EMMs with constant overhead reveal if two operations are performed on the same key or not that we denote as the global key-equality pattern. In our main result, we present strong evidence that the leakage of the global key-equality pattern is inherent for any dynamic EMM construction with $O(1)$ efficiency. In particular, we consider the slightly smaller leakage of decoupled key-equality pattern where leakage of key-equality between update and query operations is decoupled and the adversary only learns whether two operations of the same type are performed on the same key or not. We show that any EMM with at most decoupled key-equality pattern leakage incurs $\Omega(\log n)$ overhead in the leakage cell probe model. This is tight as there exist ORAM-based constructions of EMMs with logarithmic slowdown that leak no more than the decoupled key-equality pattern (and actually, much less). Furthermore, we present stronger lower bounds that encrypted multi-maps leaking at most the decoupled key-equality pattern but are able to perform one of either the update or query operations in the plaintext still require $\Omega(\log n)$ overhead. Finally, we extend our lower bounds to show that dynamic, response-hiding searchable encryption schemes must also incur $\Omega(log n)$ overhead even when one of either the document updates or searches may be performed in the plaintext.

We propose a framework for cryptanalysis of lattice-based schemes, when side information --in the form of "hints''-- about the secret and/or error is available. Our framework generalizes the so-called primal lattice reduction attack, and allows the progressive integration of hints before running a final lattice reduction step. Our techniques for integrating hints include sparsifying the lattice, projecting onto and intersecting with hyperplanes, and/or altering the distribution of the secret vector. Our main contribution is to propose a toolbox and a methodology to integrate such hints into lattice reduction attacks and to predict the performance of those lattice attacks with side information. While initially designed for side-channel information, our framework can also be used in other cases: exploiting decryption failures, or simply exploiting constraints imposed by certain schemes (LAC, Round5, NTRU), that were previously not known to (slightly) benefit from lattice attacks. We implement a Sage 9.0 toolkit to actually mount such attacks with hints when computationally feasible, and to predict their performances on larger instances. We provide several end-to-end application examples, such as an improvement of a single trace attack on Frodo by Bos et al (SAC 2018). Contrary to ad-hoc practical attacks exploiting side-channel leakage, our work is a generic way to estimate security loss even given very little side-channel information.

Triggered by the increasing deployment of embedded cryptographic devices (e.g., for the IoT), the design of authentication, encryption and authenticated encryption schemes enabling improved security against side-channel attacks has become an important research direction. Over the last decade, a number of modes of operation have been proposed and analyzed under different abstractions. In this paper, we investigate the practical consequences of these findings. For this purpose, we first translate the physical assumptions of leakage-resistance proofs into minimum security requirements for implementers. Thanks to this (heuristic) translation, we observe that (i) security against physical attacks can be viewed as a tradeoff between mode-level and implementation-level protection mechanisms, and (i}) security requirements to guarantee confidentiality and integrity in front of leakage can be concretely different for the different parts of an implementation. We illustrate the first point by analyzing several modes of operation with gradually increased leakage-resistance. We illustrate the second point by exhibiting leveled implementations, where different parts of the investigated schemes have different security requirements against leakage, leading to performance improvements when high physical security is needed. We finally initiate a comparative discussion of the different solutions to instantiate the components of a leakage-resistant authenticated encryption scheme.

Classical definitions for secure multiparty computation assume the existence of a single adversarial entity controlling the set of corrupted parties. Intuitively, the definition requires that the view of the adversary, corrupting t parties, in a real-world execution can be simulated by an adversary in an ideal model, where parties interact only via a trusted-party. No restrictions, however, are imposed on the view of honest parties in the protocol, thus, if honest parties obtain information about the private inputs of other honest parties -- it is not counted as a violation of privacy. This is arguably undesirable in many situations that fall into the MPC framework. Nevertheless, there are secure protocols (e.g., the 2-round multiparty protocol of Ishai et al. [CRYPTO 2010] tolerating a single corrupted party) that instruct the honest parties to reveal their private inputs to all other honest parties (once the malicious party is somehow identified). In this paper, we put forth a new security notion, which we call FaF-security, extending the classical notion. In essence, (t,h^*)-FaF-security requires the view of a subset of up to h^* honest parties to also be simulatable in the ideal model (in addition to the view of the malicious adversary, corrupting up to t parties). This property should still hold, even if the adversary leaks information to honest parties by sending them non-prescribed messages. We provide a thorough exploration of the new notion, investigating it in relation to a variety of existing security notions. We further investigate the feasibility of achieving FaF-security and show that every functionality can be computed with (computational) (t,h^*)-FaF full-security, if and only if 2t+ h^*<m. Interestingly, the lower-bound result actually shows that even fair FaF-security is impossible in general when 2t+ h^*\ge m (surprisingly, the view of the malicious attacker is not used as the trigger for the attack). We also investigate the optimal round complexity for (t,h^*)-Faf-secure protocols and give evidence that the leakage of private inputs of honest parties in the protocol of Ishai et al. [CRYPTO 2010] is inherent.

We present a new multiparty protocol for the distributed generation of biprime RSA moduli, with security against any subset of maliciously colluding parties assuming oblivious transfer and the hardness of factoring. Our protocol is highly modular, and its uppermost layer can be viewed as a template that generalizes the structure of prior works and leads to a simpler security proof. We introduce a combined sampling-and-sieving technique that eliminates both the inherent leakage in the approach of Frederiksen et al. (Crypto'18), and the dependence upon additively homomorphic encryption in the approach of Hazay et al. (JCrypt'19). We combine this technique with an efficient, privacy-free check to detect malicious behavior retroactively when a sampled candidate is not a biprime, and thereby overcome covert rejection-sampling attacks and achieve both asymptotic and concrete efficiency improvements over the previous state of the art.

Robust secret sharing is a strengthening of standard secret sharing that allows the shared secret to be recovered even if some of the shares being used in the reconstruction have been adversarially modified. In this work, we study the setting where out of all the $n$ shares, the adversary is allowed to adaptively corrupt and modify up to $t$ shares, where $n = 2t+1$.\footnote{Note that if the adversary is allowed to modify any more shares, then correct reconstruction would be impossible.} Further, we deal with \emph{rushing} adversaries, meaning that the adversary is allowed to see the honest parties' shares before modifying its own shares. It is known that when $n = 2t+1$, to share a secret of length $m$ bits and recover it with error less than $2^{-\sec}$, shares of size at least $m+\sec$ bits are needed. Recently, Bishop, Pastro, Rajaraman, and Wichs (EUROCRYPT 2016) constructed a robust secret sharing scheme with shares of size $m + O(\sec\cdot\polylog(n,m,\sec))$ bits that is secure in this setting against non-rushing adversaries. Later, Fehr and Yuan (EUROCRYPT 2019) constructed a scheme that is secure against rushing adversaries, but has shares of size $m + O(\sec\cdot n^{\eps}\cdot \polylog(n,m,\sec))$ bits for an arbitrary constant $\eps > 0$. They also showed a variant of their construction with share size $m + O(\sec\cdot\polylog(n,m,\sec))$ bits, but with super-polynomial reconstruction time. We present a robust secret sharing scheme that is simultaneously close-to-optimal in all of these respects -- it is secure against rushing adversaries, has shares of size $m+O(\sec \log{n} (\log{n}+\log{m}))$ bits, and has polynomial-time sharing and reconstruction. Central to our construction is a polynomial-time algorithm for a problem on semi-random graphs that arises naturally in the paradigm of local authentication of shares used by us and in the aforementioned work.

Over the last few years, there has been a surge of new cryptographic results, including laconic oblivious transfer, (anonymous/ hierarchical) identity-based encryption, trapdoor functions, chosen-ciphertext security transformations, designated-verifier zero-knowledge proofs, due to a beautiful framework recently introduced in the works of Cho et al. (Crypto 2017), and Dottling and Garg (Crypto 2017). The primitive of one-way function with encryption (OWFE) and its relatives (chameleon encryption, one-time signatures with encryption, hinting PRGs, trapdoor hash encryption, batch encryption) has been a centerpiece in all these results. While there exist multiple realizations of OWFE (and its relatives) from a variety of assumptions such as CDH, Factoring, and LWE, all such constructions fall under the same general ``missing block" framework. Although this framework has been instrumental in opening up a new pathway towards various cryptographic functionalities via the abstraction of OWFE (and its relatives), it has been accompanied by undesirable inefficiencies that might inhibit a much wider adoption in many practical scenarios. Motivated by the surging importance of the OWFE abstraction (and its relatives), a natural question to ask is whether the existing approaches can be diversified to not only obtain more constructions from different assumptions, but also in developing newer frameworks. We believe answering this question will eventually lead to important and previously unexplored performance trade-offs in the overarching applications of this novel cryptographic paradigm. In this work, we propose a new accumulation-style framework for building a new class of OWFE as well as hinting PRG constructions with a special focus on achieving shorter ciphertext size and shorter public parameter size (respectively). Such performance improvements parlay into shorter parameters in their corresponding applications. Briefly, we explore the following performance trade-offs --- (1) for OWFE, our constructions outperform in terms of ciphertext size as well as encryption time, but this comes at the cost of larger evaluation and setup times, (2) for hinting PRGs, our constructions provide a rather dramatic trade-off between evaluation time versus parameter size, with our construction leading to significantly shorter public parameter size. The trade-off enabled by our hinting PRG construction also leads to interesting implications in the CPA-to-CCA transformation provided in. We also provide concrete performance measurements for our constructions and compare them with existing approaches. We believe highlighting such trade-offs will lead to wider adoption of these abstractions in a practical sense.

The best existing traitor tracing scheme from pairings achieves $O(\sqrt{N})$-sized parameters, which has stood since 2006. This intuitively seems to be consistent with the fact that pairings allow for degree-2 computations, yielding a quadratic compression. In this work, we show that this intuition is false by building a traitor tracing scheme from pairings with $O(\sqrt[3]{N})$-sized parameters. We obtain our scheme by developing a number of new traitor tracing techniques offering various trade-offs that were not possible before, giving the first significant parameter improvements in pairings-based traitor tracing in over a decade.

We present a transformation from NIZK with inefficient provers in the uniform random string (URS) model to ZAPs (two message witness indistinguishable proofs) with inefficient provers. While such a transformation was known for the case where the prover is efficient, the security proof breaks down if the prover is inefficient. Our transformation is obtained via new applications of Nisan-Wigderson designs, a combinatorial object originally introduced in the derandomization literature. We observe that our transformation is applicable both in the setting of super-polynomial provers/poly-time adversaries, as well as a new fine-grained setting, where the prover is polynomial time and the verifier/simulator/zero knowledge distinguisher are in a lower complexity class, such as $\mathsf{NC}^1$. We also present $\mathsf{NC}^1$-fine-grained NIZK in the URS model for all of NP from the worst-case assumption $\oplus L/\poly \not\subseteq \mathsf{NC}^1$. Our techniques yield the following applications: --ZAPs for $\mathsf{AM}$ from Minicrypt assumptions (with super-polynomial time provers), --$\mathsf{NC}^1$-fine-grained ZAPs for $\mathsf{NP}$ from worst-case assumptions, --Protocols achieving an ``offline'' notion of NIZK (oNIZK) in the standard (no-CRS) model with uniform soundness in both the super-polynomial setting (from Minicrypt assumptions) and the $\mathsf{NC}^1$-fine-grained setting (from worst-case assumptions). The oNIZK notion is sufficient for use in indistinguishability-based proofs.

We present new Non-Interactive Zero-Knowledge argument systems (NIZK), based on standard assumptions that were previously not known to imply it. In particular, we rely on the hardness of both the learning parity with noise (LPN) assumption, and the existence of trapdoor hash functions (TDH, defined by Döttling et al., Crypto 2019). TDH can be based on a number of standard assumptions, including DDH, QR, DCR, and LWE. We rely on the Correlation Intractability (CI) framework for converting \Sigma-protocols into NIZK, but deviate from prior works in considering CI for searchable relations where the search function has a probabilistic representation by a simple function class (linear or constant degree in our instantiations). Namely, there is a distribution over simple functions that computes each output bit of the search function with all but small (constant) probability. We present a new tool for proving CI for such function classes via a notion that we call Approximate-Correlation Intractability. This notion requires that CI holds even against approximations of a given function class. We show that approximate-correlation intractability for just constant degree functions suffices if the underlying \Sigma-protocol is implemented using an extractable commitment scheme with approximately low-degree extraction, and that such a commitment scheme can be constructed based on LPN. We then show how to construct approximate CI hash functions for this class from any suitable rate-1 TDH (with an enhanced correctness property that is satisfied by all existing constructions).

We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover which is successful in our protocol. Finally, we show that any language in QMA has an (interactive) proof of quantum knowledge.

We present the first explicit construction of a non-malleable code that can handle tampering functions that are bounded-degree polynomials. Prior to our work, this was only known for degree-1 polynomials (affine tampering functions), due to Chattopadhyay and Li (STOC 2017). As a direct corollary, we obtain an explicit non-malleable code that is secure against tampering by bounded-size arithmetic circuits. We show applications of our non-malleable code in constructing non-malleable secret sharing schemes that are robust against bounded-degree polynomial tampering. In fact our result is stronger: we can handle adversaries that can adaptively choose the polynomial tampering function based on initial leakage of a bounded number of shares. Our results are derived from explicit constructions of seedless non-malleable extractors that can handle bounded-degree polynomial tampering functions. Prior to our work, no such result was known even for degree-2 (quadratic) polynomials.

Secret sharing enables a dealer to split a secret into a set of shares, in such a way that certain authorized subsets of share holders can reconstruct the secret, whereas all unauthorized subsets cannot. Non-malleable secret sharing (Goyal and Kumar, STOC 2018) additionally requires that, even if the shares have been tampered with, the reconstructed secret is either the original or a completely unrelated one. In this work, we construct non-malleable secret sharing tolerating $p$-time {\em joint-tampering} attacks in the plain model (in the computational setting), where the latter means that, for any $p>0$ fixed {\em a priori}, the attacker can tamper with the same target secret sharing up to $p$ times. In particular, assuming one-to-one one-way functions, we obtain: - A secret sharing scheme for threshold access structures which tolerates joint $p$-time tampering with subsets of the shares of maximal size ({\em i.e.}, matching the privacy threshold of the scheme). This holds in a model where the attacker commits to a partition of the shares into non-overlapping subsets, and keeps tampering jointly with the shares within such a partition (so-called {\em selective partitioning}). - A secret sharing scheme for general access structures which tolerates joint $p$-time tampering with subsets of the shares of size $O(\sqrt{\log n})$, where $n$ is the number of parties. This holds in a stronger model where the attacker is allowed to adaptively change the partition within each tampering query, under the restriction that once a subset of the shares has been tampered with jointly, that subset is always either tampered jointly or not modified by other tampering queries (so-called {\em semi-adaptive partitioning}). At the heart of our result for selective partitioning lies a new technique showing that every one-time {\em statistically} non-malleable secret sharing against joint tampering is in fact {\em leakage-resilient} non-malleable ({\em i.e.},\ the attacker can leak jointly from the shares prior to tampering). We believe this may be of independent interest, and in fact we show it implies lower bounds on the share size and randomness complexity of statistically non-malleable secret sharing against {\em independent} tampering.

Succinct non-interactive arguments (SNARGs) enable proofs of NP statements with very low communication. Recently, there has been significant work in both theory and practice on constructing SNARGs with very short proofs. Currently, the state-of-the-art in succinctness is due to Groth (Eurocrypt 2016) who constructed a SNARG from bilinear maps where the proof consists of just 3 group elements. In this work, we first construct a concretely-efficient designated-verifier (preprocessing) SNARG with inverse polynomial soundness, where the proof consists of just 2 group elements in a standard (generic) group. This leads to a 50% reduction in concrete proof size compared to Groth's construction. We follow the approach of Bitansky et al. (TCC 2013) who describe a compiler from linear PCPs to SNARGs in the preprocessing model. Our improvement is based on a new linear PCP packing technique that allows us to construct 1-query linear PCPs which can then be compiled into a SNARG (using ElGamal encryption over a generic group). An appealing feature of our new SNARG is that the verifier can precompute a statement-independent lookup table in an offline phase; verifying proofs then only requires 2 exponentiations and a single table lookup. This makes our new designated-verifier SNARG appealing in settings that demand fast verification and minimal communication. We then turn to the question of constructing arguments where the proof consists of a single group element. Here, we first show that any (possibly interactive) argument for a language L where the verification algorithm is "generic" (i.e., only performs generic group operations) and the proof consists of a single group element, implies a witness encryption scheme for L. We then show that under a yet-unproven, but highly plausible, hypothesis on the hardness of approximating the minimal distance of linear codes, we can construct a 2-message laconic argument for NP where the proof consists of a single group element. Under the same hypothesis, we obtain a witness encryption scheme for NP in the generic group model. Along the way, we show that under a conceptually-similar but proven hardness of approximation result, there is a 2-message laconic argument for NP with negligible soundness error where the prover's message consists of just 2 group elements. In both settings, we obtain laconic arguments (and linear PCPs) with linear decision procedures. Our constructions circumvent a previous lower bound by Groth on such argument systems with linear decision procedures by relying on imperfect completeness. Namely, our constructions have vanishing but not negligible completeness error, while the lower bound of Groth implicitly assumes negligible completeness error of the underlying argument. Our techniques thus highlight new avenues for designing linear PCPs, succinct arguments, and witness encryption schemes.

Decades of research in both cryptography and distributed systems has extensively studied the problem of state machine replication, also known as Byzantine consensus. A consensus protocol must usually satisfy two properties: {\em consistency} and {\em liveness}. These properties ensure that honest participating nodes agree on the same log and dictate when fresh transactions get added. They fail, however, to ensure against adversarial manipulation of the actual {\em ordering} of transactions in the log. Indeed, in leader-based protocols (almost all protocols used today), malicious leaders can directly choose the final transaction ordering. To rectify this problem, we propose a third consensus property: {\em transaction order-fairness}. We initiate the first formal investigation of order-fairness and explain its fundamental importance. We also provide several natural definitions for order-fairness and analyze the assumptions necessary to realize them. We also propose a new class of consensus protocols called Aequitas. Aequitas protocols are the first to achieve order-fairness in addition to consistency and liveness. They can be realized in a black-box way using existing broadcast and agreement primitives (or indeed using any consensus protocol), and work in both synchronous and asynchronous network models.

The security and performance of many integrity proof systems like SNARKs, STARKs and Bulletproofs highly depend on the underlying hash function. For this reason several new proposals have recently been developed. These primitives obviously require an in-depth security evaluation, especially since their implementation constraints have led to less standard design approaches. This work compares the security levels offered by two recent families of such primitives, namely GMiMC and HadesMiMC. We exhibit low-complexity distinguishers against the GMiMC and HadesMiMC permutations for most parameters proposed in recently launched public challenges for STARK-friendly hash functions. In the more concrete setting of the sponge construction corresponding to the practical use in the ZK-STARK protocol, we present a practical collision attack on a round-reduced version of GMiMC and a preimage attack on some instances of HadesMiMC. To achieve those results, we adapt and generalize several cryptographic techniques to fields of odd characteristic.

Composable security definitions, at times called simulation-based definitions, provide strong security guarantees that hold in any context. However, they are also met with some skepticism due to many impossibility results; goals such as commitments and zero-knowledge that are achievable in a stand-alone sense were shown to be unachievable composably (without a setup) since provably no efficient simulator exists. In particular, in the context of adaptive security, the so-called "simulator commitment problem" arises: once a party gets corrupted, an efficient simulator is unable to be consistent with its pre-corruption outputs. A natural question is whether such impossibility results are unavoidable or only artifacts of frameworks being too restrictive. In this work, we propose a novel type of composable security statement that evades the commitment problem. Our new type is able to express the composable guarantees of schemes that previously did not have a clear composable understanding. To this end, we leverage the concept of system specifications in the Constructive Cryptography framework, capturing the conjunction of several interval-wise guarantees, each specifying the guarantees between two events. We develop the required theory and present the corresponding new composition theorem. We present three applications of our theory. First, we show in the context of symmetric encryption with adaptive corruption how our notion naturally captures the expected confidentiality guarantee---the messages remain confidential until either party gets corrupted---and that it can be achieved by any standard semantically secure scheme (negating the need for non-committing encryption). Second, we present a composable formalization of (so far only known to be standalone secure) commitment protocols, which is instantiable without a trusted setup like a CRS. We show it to be sufficient for being used in coin tossing over the telephone, one of the early intuitive applications of commitments. Third, we reexamine a result by Hofheinz, Matt, and Maurer [Asiacrypt'15] implying that IND-ID-CPA security is not the right notion for identity-based encryption, unmasking this claim as an unnecessary framework artifact.

We construct a practical lattice-based zero-knowledge argument for proving multiplicative relations between committed values. The underlying commitment scheme that we use is the currently most efficient one of Baum et al. (SCN 2018), and the size of our multiplicative proof is only slightly larger than of the one for just proving knowledge of the committed values. We additionally improve on the results of Lyubashevsky and Seiler (Eurocrypt 2018) to show that the above-mentioned techniques can work over rings $Z_q[X]/(X^d+1)$ where $X^d+1$ splits into low-degree factors, which is a property necessary for many applications. In particular, we use Fourier analysis to show that the NTT coefficients of random small-norm challenges are not concentrated on any particular value.

We present a new protocol for two-party private set intersection (PSI) with semi-honest security in the plain model and one-sided malicious security in the random oracle model. Our protocol achieves a better balance between computation and communication than existing PSI protocols. Specifically, our protocol is the fastest in networks with moderate bandwidth (e.g., 30 - 100 Mbps). Considering the monetary cost (proposed by Pinkas et al. in CRYPTO 2019) to run the protocol on a cloud computing service, our protocol also compares favorably. Underlying our PSI protocol is a new lightweight multi-point oblivious pesudorandom function (OPRF) protocol based on oblivious transfer (OT) extension. We believe this new protocol may be of independent interest.

We give a framework for relating the quantitative, concrete security of a "reference'' protocol (say, one appearing in an academic paper) to that of some derived, "real'' protocol (say, appearing in a cryptographic standard). It is based on the indifferentiability framework of Maurer, Renner, and Holenstein (MRH), whose application has been exclusively focused upon non-interactive cryptographic primitives, e.g., hash functions and Feistel networks. Our extension of MRH is supported by a clearly defined execution model, and two composition lemmata, all formalized in a modern pseudocode language. Together, these allow for precise statements about game-based security properties of cryptographic objects (interactive or not) at various levels of abstraction, As a real-world application, we design and prove tight security bounds for a potential TLS 1.3 extension that integrates the SPAKE2 password-authenticated key-exchange into the existing handshake. (This is a problem of current interest to the IETF.)

Masking countermeasure is among the most powerful countermeasures to counteract side-channel attacks. Leakage models have been exhibited to theoretically reason on the security of such masked implementations. So far, the most widely used leakage model is the probing model defined by Ishai, Sahai, and Wagner at (CRYPTO 2003). While it is advantageously convenient for security proofs, it does not capture an adversary exploiting full leakage traces as, e.g., in horizontal attacks. Those attacks target the multiple manipulation of the same share to average a constant noise and recover the corresponding value. To capture a wider class of attacks another model was introduced and is referred to as the random probing model. From a leakage parameter p, each wire of the circuit leaks its value with probability p. While this model much better reflects the physical reality of side channels, it requires more complex security proofs and does not yet come with practical constructions. In this paper, we define the first framework dedicated to the random probing model. We provide an automatic tool, called VRAPS, to quantify the random probing security of a circuit from its leakage probability. We also formalize a composition property for secure random probing gadgets and exhibit its relation to the strong non-interference (SNI) notion used in the context of probing security. We then revisit the expansion idea proposed by Ananth, Ishai, and Sahai (CRYPTO 2018) and introduce a compiler that builds a random probing secure circuit from small base gadgets achieving a random probing expandability property. We instantiate this compiler with small gadgets for which we verify the expected properties directly from our automatic tool. Our construction can tolerate a leakage probability up to 2^−8, against 2^−25 for the previous construction, with a better asymptotic complexity.

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 has more to offer. We start with the development of a new versatile tool: we prove that, subject to the Riemann Hypothesis for Hecke L-functions, certain random walks on the Arakelov class group have a rapid mixing property. We then exploit this result to relate the average-case and the worst-case of the Shortest Vector Problem in ideal lattices. Our reduction appears particularly sharp: for Hermite-SVP in ideal lattices of certain cyclotomic number fields, it loses no more than a $\tilde O(\sqrt n)$ factor on the Hermite approximation factor. Furthermore, we suggest that this rapid-mixing theorem should find other applications in cryptography and in algorithmic number theory.

Reverse firewalls were introduced at Eurocrypt 2015 by Miro-nov and Stephens-Davidowitz, as a method for protecting cryptographic protocols against attacks on the devices of the honest parties. In a nutshell: a reverse firewall is placed outside of a device and its goal is to ``sanitize'' the messages sent by it, in such a way that a malicious device cannot leak its secrets to the outside world. It is typically assumed that the cryptographic devices are attacked in a ``functionality-preserving way'' (i.e.~informally speaking, the functionality of the protocol remains unchanged under this attacks). In their paper, Mironov and Stephens-Davidowitz construct a protocol for passively-secure two-party computations with firewalls, leaving extension of this result to stronger models as an open question. In this paper, we address this problem by constructing a protocol for secure computation with firewalls that has two main advantages over the original protocol from Eurocrypt 2015. Firstly, it is a \emph{multi}party computation protocol (i.e.~it works for an arbitrary number $n$ of the parties, and not just for $2$). Secondly, it is secure in much stronger corruption settings, namely in the \emph{actively corruption model}. More precisely: we consider an adversary that can fully corrupt up to $n-1$ parties, while the remaining parties are corrupt in a functionality-preserving way. Our core techniques are: malleable commitments and malleable non-interactive zero-knowledge, which in particular allow us to create a novel protocol for multiparty augmented coin-tossing into the well with reverse firewalls (that is based on a protocol of Lindell from Crypto 2001).

We give a four-round black-box construction of a commit-and-prove protocol with succinct communication. Our construction is WI and has constant soundness error, and it can be upgraded into a one that is ZK and has negligible soundness error by relying on a round-preserving transformation of Khurana et al. (TCC 2018). Our construction is obtained by combining the MPC-in-the-head technique of Ishai et al. (SICOMP 2009) with the two-round succinct argument of Kalai et al. (STOC 2014), and the main technical novelty lies in the analysis of the soundness---we show that, although the succinct argument of Kalai et al. does not necessarily provide soundness for NP statements, it can be used in the MPC-in-the-head technique for proving the consistency of committed MPC views. Our construction is based on sub-exponentially hard collision-resistant hash functions, two-round PIRs, and two-round OTs.

In this work, we conduct a comprehensive study on establishing hardness reductions for (Module) Learning with Rounding over rings (RLWR). Towards this, we present an algebraic framework of LWR, inspired by a recent work of Peikert and Pepin (TCC '19). Then we show a search-to-decision reduction for Ring-LWR, generalizing a result in the plain LWR setting by Bogdanov et al. (TCC '15). Finally, we show a reduction from Ring-LWE to Module Ring-LWR (even for leaky secrets), generalizing the plain LWE to LWR reduction by Alwen et al. (Crypto '13). One of our central techniques is a new ring leftover hash lemma, which might be of independent interests.

Efficiently sampling a quantum state that is hard to distinguish from a truly random quantum state is an elementary task in quantum information theory that has both computational and physical uses. This is often referred to as pseudorandom (quantum) state generator, or PRS generator for short. In existing constructions of PRS generators, security scales with the number of qubits in the states, i.e.\ the (statistical) security parameter for an $n$-qubit PRS is roughly $n$. Perhaps counter-intuitively, $n$-qubit PRS are not known to imply $k$-qubit PRS even for $k<n$. Therefore the question of \emph{scalability} for PRS was thus far open: is it possible to construct $n$-qubit PRS generators with security parameter $\secp$ for all $n, \secp$. Indeed, we believe that PRS with tiny (even constant) $n$ and large $\secp$ can be quite useful. We resolve the problem in this work, showing that any quantum-secure one-way function implies scalable PRS. We follow the paradigm of first showing a \emph{statistically} secure construction when given oracle access to a random function, and then replacing the random function with a quantum-secure (classical) pseudorandom function to achieve computational security. However, our methods deviate significantly from prior works since scalable pseudorandom states require randomizing the amplitudes of the quantum state, and not just the phase as in all prior works. We show how to achieve this using Gaussian sampling.

Secure messaging (SM) protocols allow users to communicate securely over untrusted infrastructure. In contrast to most other secure communication protocols (such as TLS, SSH, or Wireguard), SM sessions may be long-lived (e.g., years) and highly asynchronous. In order to deal with likely state compromises of users during the lifetime of a session, SM protocols do not only protect authenticity and privacy, but they also guarantee forward secrecy (FS) and post-compromise security (PCS). The former ensures that messages sent and received before a state compromise remain secure, while the latter ensures that users can recover from state compromise as a consequence of normal protocol usage. SM has received considerable attention in the two-party case, where prior work has studied the well-known double-ratchet paradigm, in particular, and SM as a cryptographic primitive, in general. Unfortunately, this paradigm does not scale well to the problem of secure group messaging (SGM). In order to address the lack of satisfactory SGM protocols, the IETF has launched the message-layer security (MLS) working group, which aims to standardize an eponymous SGM protocol. In this work we analyze the TreeKEM protocol, which is at the core of the SGM protocol proposed by the MLS working group. On a positive note, we show that TreeKEM achieves PCS in isolation (and slightly more). However, we observe that the current version of TreeKEM does not provide an adequate form of FS. More precisely, our work proceeds by formally capturing the exact security of TreeKEM as a so-called continuous group key agreement (CGKA) protocol, which we believe to be a primitive of independent interest. To address the insecurity of TreeKEM, we propose a simple modification to TreeKEM inspired by recent work of Jost et al. (EUROCRYPT '19) and an idea due to Kohbrok (MLS Mailing List). We then show that the modified version of TreeKEM comes with almost no efficiency degradation but achieves optimal (according to MLS specification) CGKA security, including FS and PCS. Our work also lays out how a CGKA protocol can be used to design a full SGM protocol.

We study the security of CTR-DRBG, one of NIST’s recommended Pseudorandom Number Generator (PRNG) designs. Recently, Woodage and Shumow (Eurocrypt’ 19), and then Cohney et al. (S&P’ 20) point out some potential vulnerabilities in both NIST specification and common implementations of CTR-DRBG. While these researchers do suggest counter-measures, the security of the patched CTR-DRBG is still questionable. Our work fills this gap, proving that CTR-DRBG satisfies the robustness notion of Dodis et al. (CCS’13), the standard security goal for PRNGs.

We put forth a new framework for building pairing-based non-interactive zero-knowledge (NIZK) arguments for a wide class of algebraic languages, which are an extension of linear languages, containing disjunctions of linear languages and more. Our approach differs from the Groth-Sahai methodology, in that we rely on pairings to compile a Sigma-protocol into a NIZK. Our framework enjoys a number of interesting features: - conceptual simplicity, parameters derive from the Sigma-protocol; - proofs as short as resulting from the Fiat-Shamir heuristic applied to the underlying Sigma-protocol; - fully adaptive soundness and perfect zero-knowledge in the common random string model with a single random group element as CRS; - yields simple and efficient two-round, public coin, publicly-verifiable perfect witness- indistinguishable (WI) arguments(ZAPs) in the plain model. To our knowledge, this is the first construction of two-rounds statistical witness-indistinguishable arguments from pairing assumptions. Our proof system relies on a new (static, falsifiable) assumption over pairing groups which generalizes the standard kernel Diffie-Hellman assumption in a natural way and holds in the generic group model (GGM) and in the algebraic group model (AGM). Replacing Groth-Sahai \NIZKs with our new proof system allows to improve several important cryptographic primitives. In particular, we obtain the shortest tightly-secure structure-preserving signature scheme (which are a core component in anonymous credentials), the shortest tightly-secure quasi-adaptive \NIZK with unbounded simulation soundness (which in turns implies the shortest tightly-mCCA-secure cryptosystem), and shorter ring signatures.

We show how to generalize Gama and Nguyen's slide reduction algorithm [STOC '08] for solving the approximate Shortest Vector Problem over lattices (SVP) to allow for arbitrary block sizes, rather than just block sizes that divide the rank n of the lattice. This leads to significantly better running times for most approximation factors. We accomplish this by combining slide reduction with the DBKZ algorithm of Micciancio and Walter [Eurocrypt '16]. We also show a different algorithm that works when the block size is quite large---at least half the total rank. This yields the first non-trivial algorithm for sublinear approximation factors. Together with some additional optimizations, these results yield significantly faster provably correct algorithms for \delta-approximate SVP for all approximation factors n^{1/2+\eps} \leq \delta \leq n^{O(1)}, which is the regime most relevant for cryptography. For the specific values of \delta = n^{1-\eps} and \delta = n^{2-\eps}, we improve the exponent in the running time by a factor of 2 and a factor of 1.5 respectively.

This paper introduces Spartan, a new family of zero-knowledge succinct non- interactive arguments of knowledge (zkSNARKs) for the rank-1 constraint satisfiabil- ity (R1CS), an NP-complete language that generalizes arithmetic circuit satisfiability. A distinctive feature of Spartan is that it offers the first zkSNARKs without trusted setup (i.e., transparent zkSNARKs) for NP where verifying a proof incurs sub-linear costs—without requiring uniformity in the NP statement’s structure. Furthermore, Spartan offers zkSNARKs with a time-optimal prover, a property that has remained elusive for nearly all zkSNARKs in the literature. To achieve these results, we introduce new techniques that we compose with the sum-check protocol, a seminal interactive proof protocol: (1) computation commit- ments, a primitive to create a succinct commitment to a description of a computation; this technique is crucial for a verifier to achieve sub-linear costs after investing a one-time, public computation to preprocess a given NP statement; (2) SPARK, a cryptographic compiler to transform any existing extractable polynomial commitment scheme for multilinear polynomials to one that efficiently handles sparse multilinear polynomials; this technique is critical for achieving a time-optimal prover; and (3) a compact encoding of an R1CS instance as a low-degree polynomial. The end result is a public-coin succinct interactive argument of knowledge for NP (which can be viewed as a succinct variant of the sum-check protocol); we transform it into a zkSNARK using prior techniques. By applying SPARK to different commitment schemes, we obtain several zkSNARKs where the verifier’s costs and the proof size range from $O(log^2{n})$ to $O(\sqrt{n})$ depending on the underlying commitment scheme ($n$ denotes the size of the NP statement). These schemes do not require a trusted setup except for one that requires a universal trusted setup. We implement Spartan as a library in about 8,000 lines of Rust. We use the library to build a transparent zkSNARK in the random oracle model where security holds under the discrete logarithm assumption. We experimentally evaluate it and compare with recent zkSNARKs for R1CS instance sizes up to 220 constraints. Among schemes without trusted setup, Spartan offers the fastest prover with speedups of 36--$152\times$ depending on the baseline, produces proofs that are shorter by 1.2--$416\times$, and incurs the lowest verification times with speedups of 3.6--$1326\times$. When compared to the state-of-the-art zkSNARK with trusted setup, Spartan’s prover is $2\times$ faster for arbitrary R1CS instances and $16\times$ faster for data-parallel workloads.

Secure two party computation (2PC) of arbitrary programs can be efficiently achieved using garbled circuits (GC). The bottleneck of GC efficiency is communication. It is widely believed that it is necessary to transmit the entire GC during 2PC, even for conditional branches that are not taken. This folklore belief is false. We propose a novel GC technique, stacked garbling, that eliminates the communication cost of inactive conditional branches. We extend the ideas of conditional GC evaluation explored in (Kolesnikov, Asiacrypt 18) and (Heath and Kolesnikov, Eurocrypt 20). Unlike these works, ours is for general 2PC where no player knows which conditional branch is taken. Our garbling scheme, Stack, requires communication proportional to the longest execution path rather than to the entire circuit. Stack is compatible with state-of-the-art techniques, such as free XOR and half-gates. Stack is a garbling scheme. As such, it can be plugged into a variety of existing protocols, and the resulting round complexity is the same as that of standard GC. The approach does incur computation cost quadratic in the conditional branching factor vs linear in standard schemes, but the tradeoff is beneficial for most programs: GC computation even on weak hardware is faster than GC transmission on fast channels. We implemented Stack in C++. Stack reduces communication cost by approximately the branching factor: for 16 branches, communication is reduced by 10.5x. In terms of wall-clock time for circuits with branching factor 16 over a 50 Mbps WAN on a laptop, Stack outperforms state-of- the-art half-gates-based 2PC by more than 4x.

Inserting backdoors in encryption algorithms has long seemed like a very interesting, yet difficult problem. Most attempts have been unsuccessful for symmetric-key primitives so far and it remains an open problem of how to build such ciphers. In this work, we propose the MALICIOUS framework, a new method to build tweakable block ciphers that have a backdoor hidden, which allows to retrieve the secret key. Our backdoor is differential in nature: a specific related-tweak differential path with high probability is hidden during design phase of the cipher. We explain how the backdoor can be used to practically recover the secret key of a user for any entity knowing the backdoor and we also argue why even knowing the presence of the backdoor and the workings of the cipher will not permit to retrieve the backdoor for an external user. We analyze the security of our construction in the classical black-box model and we show that retrieving the backdoor (the hidden high-probability differential path) is very difficult. We instantiate our framework by proposing the LowMC-M construction, a new family of tweakable block ciphers based on instances of the LowMC cipher, which allow such backdoor embedding. Generating LowMC-M instances is trivial and the LowMC-M family has basically the same efficiency as the LowMC instances it is based on.

We revisit recent works by Don, Fehr, Majenz and Schaffner and by Liu and Zhandry on the security of the Fiat-Shamir transformation of sigma-protocols in the quantum random oracle model (QROM). Two natural questions that arise in this context are: (1) whether the results extend to the Fiat-Shamir transformation of {\em multi-round} interactive proofs, and (2) whether Don et al.'s O(q^2) loss in security is optimal. Firstly, we answer question (1) in the affirmative. As a byproduct of solving a technical difficulty in proving this result, we slightly improve the result of Don et al., equipping it with a cleaner bound and an even simpler proof. We apply our result to digital signature schemes showing that it can be used to prove strong security for schemes like MQDSS in the QROM. As another application we prove QROM-security of a non-interactive OR proof by Liu, Wei and Wong. As for question (2), we show via a Grover-search based attack that Don et al.'s quadratic security loss for the Fiat-Shamir transformation of sigma-protocols is optimal up to a small constant factor. This extends to our new multi-round result, proving it tight up to a factor that depends on the number of rounds only, i.e. is constant for any constant-round interactive proof.

This paper initiates the study of the provable security of authenticated encryption (AE) in the memory-bounded setting. Recent works -- Tessaro and Thiruvengadam (TCC '18), Jaeger and Tessaro (EUROCRYPT '19), and Dinur (EUROCRYPT '20) -- focus on confidentiality, and look at schemes for which trade-offs between the attacker's memory and its data complexity are inherent. Here, we ask whether these results and techniques can be lifted to the full AE setting, which additionally asks for integrity. We show both positive and negative results. On the positive side, we provide tight memory-sensitive bounds for the security of GCM and its generalization, CAU (Bellare and Tackmann, CRYPTO '16). Our bounds apply to a restricted case of AE security which abstracts the deployment within protocols like TLS, and rely on a new memory-tight reduction to corresponding restricted notions of confidentiality and integrity. In particular, our reduction uses an amount of memory which linearly depends on that of the given adversary, as opposed to only imposing a constant memory overhead as in earlier works (Auerbach et al, CRYPTO '17). On the negative side, we show that a large class of black-box reductions cannot generically lift confidentiality and integrity security to a joint definition of AE security in a memory-tight way.

A well-established PRP-to-PRF conversion design is truncation: one evaluates an $n$-bit pseudorandom permutation on a certain input, and truncates the result to $a$ bits. The construction is known to achieve tight $2^{n-a/2}$ security. Truncation has gained popularity due to its appearance in the GCM-SIV key derivation function (ACM CCS 2015). This key derivation function makes four evaluations of AES, truncates the outputs to $n/2$ bits, and concatenates these to get a $2n$-bit subkey. In this work, we demonstrate that truncation is wasteful. In more detail, we present the Summation-Truncation Hybrid (STH). At a high level, the construction consists of two parallel evaluations of truncation, where the truncated $(n-a)$-bit chunks are not discarded but rather summed together and appended to the output. We prove that STH achieves a similar security level as truncation, and thus that the $n-a$ bits of extra output is rendered for free. In the application of GCM-SIV, the current key derivation can be used to output $3n$ bits of random material, or it can be reduced to three primitive evaluations. Both changes come with no security loss.

We study collision-finding against Merkle-Damgård hashing in the random-oracle model by adversaries with an arbitrary $S$-bit auxiliary advice input about the random oracle and $T$ queries. Recent work showed that such adversaries can find collisions (with respect to a random IV) with advantage $\Omega(ST^2/2^n)$, where $n$ is the output length, beating the birthday bound by a factor of $S$. These attacks were shown to be optimal. We observe that the collisions produced are very long, on the order $T$ blocks, which would limit their practical relevance. We prove several results related to improving these attacks to find short collisions. We first exhibit a simple attack for finding $B$-block-long collisions achieving advantage $\tilde{\Omega}(STB/2^n)$. We then study if this attack is optimal. We show that the prior technique based on the bit-fixing model (used for the $ST^2/2^n$ bound) provably cannot reach this bound, and towards a general result we prove there are qualitative jumps in the optimal attacks for finding length $1$, length $2$, and unbounded-length collisions. Namely, the optimal attacks achieve (up to logarithmic factors) order of $(S+T)/2^n$, $ST/2^n$ and $ST^2/2^n$ advantage. We also give an upper bound on the advantage of a restricted class of short-collision finding attacks via a new analysis on the growth of trees in random functional graphs that may be of independent interest.

Private intersection-sum with cardinality allows two parties, where each party holds a private set and one of the parties additionally holds a private integer value associated with each element in her set, to jointly compute the cardinality of the intersection of the two sets as well as the sum of the associated integer values for all the elements in the intersection, and nothing beyond that. We present a new construction for private intersection sum with cardinality that provides malicious security with abort and guarantees that both parties receive the output upon successful completion of the protocol. A central building block for our constructions is a primitive called shuffled distributed oblivious PRF (DOPRF), which is a PRF that offers oblivious evaluation using a secret key shared between two parties, and in addition to this allows obliviously permuting the PRF outputs of several parallel oblivious evaluations. We present the first construction for shuffled DOPRF with malicious security. We further present several new sigma proof protocols for relations across Pedersen commitments, ElGamal encryptions, and Camenisch-Shoup encryptions that we use in our main construction, for which we develop new batching techniques to reduce communication. We implement and evaluate the efficiency of our protocol and show that we can achieve communication cost that is only 4-5x greater than the most efficient semi-honest protocol. When measuring monetary cost of executing the protocol in the cloud, our protocol is 25x more expensive than the semi-honest protocol. Our construction also allows for different parameter regimes that enable trade-offs between communication and computation.

Protocols for password authenticated key exchange (PAKE) allow two parties who share only a weak password to agree on a cryptographic key. We revisit the notion of PAKE in the universal composability (UC) framework, and propose a relaxation of the PAKE functionality of Canetti et al. that we call lazy-extraction PAKE (lePAKE). Our relaxation allows the ideal-world adversary to postpone its password guess until after a session is complete. We argue that this relaxed notion still provides meaningful security in the password-only setting. As our main result, we show that several PAKE protocols that were previously only proven secure with respect to a ``game-based'' definition of security can be shown to UC-realize the lePAKE functionality in the random-oracle model. These include SPEKE, SPAKE2, and TBPEKE, the most efficient PAKE schemes currently known.

In recent work, Garg, Hajiabadi, Mahmoody, and Rahimi (TCC 2018) introduced a new encryption framework, which they referred to as Registration-Based Encryption (RBE). The central motivation behind RBE was to provide a novel methodology for solving the well-known key-escrow problem in Identity-Based Encryption (IBE) systems. Informally, in an RBE system, there is no private-key generator unlike IBE systems, but instead, it is replaced with a public key accumulator. Every user in an RBE system samples its own public-secret key pair and sends the public key to the accumulator for registration. The key accumulator has no secret state and is only responsible for compressing all the registered user identity-key pairs into a short public commitment. Here the encryptor only requires the compressed parameters along with the target identity, whereas a decryptor requires supplementary key material along with the secret key associated with the registered public key. The initial construction by Garg et al. based on standard assumptions only provided weak efficiency properties. In a follow-up work by Garg, Hajiabadi, Mahmoody, Rahimi, and Sekar (PKC 2019), they gave an efficient RBE construction from standard assumptions. However, both these works considered the key accumulator to be honest which might be too strong an assumption in real-world scenarios. In this work, we initiate a formal study of RBE systems with malicious key accumulators. To that end, we introduce a strengthening of the RBE framework which we call Verifiable RBE (VRBE). A VRBE system additionally gives the users an extra capability to obtain short proofs from the key accumulator proving correct (and unique) registration for every registered user as well as proving non-registration for any yet unregistered identity. We construct VRBE systems that provide succinct proofs of registration and non-registration from standard assumptions (such as CDH, Factoring, LWE). Our proof systems also naturally allow a much more efficient audit process which can be performed by any non-participating third party as well. A by-product of our approach is that we provide a more efficient RBE construction than that provided in the prior work of Garg \etal\cite{GHMRS19}. And, lastly, we initiate a study on the extension of VRBE to a wider range of access and trust structures.