International Association
for Cryptologic Research

Due to the adoption of the horizontal business model with the globalization of semiconductor manufacturing, the overproduction of integrated circuits (ICs) and the piracy of intellectual properties (IPs) have become a significant threat to the semiconductor supply chain. Logic locking has emerged as a primary design-for-security measure to counter these threats. In logic locking, ICs become fully functional after fabrication only when unlocked with the correct key. However, Boolean satisfiability-based attacks have rendered most locking schemes ineffective. This gives rise to the numerous defenses and new locking methods to achieve SAT resiliency. This paper provides a unique perspective on the SAT attack efficiency based on conjunctive normal form (CNF) stored in the SAT solver. First, we show that the attack learns a new relation between key bits upon every distinguishing pattern. After each iteration, these additional clauses appended to the solver could significantly decrease the key search complexity. Second, we demonstrate that the SAT attack can break the locking scheme within the linear complexity of key size. The deviation away from linear search can be explained by the oracle's output and different logic gate types. This helps to answer how different distinguishing input eliminates fewer or more incorrect keys. Moreover, we show how key constraints on point functions affect the complexity of SAT attack. The proper key constraint on AntiSAT locking can effectively reduce the SAT attack complexity to constant 1. The same constraint minimizes the complexity of breaking CAS-Lock down to the linear range. Our analysis provides fresh perspectives on the capabilities of SAT attack, and we offer new directions to achieve SAT resiliency.

In the first part of the paper, we show a generic compiler that transforms any oracle algorithm that can query multiple oracles adaptively, i.e., can decide on which oracle to query at what point dependent on previous oracle responses, into a static algorithm that fixes these choices at the beginning of the execution. Compared to naive ways of achieving this, our compiler controls the blow-up in query complexity for each oracle individually, and causes a very mild blow-up only.

In the second part of the paper, we use our compiler to show the security of the very efficient hash-based split-key PRF proposed by Giacon, Heuer and Poettering (PKC 2018), in the quantum random-oracle model. Using a split-key PRF as the key-derivation function gives rise to a secure KEM combiner. Thus, our result shows that the hash-based construction of Giacon et al. can be safely used in the context of quantum attacks, for instance to combine a well-established but only classically-secure KEM with a candidate KEM that is believed to be quantum-secure.

Our security proof for the split-key PRF crucially relies on our adaptive-to-static compiler, but we expect our compiler to be useful beyond this particular application. Indeed, we discuss a couple of other, known results from the literature that would have profitted from our compiler, in that these works had to go though serious complications in oder to deal with adaptivity.

Private Set Intersection (PSI) allows a set of mutually distrustful parties, each holds a private data set, to compute the intersection of all sets, such that no information is revealed except for the intersection. The state-of-the-art PSI protocol (Garimella et al., CRYPTO'21) in the multi-party setting tolerating any number of malicious corruptions requires the communication bandwidth of $O(n\ell|\mathbb{F}|)$ bits for the central party $P_0$ due to the star architecture, where $n$ is the number of parties, $\ell$ is the size of each set and $|\mathbb{F}|$ is the size of an exponentially large field $\mathbb{F}$. When $n$ and $\ell$ are large, this forms an efficiency bottleneck (especially for networks with restricted bandwidthes). In this paper, we present a new multi-party PSI protocol in dishonest-majority malicious setting, which reduces the communication bandwidth of the central party $P_0$ from $O(n\ell|\mathbb{F}|)$ bits to $O(\ell|\mathbb{F}|)$ bits using a tree architecture. Furthermore, our PSI protocol reduces the expensive LPN encoding operations performed by $P_0$ by a factor of $n$ as well as the computational cost by $2n\ell$ hash operations in total. Additionally, while the multi-party PSI protocol (Garimella et al., CRYPTO'21) with a single output is secure, we present a simple attack against its multi-output extension, which allows an adversary to learn more information on the sets of honest parties beyond the intersection of all sets.

Fully homomorphic encryption~(FHE) has flourished since it was first constructed by Gentry~(STOC 2009). Single instruction multiple data~(SIMD) gave rise to efficient homomorphic operations on vectors in \((\mathbb{F}_{t^d})^\ell\), for prime \(t\). RLWE instantiated with cyclotomic polynomials of the form \(X^{2^N}+1\) dominate implementations of FHE due to highly efficient fast Fourier transformations. However, this choice yields very short SIMD plaintext vectors and high degree extension fields, e.g. \(\ell < 100, d > 100\) for small primes~(\(t = 3, 5, \dots\)).

In this work, we describe a method to encode more data on top of SIMD, \emph{Field Instruction Multiple Data}, applying reverse multiplication friendly embedding~(RMFE) to FHE. With RMFE, length-\(k\) \(\mathbb{F}_{t}\) vectors can be encoded into \(\mathbb{F}_{t^d}\) and multiplied once. The results have to be recoded~(decoded and then re-encoded) before further multiplications can be done. We introduce an FHE-specific technique to additionally evaluate arbitrary linear transformations on encoded vectors for free during the FHE recode operation. On top of that, we present two optimizations to unlock high degree extension fields with small \(t\) for homomorphic computation: \(r\)-fold RMFE, which allows products of up to \(2^r\) encoded vectors before recoding, and a three-stage recode process for RMFEs obtained by composing two smaller RMFEs. Experiments were performed to evaluate the effectiveness of FIMD from various RMFEs compared to standard SIMD operations. Overall, we found that FIMD generally had \(>2\times\) better (amortized) multiplication times compared to FHE for the same amount of data, while using almost \(k/2 \times\) fewer ciphertexts required.

We present two provably secure password-authenticated key exchange (PAKE) protocols based on a commutative group action. To date the most important instantiation of isogeny-based group actions is given by CSIDH. To model the properties more accurately, we extend the framework of cryptographic group actions (Alamati et al., ASIACRYPT 2020) by the ability of computing the quadratic twist of an elliptic curve. This property is always present in the CSIDH setting and turns out to be crucial in the security analysis of our PAKE protocols. Despite the resemblance, the translation of Diffie-Hellman based PAKE protocols to group actions either does not work with known techniques or is insecure ("How not to create an isogeny-based PAKE", Azarderakhsh et al., ACNS 2020). We overcome the difficulties mentioned in previous work by using a "bit-by-bit" approach, where each password bit is considered separately. Our first protocol $\mathsf{X\text{-}GA\text{-}PAKE}_\ell$ can be executed in a single round. Both parties need to send two set elements for each password bit in order to prevent offline dictionary attacks. The second protocol $\mathsf{Com\text{-}GA\text{-}PAKE}_\ell$ requires only one set element per password bit, but one party has to send a commitment on its message first. We also discuss different optimizations that can be used to reduce the computational cost. We provide comprehensive security proofs for our base protocols and deduce security for the optimized versions.

Since the advent of pairing-based cryptography, various optimization methods that increase the speed of pairing computations have been exploited, as well as new types of pairings. This paper extends the work of Kinoshita and Suzuki who proposed a new formula for the $ \beta$-Weil pairing on curves with even embedding degree by eliminating denominators and exponents during the computation of the Weil pairing. We provide novel formulas suitable for the parallel computation for the $\beta$-Weil pairing on curves with odd embedding degree which involve vertical line functions useful for sparse multiplications. For computations we used Miller's algorithm combined with storage and multifunction methods. Applying our framework to BLS-$27$, BLS-$15$ and BLS-$9$ curves at respectively the $256$ bit, the $192$ bit and the $128$ bit security level, we obtain faster $\beta$-Weil pairings than the previous state-of-the-art constructions. The correctness of all the formulas and bilinearity of pairings obtained in this work is verified by a SageMath code.

A software watermarking scheme can embed a message into a program while preserving its functionality. The embedded message can be extracted later by an extraction algorithm, and no one could remove it without significantly changing the functionality of the program. A watermarking scheme is public key if neither the marking procedure nor the extraction procedure needs a watermarking secret key. Prior constructions of watermarking schemes mainly focus on watermarking pseudorandom functions (PRFs), and the major open problem in this direction is to construct a public-key watermarkable PRF.

In this work, we solve the open problem via constructing public-key watermarkable PRFs with different trade-offs from various assumptions, ranging from standard lattice assumptions to the existence of indistinguishability obfuscation. To achieve the results, we first construct watermarking schemes in a weaker model, where the extraction algorithm is provided with a “hint” about the watermarked PRF key. Then we upgrade the constructions to standard watermarking schemes using a robust unobfuscatable PRF. We also provide the first construction of robust unobfuscatable PRF in this work, which is of independent interest.

Non-malleable zero-knowledge, originally introduced in the context of man-in-the-middle attacks, serves as an important building block to protect against concurrent attacks where different protocols may coexist and interleave. While this primitive admits almost optimal constructions in the plain model, they are several orders of magnitude slower in practice than standalone zero-knowledge. This is in sharp contrast to non-malleable commitments where practical constructions (under the DDH assumption) have been known for a while.

We present a new approach for constructing efficient non-malleable zero-knowledge for all languages in NP, based on a new primitive called instance-based non-malleable commitment (IB-NMC). We show how to construct practical IB-NMC by leveraging the fact that simulators of sub-linear zero-knowledge protocols can be much faster than the honest prover algorithm. With an efficient implementation of IB-NMC, our approach yields the first general-purpose non-malleable zero-knowledge protocol that achieves practical efficiency in the plain model.

All of our protocols can be instantiated from symmetric primitives such as block-ciphers and hash functions, have reasonable efficiency in practice, and are general-purpose. Our techniques also yield the first efficient non-malleable commitment scheme without public-key assumptions.

In recent years, the number of applications of the repeated squaring assumption has been growing rapidly. The assumption states that, given a group element $x$, an integer $T$, and an RSA modulus $N$, it is hard to compute $x^{2^T} \mod N$---or even decide whether $y\stackrel{?}{=}x^{2^T} \mod N$---in parallel time less than the trivial approach of computing $T$ sequential squarings. This rise has been driven by efficient interactive proofs for repeated squaring, opening the door to more efficient constructions of verifiable delay functions, various secure computation primitives, and proof systems for more general languages.

In this work, we study the complexity of statistically-sound interactive proofs for the repeated squaring relation. Technically, we consider interactive proofs where the prover sends at most $k \ge 0$ elements per round and the verifier performs generic group operations over the group $\mathbb{Z}_N^\star$. As our main contribution, we show that for any one-round proof with a randomized verifier (i.e., an MA proof) the verifier either runs in parallel time $\Omega(T/(k+1))$ with high probability, or is able to factor $N$ given the proof provided by the prover. This shows that either the prover essentially sends $p,q$ such that $N = p\cdot q$ (which is infeasible or undesirable in most applications), or a variant of Pietrzak's proof of repeated squaring (ITCS 2019) has optimal verifier complexity $O(T/(k+1))$. In particular, it is impossible to obtain a statistically-sound one-round proof of repeated squaring with efficiency on par with the computationally-sound protocol of Wesolowski (EUROCRYPT 2019), with a generic group verifier. We further extend our one-round lower bound to a natural class of recursive (multi-round) interactive proofs for repeated squaring.

The rotational differential-linear attacks, proposed at EUROCRYPT 2021, is a generalization of differential-linear attacks by replacing the differential part of the attacks with rotational differentials. At EUROCRYPT 2021, Liu et al. presented a method based on Morawiecki et al.’s technique (FSE 2013) for evaluating the rotational differential-linear correlations for the special cases where the output linear masks are unit vectors. With this method, some powerful (rotational) differential-linear distinguishers with output linear masks being unit vectors against Friet, Xoodoo, and Alzette were discovered. However, how to compute the rotational differential-linear correlations for arbitrary output masks was left open. In this work, we partially solve this open problem by presenting an efficient algorithm for computing the (rotational) differential-linear correlation of modulo additions for arbitrary output linear masks, based on which a technique for evaluating the (rotational) differential-linear correlation of ARX ciphers is derived. We apply the technique to Alzette, Siphash, Chacha, and Speck. As a result, significantly improved (rotational) differential-linear distinguishers including deterministic ones are identified. All results of this work are practical and experimentally verified to confirm the validity of our methods. In addition, we try to explain the experimental distinguishers employed in FSE 2008, FSE 2016, and CRYPTO 2020 against Chacha. The predicted correlations are close to the experimental ones.

Proofs of Retrievability (PoR) protocols ensure that a client can fully retrieve a large outsourced file from an untrusted server. Good PoRs should have low communication complexity, small storage overhead and clear security guarantees. We design a good PoR based on a family of graph codes called expander codes. We use expander codes based on graphs derived from point-line incidence relations of finite affine planes. Høholdt et al. showed that, when using Reed-Solomon codes as inner codes, these codes have good dimension and minimum distance over a relatively small alphabet. Moreover, expander codes possess very efficient unique decoding algorithms. We take advantage of these results to de- sign a PoR scheme that extracts the outsourced file in quasi-linear time and features better concrete parameters than state-of-the-art schemes w.r.t storage overhead and size of the outsourced file. Using the Con- structive Cryptography framework of Maurer, we get sharper and more rigourous security guarantees for our scheme than the ones given by the usual epsilon-adversary model. We follow an unbounded-use audit procedure to ensure that the extraction of the outsourced file will succeed w.h.p.. The properties of our expander codes yield an audit with communication complexity comparable to other code-based PoRs.

In recent years, cryptocurrencies have increasingly been used in cybercrime and have become the key means of payment in darknet marketplaces, partly due to their alleged anonymity. Furthermore, the research attacking the anonymity of even those cryptocurrencies that claim to offer anonymity by design is growing and is being applied by law enforcement agencies in the fight against cybercrime. Their investigative measures require a certain degree of suspicion and it is unclear whether findings resulting from attacks on cryptocurrencies' anonymity can indeed establish that required degree of suspicion. The reason for this is that these attacks are partly based upon uncertain assumptions which are often not properly addressed in the corresponding papers. To close this gap, we extract the assumptions in papers that are attacking Bitcoin, Monero and Zcash, major cryptocurrencies used in darknet markets which have also received the most attention from researchers. We develop a taxonomy to capture the different nature of those assumptions in order to help investigators to better assess whether the required degree of suspicion for specific investigative measures could be established. We found that assumptions based on user behaviour are in general the most unreliable and thus any findings of attacks based on them might not allow for intense investigative measures such as pre-trial detention. We hope to raise awareness of the problem so that in the future there will be fewer unlawful investigations based upon uncertain assumptions and thus fewer human rights violations.

Verifiable random functions (VRFs) are a useful extension of pseudorandom functions for which it is possible to generate a proof that a certain image is indeed the correct function value (relative to a public verification key). Due to their strong soundness requirements on such proofs, VRFs are notoriously hard to construct, and existing constructions suffer either from complex proofs (for function images), or rely on complex and non-standard assumptions.

In this work, we attempt to explain this phenomenon. We show that for a large class of pairing-based VRFs, it is not possible to obtain short proofs and a reduction to a simple assumption simultaneously. Since the class of "consecutively verifiable" VRFs we consider contains in particular the VRF of Lysyanskaya and that of Dodis-Yampolskiy, our results explain the large proof size, resp. the complex assumption of these VRFs.

In this paper we study search problems that arise very often in cryptanalysis: nested search problems, where each search layer has known degrees of freedom as well as constraints. Classical nested searches can be transformed into quantum algorithms, using Grover's quantum search or amplitude amplification by Brassard et al., obtaining up to a square-root speedup. However, the nesting introduces technicalities in the quantum complexity analysis that are complex to handle and have been so far analyzed in previous works in a case-by-case manner. In this paper, we aim to simplify the quantum transformation and corresponding analysis.

We introduce a framework to transform classical nested searches into a quantum procedure and to analyze its complexity. The resulting quantum procedure is easier to describe and analyze compared to previous works, both in the asymptotic setting and for concrete instantiations. Its time complexity and success probability can be bounded using a generic formula, or more precisely with numerical optimization.

Along the way to this result, we introduce an algorithm for variable-time amplitude amplification of independent interest. It allows to obtain essentially the same asymptotic complexity as a previous algorithm by Ambainis (STACS 2012) using only several layers of amplitude amplification, and without relying on amplitude estimation.

Moreover, we present some direct applications of our results in cryptanalysis.

There are numerous settings in which people's preferences are aggregated outside of formal elections, and where privacy and verification are important but the stringent authentication and coercion-resistant properties of government elections do not apply, a prime example being social media platforms. These systems are often iterative and have no trusted authority, in contrast to the centrally organised, single-shot elections on which most of the literature is focused. Moreover, they require a continuous flow of aggregation to take place and become available even as input is still collected from the participants which is in contrast to "fairness" in classical elections where partial results should never be revealed.

In this work, we explore opinion aggregation in a decentralised, iterative setting by proposing a novel protocol in which randomly-chosen participants take turns to act in an incentive-driven manner as decryption authorities. Our construction provides public verifiability, robust vote privacy and liveness guarantees, while striving to minimise the resources each participant needs to contribute.

Hashing arbitrary values to points on an elliptic curve is a required step in many cryptographic constructions, and a number of techniques have been proposed to do so over the years. One of the first ones was due to Shallue and van de Woestijne (ANTS-VII), and it had the interesting property of applying to essentially all elliptic curves over finite fields. It did not, however, have the desirable property of being indifferentiable from a random oracle when composed with a random oracle to the base field.

Various approaches have since been considered to overcome this limitation, starting with the foundational work of Brier et al. (CRYPTO 2011). For example, if $f\colon \mathbb{F}_q\to E(\mathbb{F}_q)$ is the Shallue--van de Woestijne (SW) map and $\mathfrak{h}_1,\mathfrak{h}_2$ are two independent random oracles to $\mathbb{F}_q$, we now know that $m\mapsto f\big(\mathfrak{h}_1(m)\big)+f\big(\mathfrak{h}_2(m)\big)$ is indifferentiable from a random oracle. Unfortunately, this approach has the drawback of being twice as expensive to compute than the straightforward, but not indifferentiable, $m\mapsto f\big(\mathfrak{h}_1(m)\big)$. Most other solutions so far have had the same issue: they are at least as costly as two base field exponentiations, whereas plain encoding maps like $f$ cost only one exponentiation. Recently, Koshelev (DCC 2022) provided the first construction of indifferentiable hashing at the cost of one exponentiation, but only for a very specific class of curves (some of those with $j$-invariant $0$), and using techniques that are unlikely to apply more broadly.

In this work, we revisit this long-standing open problem, and observe that the SW map actually fits in a one-parameter family $(f_u)_{u\in\mathbb{F}_q}$ of encodings, such that for independent random oracles $\mathfrak{h}_1, \mathfrak{h}_2$ to $\mathbb{F}_q$, $F\colon m\mapsto f_{\mathfrak{h}_2(m)}\big(\mathfrak{h}_1(m)\big)$ is indifferentiable. Moreover, on a very large class of curves (essentially those that are either of odd order or of order divisible by 4), the one-parameter family admits a rational parametrization, which let us compute $F$ at almost the same cost as small $f$, and finally achieve indifferentiable hashing to most curves with a single exponentiation.

Our new approach also yields an improved variant of the Elligator Squared technique of Tibouchi (FC 2014) that represents points of arbitrary elliptic curves as close-to-uniform random strings.

Adaptive security is a highly desirable property in the design of secure protocols. It tolerates adversaries that corrupt parties as the protocol proceeds, as opposed to static security where the adversary corrupts the parties at the onset of the execution. The well-accepted folklore is that static and adaptive securities are equivalent for perfectly secure protocols. Indeed, this folklore is backed up by a transformation by Canetti et al. (EUROCRYPT'01), showing that any perfectly secure protocol that is statically secure and satisfies some basic requirements is also adaptively secure. Yet, the transformation results in an adaptively secure protocol with inefficient simulation (i.e., where the simulator might run in super-polynomial time even if the adversary runs just in polynomial time). Inefficient simulation is problematic when using the protocol as a sub-routine in the computational setting.

Our main question is whether an alternative efficient transformation from static to adaptive security exists. We show an inherent difficulty in achieving this goal generically. In contrast to the folklore, we present a protocol that is perfectly secure with efficient static simulation (therefore also adaptively secure with inefficient simulation), but for which efficient adaptive simulation does not exist (assuming the existence of one-way permutations).

In addition, we prove that the seminal protocol of Ben-Or, Goldwasser, and Wigderson (STOC'88) is secure against adaptive, semi-honest corruptions with efficient simulation. Previously, adaptive security of the protocol, as is, was only known either for a restricted class of circuits or for all circuits but with inefficient simulation.

Homomorphic encryption (HE) schemes enable computing functions on encrypted data, by means of a public $\Eval$ procedure that can be applied to ciphertexts. But the evaluated ciphertexts so generated may differ from freshly encrypted ones. This brings up the question of whether one can keep computing on evaluated ciphertexts. An \emph{$i$-hop} homomorphic encryption scheme is one where $\Eval$ can be called on its own output up to $i$~times, while still being able to decrypt the result. A \emph{multi-hop} homomorphic encryption is a scheme which is $i$-hop for all~$i$. In this work we study $i$-hop and multi-hop schemes in conjunction with the properties of function-privacy (i.e., $\Eval$'s output hides the function) and compactness (i.e., the output of $\Eval$ is short). We provide formal definitions and describe several constructions.

First, we observe that "bootstrapping" techniques can be used to convert any (1-hop) homomorphic encryption scheme into an $i$-hop scheme for any~$i$, and the result inherits the function-privacy and/or compactness of the underlying scheme. However, if the underlying scheme is not compact (such as schemes derived from Yao circuits) then the complexity of the resulting $i$-hop scheme can be as high as $k^{O(i)}$.

We then describe a specific DDH-based multi-hop homomorphic encryption scheme that does not suffer from this exponential blowup. Although not compact, this scheme has complexity linear in the size of the composed function, independently of the number of hops. The main technical ingredient in this solution is a \emph{re-randomizable} variant of the Yao circuits. Namely, given a garbled circuit, anyone can re-garble it in such a way that even the party that generated the original garbled circuit cannot recognize it. This construction may be of independent interest.

Machine learning as a service scenario typically requires the client to trust the server and provide sensitive data in plaintext. However, with the recent improvements in fully homomorphic encryption (FHE) schemes, many such applications can be designed in a privacy preserving way. In this work, we focus on such a problem, private decision tree evaluation (PDTE) --- where a server has a decision tree classification model, and a client wants to use the model to classify her private data without revealing the data or the classification result to the server. We present an efficient non-interactive design of PDTE, that we call SortingHat, based on FHE techniques. As part of our design, we solve multiple cryptographic problems related to FHE: (1) we propose a fast homomorphic comparison function where one input can be in plaintext format; (2) we design an efficient binary decision tree evaluation technique in the FHE setting, which we call homomorphic traversal, and apply it together with our homomorphic comparison to evaluate private decision tree classifiers, obtaining running times orders of magnitude faster than the state of the art; (3) we improve both the communication cost and the time complexity of transciphering, by applying our homomorphic comparison to the FiLIP stream cipher. Through a prototype implementation, we demonstrate that our improved transciphering solution runs around 400 times faster than previous works. We finally present a choice in terms of PDTE design: we present a version of SortingHat without transciphering that achieves significant improvement in terms of computation cost comparing to prior works; and another version with transciphering that has a communication cost about 20 thousand times smaller but comparable running time.

In this work we propose a new accumulator construction and efficient ways to prove knowledge of some element in a set without leaking anything about the element. This problem arises in several applications including privacy-preserving distributed ledgers (e.g., Zcash) and anonymous credentials. Our approaches do not require a trusted setup and significantly improve on the efficiency state of the of the art. We introduce new techniques inspired by commit-and-prove techniques and combine shallow Merkle trees, 2-cycles of elliptic curves to obtain constructions that are highly practical. Our basic construction—which we dub $\mathsf{Curve} \ \mathsf{Trees}$—is completely transparent (does not require a trusted setup) and is based on simple standard assumptions (DLOG and Random Oracle Model). It has small proofs and commitments and very efficient proving and verification time. Curve trees can be instantiated to be efficient in practice: the commitment to a set (accumulator) is 256 bits for any set size; for a set of size $2^{32}$ a proof is approximately 2KB, a verifier runs in $\approx 160$ms (easily parallelizable to $\approx 80$ms) and a prover in $\approx 3.6$s on an ordinary laptop. Using our construction as a building block we can construct a simple and concretely efficient anonymous cryptocurrency with full anonymity set. We estimate the verification time to be $\approx 320$ms (and trivially parallelizable to run in $\approx 160$ms) or $< 10$ms when batch-verifying multiple ($> 100$) transactions simultaneously. Transaction sizes are $< 3$KB. Our timings are competitive with those of the approach in Zcash Sapling and trade slightly larger proofs (proofs in Zcash are 0.2KB) for a completely transparent setup.