International Association
for Cryptologic Research

Succinct non-interactive zero-knowledge arguments of knowledge (zk-SNARKs) are a type of non-interactive proof system enabling efficient privacy-preserving proofs of membership for NP languages. A great deal of works has studied candidate constructions that are secure against quantum attackers, which are based on either lattice assumptions, or post-quantum collision-resistant hash functions. In this paper, we propose a code-based zk-SNARK scheme, whose security is based on the rank support learning (RSL) problem, a variant of the random linear code decoding problem in the rank metric. Our construction follows the general framework of Gennaro et al. (CCS'18), which is based on square span programs (SSPs). Due to the fundamental differences between the hardness assumptions, our proof of security cannot apply the techniques from the lattice-based constructions, and indeed, it distinguishes itself by the use of techniques from coding theory. We also provide the scheme with a set of concrete parameters.

The number theoretic transform (NTT) permits a very efficient method to perform multiplication of very large degree polynomials, which is the most time-consuming operation in fully homomorphic encryption (FHE) schemes and a class of non-interactive succinct zero-knowledge proof systems such as zk-SNARK. Efficient modular arithmetic plays an important role in the performance of NTT, and therefore it is studied extensively. The access pattern to the memory, on the other hand, may play much greater role, as the NTT execution time is mostly memory-bound due to large degree polynomials. In this paper, we propose two algorithms for fast computation of NTT on a class of graphical processing units (GPU) by optimizing the memory access patterns. We present an approach i) to optimize the number of accesses to slow global memory for thread synchronization, and ii) to make better use of spatial locality in global memory accesses. It turns out that by controlling certain parameters in CUDA platform for general-purpose GPU computing (GPGPU) such as kernel count, block size and block shape, we can affect the performance of NTT. To best of our knowledge, this work is unique for it suggests a recipe for selecting optimum CUDA parameters to obtain the best NTT performance for a given polynomial degree. Our implementation results on various GPU devices for all power-of-two polynomial degrees from $2^{12}$ to $2^{28}$ show that our algorithms compare favorably with the other state-of-the-art GPU implementations in the literature with the optimum selection of these three CUDA parameters.

We define and analyze the Commutative Isogeny Hidden Number Problem which is the natural analogue of the Hidden Number Problem in the CSIDH and CSURF setting. In short, the task is as follows: Given two supersingular elliptic curves \(E_A\), \(E_B\) and access to an oracle that outputs some of the most significant bits of the \({\mathsf{CDH}}\) of two curves, an adversary must compute the shared curve \(E_{AB}={\mathsf{CDH}}(E_A,E_B)\). We show that we can recover \(E_{AB}\) in polynomial time by using Coppersmith's method as long as the oracle outputs \({\frac{13}{24}} + \varepsilon \approx 54\%\) (CSIDH) and \({\frac{31}{41}} + \varepsilon \approx 76\%\) (CSURF) of the most significant bits of the \({\mathsf{CDH}}\), where $\varepsilon > 0$ is an arbitrarily small constant. To this end, we give a purely combinatorial restatement of Coppersmith's method, effectively concealing the intricate aspects of lattice theory and allowing for near-complete automation. By leveraging this approach, we attain recovery attacks with $\varepsilon$ close to zero within a few minutes of computation.

In this paper, we improve the cube attack by exploiting low-degree factors of the superpoly w.r.t. certain "special" index set of cube (ISoC). This can be viewed as a special case of the correlation cube attack proposed at Eurocrypt 2018, but under our framework more beneficial equations on the key variables can be obtained in the key-recovery phase. To mount our attack, one has two challenging problems: (1) effectively recover algebraic normal form of the superpoly and extract out its low-degree factors; and (2) efficiently search a large quantity of good ISoCs. We bring in new techniques to solve both of them.

First, we propose the variable substitution technique for middle rounds of a cipher, in which polynomials on the key variables in the algebraic expressions of internal states are substituted by new variables. This will improve computational complexity of the superpoly recovery and promise more compact superpolys that can be easily decomposed with respect to the new variables. Second, we propose the vector numeric mapping technique, which seeks out a tradeoff between efficiency of the numeric mapping technique (Crypto 2019) and accuracy of the monomial prediction technique (Asiacrypt 2020) in degree evaluation of superpolys. Combining with this technique, a fast pruning method is given and modeled by MILP to filter good ISoCs of which the algebraic degree satisfies some fixed threshold. Thanks to automated MILP solvers, it becomes practical to comprehensively search for good cubes across the entire search space.

To illustrate the power of our techniques, we apply all of them to Trivium stream cipher. As a result, we have recovered the superpolys of three cubes given by Kesarwani et al. in 2020, only to find they do not have zero-sum property up to 842 rounds as claimed in their paper. To our knowledge, the previous best practical key recovery attack was on 820-round Trivium with complexity $2^{53.17}$. We put forward 820-, 825- and 830-round practical key-recovery attacks, in which there are $\mathbf{2^{80}\times 87.8\%}$, $\mathbf{2^{80}\times 83\%}$ and $\mathbf{2^{80}\times 65.7\%}$ keys that could be practically recovered, respectively, if we consider $\mathbf{2^{60}}$ as the upper bound for practical computational complexity. Besides, even for computers with computational power not exceeding $\mathbf{2^{52}}$ (resp. $\mathbf{2^{55}}$), we can still recover $\mathbf{58\%}$ (resp. $\mathbf{46.6\%}$) of the keys in the key space for 820 rounds (resp. 830 rounds). Our attacks have led 10 rounds more than the previous best practical attack.

In this paper, we consider how to use fully homomorphic encryptions (FHEs) to solve the problem of secure computations over set intersection where one party holds a relatively set of size $N_s$ and the other party holds a relatively small set of size $N_r$ collaboratively compute some functionality over their set intersection without revealing other information. This problem has many applications for online collaboration, for example, fingerprint matching, online dating, and shareriding.

This paper introduces Sigmabus, a technique designed to enhance the efficiency of zero-knowledge circuits by relocating computationally expensive operations outside the circuit. Specifically, Sigmabus focuses on moving elliptic curve group operations, typically proven with expensive non-native field arithmetic, to external computations. By leveraging Sigma protocols, elliptic curve group operations are proven outside the circuit, while additional constraints are applied to the circuit to ensure correct execution of the Sigma protocol. This approach can achieve significant performance improvements in zero-knowledge circuits. This paper presents the Sigmabus protocol along with its security proofs, and demonstrates its practical implications through various use cases.

Succinct arguments allow a prover to convince a verifier of the validity of any statement in a language, with minimal communication and verifier's work. Among other approaches, lattice-based protocols offer solid theoretical foundations, post-quantum security, and a rich algebraic structure. In this work, we present some new approaches to constructing efficient lattice-based succinct arguments. Our main technical ingredient is a new commitment scheme based on vanishing polynomials, a notion borrowed from algebraic geometry. We analyse the security of such a commitment scheme, and show how to take advantage of the additional algebraic structure to build new lattice-based succinct arguments. A few highlights amongst our results are:

- The first recursive folding (i.e. Bulletproofs-like) protocol for linear relations with polylogarithmic verifier runtime. Traditionally, the verifier runtime has been the efficiency bottleneck for such protocols (regardless of the underlying assumptions). - The first verifiable delay function (VDF) based on lattices, building on a recently introduced sequential relation. - The first lattice-based \emph{linear-time prover} succinct argument for NP, in the preprocessing model. The soundness of the scheme is based on (knowledge)-k-R-ISIS assumption [Albrecht et al., CRYPTO'22].

Lucas sequences are constant-recursive integer sequences with a long history of applications in cryptography, both in the design of cryptographic schemes and cryptanalysis. In this work, we study the sequential hardness of computing Lucas sequences over an RSA modulus.

First, we show that modular Lucas sequences are at least as sequentially hard as the classical delay function given by iterated modular squaring proposed by Rivest, Shamir, and Wagner (MIT Tech. Rep. 1996) in the context of time-lock puzzles. Moreover, there is no obvious reduction in the other direction, which suggests that the assumption of sequential hardness of modular Lucas sequences is strictly weaker than that of iterated modular squaring. In other words, the sequential hardness of modular Lucas sequences might hold even in the case of an algorithmic improvement violating the sequential hardness of iterated modular squaring. Moreover, we note that modular Lucas sequences also yield a time-lock puzzle, similar to the classical construction of Rivest, Shamir and Wagner.

Second, we demonstrate the feasibility of constructing practically-efficient verifiable delay functions based on the sequential hardness of modular Lucas sequences. Our construction builds on the work of Pietrzak (ITCS 2019) by leveraging the intrinsic connection between the problem of computing modular Lucas sequences and exponentiation in an appropriate extension field.

Extremely Lossy Functions (ELFs) are families of functions that, depending on the choice during key generation, either operate in injective mode or instead have only a polynomial image size. The choice of the mode is indistinguishable to an outsider. ELFs were introduced by Zhandry (Crypto 2016) and have been shown to be very useful in replacing random oracles in a number of applications.

One open question is to determine the minimal assumption needed to instantiate ELFs. While all constructions of ELFs depend on some form of exponentially-secure public-key primitive, it was conjectured that exponentially-secure secret-key primitives, such as one-way functions, hash functions or one-way product functions, might be sufficient to build ELFs. In this work we answer this conjecture mostly negative: We show that no primitive, which can be derived from a random oracle (which includes all secret-key primitives mentioned above), is enough to construct even moderately lossy functions in a black-box manner. However, we also show that (extremely) lossy functions themselves do not imply public-key cryptography, leaving open the option to build ELFs from some intermediate primitive between the classical categories of secret-key and public-key cryptography.

This paper offers a mathematical introduction to fully homomorphic encryption, a concept that enables computation on encrypted data. We trace the historical development of FHE, describe Fully Homomorphic Encryption over the Torus (TFHE) and how it performs certain mathematical operations, and explore bootstrapping and the possibility for adjusting computational depth. This paper equips readers with a brief understanding of FHE's evolution and the essential mechanisms facilitating practical implementation.

Non-interactive key exchange (NIKE) schemes like the Diffie-Hellman key exchange are a widespread building block in several cryptographic protocols. Since the Diffie-Hellman key exchange is not post-quantum secure, it is important to investigate post-quantum alternatives.

We analyze the security of the LWE-based NIKE by Ding et al. (ePrint 2012) and Peikert (PQCrypt 2014) in a multi-user setting where the same public key is used to generate shared keys with multiple other users. The Diffie-Hellman key exchange achieves this security notion. The mentioned LWE-based NIKE scheme comes with an inherent correctness error (Guo et al., PKC 2020), and this has significant implications for the multi-user security, necessitating a closer examination.

Single-user security generically implies multi-user security when all users generate their keys honestly for NIKE schemes with negligible correctness error. However, the LWE-based NIKE requires a super-polynomial modulus to achieve a negligible correctness error, which makes the scheme less efficient. We show that - generically, single-user security does not imply multi-user security when the correctness error is non-negligible, but despite this - the LWE-based NIKE with polynomial modulus is multi-user secure for honest users when the number of users is fixed in advance. This result takes advantage of the leakage-resilience properties of LWE.

We then turn to a stronger model of multi-user security that allows adversarially generated public keys. For this model, we consider a variant of the LWE-based NIKE where each public key is equipped with a NIZKPoK of the secret key. Adding NIZKPoKs is a standard technique for this stronger model and Hesse et al. (Crypto 2018) showed that this is sufficient to achieve security in the stronger multi-user security model for perfectly correct NIKEs (which the LWE-based NIKE is not). We show that - for certain parameters that include all parameters with polynomial modulus, the LWE-based NIKE can be efficiently attacked with adversarially generated public keys, despite the use of NIZKPoKs, but - for suitable parameters (that require a super-polynomial modulus), this security notion is achieved by the LWE-based NIKE with NIZKPoKs. This stronger security notion has been previously achieved for LWE-based NIKE only in the QROM, while all our results are in the standard model.

Updatable public key encryption has recently been introduced as a solution to achieve forward-security in the context of secure group messaging without hurting efficiency, but so far, no efficient lattice-based instantiation of this primitive is known.

In this work, we construct the first LWE-based UPKE scheme with polynomial modulus-to-noise rate, which is CPA-secure in the standard model. At the core of our security analysis is a generalized reduction from the standard LWE problem to (a stronger version of) the Extended LWE problem. We further extend our construction to achieve stronger security notions by proposing two generic transforms. Our first transform allows to obtain CCA security in the random oracle model and adapts the Fujisaki-Okamoto transform to the UPKE setting. Our second transform allows to achieve security against malicious updates by adding a NIZK argument in the update mechanism. In the process, we also introduce the notion of Updatable Key Encapsulation Mechanism (UKEM), as the updatable variant of KEMs. Overall, we obtain a CCA-secure UKEM in the random oracle model whose ciphertext sizes are of the same order of magnitude as that of CRYSTALS-Kyber.

The supersingular Endomorphism Ring problem is the following: given a supersingular elliptic curve, compute all of its endomorphisms. The presumed hardness of this problem is foundational for isogeny-based cryptography. The One Endomorphism problem only asks to find a single non-scalar endomorphism. We prove that these two problems are equivalent, under probabilistic polynomial time reductions.

We prove a number of consequences. First, assuming the hardness of the endomorphism ring problem, the Charles–Goren–Lauter hash function is collision resistant, and the SQIsign identification protocol is sound. Second, the endomorphism ring problem is equivalent to the problem of computing arbitrary isogenies between supersingular elliptic curves, a result previously known only for isogenies of smooth degree. Third, there exists an unconditional probabilistic algorithm to solve the endomorphism ring problem in time $\tilde O(p^{1/2})$, a result that previously required to assume the generalized Riemann hypothesis.

To prove our main result, we introduce a flexible framework for the study of isogeny graphs with additional information. We prove a general and easy-to-use rapid mixing theorem.

FIDO2 is currently the main initiative for passwordless authentication in web servers. It mandates the use of secure hardware authenticators to protect the authentication protocol’s secrets from compromise. However, to ensure that only secure authenticators are being used, web servers need a method to attest their properties. The FIDO2 specifications allow for authenticators and web servers to choose between different attestation modes to prove the characteristics of an authenticator, however the properties of most these modes have not been analysed in the context of FIDO2. In this work, we analyse the security and privacy properties of FIDO2 when different attestation modes included in the standard are used, and show that they lack good balance between security, privacy and revocation of corrupted devices. For example, the basic attestation mode prevents remote servers from tracing user’s actions across different services while requiring reduced trust assumptions. However in case one device is compromised, all the devices from the same batch (e.g., of the same brand or model) need to be recalled, which can be quite complex (and arguably impractical) in consumer scenarios. As a consequence we suggest a new attestation mode based on the recently proposed TokenWeaver, which provides more convenient mechanisms for revoking a single token while maintaining user privacy.

Picnic is a NIST PQC Round 3 Alternate signature candidate that builds upon symmetric primitives following the MPC-in-the-head paradigm. Recently, researchers have been exploring more secure/efficient signature schemes from conservative one-way functions based on AES, or new low complexity one-way functions like Rain (CCS 2022) and AIM (CCS 2023). The signature schemes based on Rain and AIM are currently the most efficient among MPC-in-the-head-based schemes, making them promising post-quantum digital signature candidates.

However, the exact hardness of these new one-way functions deserves further study and scrutiny. This work presents algebraic attacks on RAIN and AIM for certain instances, where one-round Rain can be compromised in $2^{n/2}$ for security parameter $n\in \{128,192,256\}$, and two-round Rain can be broken in $2^{120.3}$, $2^{180.4}$, and $2^{243.1}$ encryptions, respectively. Additionally, we demonstrate an attack on AIM-III (which aims at 192-bit security) with a complexity of $2^{186.5}$ encryptions. These attacks exploit the algebraic structure of the power function over fields with characteristic 2, which provides potential insights into the algebraic structures of some symmetric primitives and thus might be of independent interest.

The long running time of isogeny-based cryptographic constructions has proved to be a boon in disguise for one particular type of primitive called Verifiable Delay Functions (VDFs). VDFs are characterised by sequential function evaluation but an immediate output verification. In order to ensure secure use of VDFs in real-world applications, it is important to determine the fastest implementation. Considering the point of view of an attacker (say with unbounded resources), this paper aims to achieve the fastest possible hardware implementation of isogeny-based VDFs. It is the first work that implements the $2^T$-isogeny walk involved in the evaluation step of an isogeny VDF. To meet our goal, we use redundant representations of integers and introduce a new lookup table-based algorithm for modular reduction. We also provide a survey of elliptic curve arithmetic to arrive at the most cost-effective curve computations and propose an improvement of the point doubling algorithm for better parallelism. The evaluation step of a VDF is defined to be sequential, which means that there is limited scope for parallelism. Nevertheless, taking this constraint into account our proposed design targets the highest levels of parallelism possible on an architectural level of an isogeny VDF implementation. We provide detailed analysis of all our arithmetic modules as well as estimates for their critical path delays and area consumption. Our 28nm ASIC design computes a $4^{100} = 2^{200}$-isogeny in 7.1$\mu s$. It is the first high-performance ASIC implementation for evaluation of isogeny VDFs.

We introduce a novel side-channel-based reverse engineering technique capable of reconstructing a procedure solely from inputs, outputs, and traces of execution. Beyond generic restrictions, we do not assume any prior knowledge of the procedure or the chip it operates on. These restrictions confine our analysis to 8-bit RISC constant-time software implementations.

Specifically, we demonstrate the feasibility of reconstructing a symmetric cryptographic cipher, even in scenarios where traces are sampled with information loss and noise, such as when measuring the power consumption of the chip.

EURECOM is seeking a highly motivated and talented PhD student to join our research team under the supervision of Prof. Antonio Faonio. The selected candidate will work on a challenging and innovative research project focused on zero-knowledge proofs (ZKPs) and their applications in cloud scenarios. The primary objectives of this position are to advance the state-of-the-art in ZKPs, with a particular emphasis on zkSNARKs, and to explore novel applications of such protocols.

Closing date for applications:

Contact: Antonio Faonio

More information: https://www.eurecom.fr/en/job/zero-knowledge-proofs

NUS-Singapore is seeking a highly motivated PhD candidate in the field of cyber security with an emphasis on embedded and hardware security. Candidates with experience in one or more of the following are preferred: PUF, lightweight cryptography, post-quantum cryptography; designing novel cryptographic primitives and protocols; digital design on ASIC or FPGA platforms using hardware description languages; computer architectures and embedded software; side-channel analysis and fault attacks; and machine learning and artificial intelligence especially for security applications. Research topics include but are not limited to Physical unclonable function (PUF) secure cryptographic implementations in hardware and software; mechanisms against side-channel analysis and fault attacks security and privacy for IoT systems; post-quantum cryptography; In the first instance, candidates can discuss applications with Dr. Prosanta Gope via email (p.gope@sheffield.ac.uk). Who can Apply? (i) You should possess a strong research background, evidenced by accomplishments like publications in reputable research venues. (ii) You should exhibit high levels of motivation.

Closing date for applications:

Contact: Dr Prosanta Gope (p.gope@sheffield.ac.uk)

The CryptoLux group of the University of Luxembourg has a vacancy for a post-doctoral researcher in the area of security/privacy of blockchains and smart contracts. The successful candidate will contribute to a research project entitled Advanced Cryptography for Finance and Privacy (CryptoFin), which is funded by the Fonds National de la Recherche (FNR). Starting in September 2023, CryptoFin will run over a period of 3 years and be carried out in collaboration with the cryptography teams of Stanford University and Ethereum foundation. The mission of the CryptoFin project is to develop innovative solutions for some of the most pressing research problems in the blockchain domain, especially in the context of layer-2 protocols for off-chain transactions and the design of advanced cryptographic techniques like verifiable delay functions, proof-of-X systems with special features, and new MPC/SNARK-friendly primitives.

Candidates must hold a Ph.D. degree in cryptography, IT security, or a related field. Preference will be given to candidates with a strong publication record that includes at least one paper at an IACR conference/workshop or one of the top-4 security conferences. Experience in blockchains and/or smart contracts is a plus. Candidates with an interest to conduct research in one of the following areas are particularly encouraged to apply:

• Applied cryptography (especially design/analysis of symmetric cryptosystems)
• Cryptofinance and cryptoeconomics
• Privacy and anonymity on the Internet

The position is initially offered for 1 year, but an extension by 2 years is possible. The University of Luxembourg offers excellent working conditions and a highly competitive salary. Interested candidates are invited to send their application by email to Prof. Alex Biryukov before October 15, 2023 (early submission is encouraged). The application material should contain a cover letter explaining the candidate's research interests, a CV (incl. photo), a list of publications, scans of diploma certificates, and contact details of 3 references.

Closing date for applications:

Contact: Prof. Alex Biryukov (alex.biryukov@uni.lu)