International Association
for Cryptologic Research

The Key Management team at Copper is tasked with developing the core components of Copper’s custodial solution. This entails the constructions of tools and services essential for secure storage and transfer of digital assets. A focal point of the team’s responsibilities includes developing and integrating MPC-based cryptographic signature algorithms.

Key Responsibilities of the role

You will conduct cryptography literature review, select relevant papers, and produce a specification and pseudocode from the paper(s). If situation demands it, you will produce original peer-reviewed research in close collaboration with academia.

You will implement cryptography specifications into secure, production-level code.

You will integrate cryptography primitives into Copper’s internal MPC framework.

You will participate in the design and implementation of Copper’s next generation MPC framework.

You will interact with auditors and academia, as well as cryptographers on the client side.

You will provide cryptography support for other Copper development teams.

Your experience, skills and knowledge

This role is right for you if:

You have a strong foundation in modern/provable cryptography.

You consider yourself an applied cryptographer who doesn’t shy away from implementing your constructions.

You are interested in a unique intersection between academia and industry wherein you will be working with both theory and real-world implementations.

Essential

Research experience in modern cryptography -preferably in MPC, Threshold Cryptography, [Zero-Knowledge] proof systems etc- and its tools e.g. Proving your constructions secure in UC paradigm.

Engineering experience in implementation of modern cryptography primitives.

Fluency in a systems language like C, C++, Rust, or Go.

Familiarity with algorithms and aptitude in cryptographic system design.

Desirable

PhD in Cryptography with research track record.

Familiarity/Experience with distributed systems.

Prior knowledge/experience with MPC or crypto custody.

Closing date for applications:

Contact:

Clara Luna

clara.luna@copper.co

More information: https://www.linkedin.com/jobs/view/3922104775

The Cheon-Kim-Kim-Song (CKKS) fully homomorphic encryption scheme is designed to efficiently perform computations on real numbers in an encrypted state. Recently, Drucker et al. [J. Cryptol.] proposed an efficient strategy to use CKKS in a black-box manner to perform computations on binary data.

In this work, we introduce several CKKS bootstrapping algorithms designed specifically for ciphertexts encoding binary data. Crucially, the new CKKS bootstrapping algorithms enable to bootstrap ciphertexts containing the binary data in the most significant bits. First, this allows to decrease the moduli used in bootstrapping, saving a larger share of the modulus budget for non-bootstrapping operations.

In particular, we obtain full-slot bootstrapping in ring degree $2^{14}$ for the first time. Second, the ciphertext format is compatible with the one used in the DM/CGGI fully homomorphic encryption schemes. Interestingly, we may combine our CKKS bootstrapping algorithms for bits with the fast ring packing technique from Bae et al. [CRYPTO'23]. This leads to a new bootstrapping algorithm for DM/CGGI that outperforms the state-of-the-art approaches when the number of bootstraps to be performed simultaneously is in the low hundreds.

A verifiable delay function (VDF) is a cryptographic primitive that takes a long time to compute, but produces a unique output that is efficiently and publicly verifiable.

Mahmoody, Smith and Wu (ICALP 2020) prove that VDFs satisfying both perfect completeness and adaptive perfect uniqueness do not exist in the random oracle model. Moreover, Ephraim, Freitag, Komargodski, and Pass (EUROCRYPT 2020) construct a VDF with perfect completeness and computational uniqueness, a much weaker guarantee compare to perfect uniqueness, in the random oracle model under the repeated squaring assumption.

In this work, we close the gap between existing constructions and known lower bounds by showing that VDFs with imperfect completeness and non-adaptive computational uniqueness cannot be constructed in the pure random oracle model (without additional computational assumptions).

We propose a new unified framework to construct multi-server, information-theoretic Private Information Retrieval (PIR) schemes that leverage global preprocesing to achieve sublinear computation per query. Despite a couple earlier attempts, our understanding of PIR schemes in the global preprocessing model remains limited, and so far, we only know a few sparse points in the broad design space. With our new unified framework, we can generalize the results of Beimel, Ishai, and Malkin to broader parameter regimes, thus enabling a tradeoff between bandwidth and computation. Specifically, for any constant $S > 1$, we can get an $S$-server scheme whose bandwidth consumption is as small as $n^{1/(S+1) + \epsilon}$ while achieving computation in the $n^\delta$ regime for some constant $\delta \in (0, 1)$. Moreover, we can get a scheme with polylogarithmic bandwidth and computation, requiring only polylogarithmic number of servers.

Decentralized Multi-Client Functional Encryption (DMCFE) extends the basic functional encryption to multiple clients that do not trust each other. They can independently encrypt the multiple plaintext-inputs to be given for evaluation to the function embedded in the functional decryption key, defined by multiple parameter-inputs. And they keep control on these functions as they all have to contribute to the generation of the functional decryption keys. Tags can be used in the ciphertexts and the keys to specify which inputs can be combined together. As any encryption scheme, DMCFE provides privacy of the plaintexts. But the functions associated to the functional decryption keys might be sensitive too (e.g. a model in machine learning). The function-hiding property has thus been introduced to additionally protect the function evaluated during the decryption process.

In this paper, we provide new proof techniques to analyze a new concrete construction of function-hiding DMCFE for inner products, with strong security guarantees in the random oracle model: the adversary can adaptively query multiple challenge ciphertexts and multiple challenge keys, with unbounded repetitions of the same message tags in the ciphertext-queries and a fixed polynomially-large number of repetitions of the same key tags in the key-queries, allowing static corruption of the secret encryption keys. Previous constructions were proven secure in the selective setting only.

Provable security based on a robust mathematical framework is the gold standard for security evaluation in cryptography. Several provable secure cryptosystems have been studied for public key cryptography. However, provably secure symmetric-key cryptography has received little attention. Although there are known provably secure symmetric-key cryptosystems based on the hardness of factorization and discrete logarithm problems, they are not only slower than conventional block ciphers but can also be broken by quantum computers. Our study aims to tackle this latter problem by proposing a new provably secure Feistel cipher using collision resistant hash functions based on a Short Integer Solution problem (SIS). Even if cipher primitives are resistant to quantum algorithms, it is crucial to determine whether the cipher is resilient to differential cryptanalysis, a fundamental and powerful attack against symmetric-key cryptosystems. In this paper, we demonstrate that the proposed cipher family is secure against differential cryptanalysis by deriving an upper bound on the maximum differential probability. In addition, we demonstrate the potential success of differential cryptanalysis for short block sizes and statistically evaluate the average resistance of cipher instances based on differential characteristic probabilities. This method approximates the S-box output using a folded two-dimensional normal distribution and employs a generalized extreme value distribution. This evaluation method is first introduced in this paper and serves as the basis for studying the differential characteristics of lattice matrices and the number of secure rounds. This study is foundational research on differential cryptanalysis against block ciphers using a lattice matrix based on SIS.

Witness encryption (WE) allows a ciphertext to be encrypted under an NP problem such that anyone holding a valid witness for that problem can decrypt it (flexible decryptors), without interaction with others (non-interaction). However, existing schemes are either impractical or achieve only a part of these WE features. We propose a novel WE scheme that 1) is based on bilinear maps such as pairings, 2) achieves the property of flexible decryptors, and 3) still requires the decryptor's communication with a trusted signer, who only performs a fixed amount of computation and communication at regular intervals, regardless of the number of ciphertexts. It provides extractable security and can be extended to a threshold multiple signers setting, avoiding reliance on a single signer. As a significant application of our WE scheme, we build a novel one-time program (OTP) scheme in which the signers' computational and communication costs remain constant, independent of the number of OTPs to be evaluated simultaneously. This feature ensures scalable OTP evaluations without risking decreased signer participation or compromised decentralization due to increased operational costs for the signers.

The extensive use of cloud storage has created an urgent need to search and share data. Public key authenticated encryption with keyword search (PAEKS) allows for the retrieval from encrypted data, while resisting the insider keyword guessing attacks (IKGAs). Most PAEKS schemes only work with single-receiver model, exhibiting very limited applicability. To address this concern, there have been researches on broadcast authenticated encryption with keyword search (BAEKS) to achieve multi-receiver ciphertext search. But to our best knowledge, existing BAEKS schemes are susceptible to quantum computing attacks. In this paper, we propose lattice-based BAEKS, the first post-quantum broadcast authenticated encryption with keyword search, providing robust quantum-safety in multi-receiver model. Specifically, we leverage several lattice sampling algorithms and rejection sampling technique to construct our BAEKS scheme. Furthermore, we incorporate minimal cover set technique and lattice basis extension algorithm to construct an enhanced version, namely FS-BAEKS. Moreover, we give a rigorous security analysis of our scheme. Ultimately, the best computational overhead of BAEKS and Test algorithms in our BAEKS scheme delivers up to approximately 12-x and 402-x faster over prior arts when the number of receivers is six, respectively, which is practical for cloud storage systems.

We introduce SQIsign2D-West, a variant of SQIsign using two-dimensional isogeny representations. SQIsignHD was the first variant of SQIsign to use higher dimensional isogeny representations. Its eight-dimensional variant is geared towards provable security but is deemed unpractical. Its four-dimensional variant is geared towards efficiency and has significantly faster signing times than SQIsign, but slower verification owing to the complexity of the four-dimensional representation. Its authors commented on the apparent difficulty of getting any improvement over SQIsign by using two-dimensional representations. In this work, we introduce new algorithmic tools that make two-dimensional representations a viable alternative. These lead to a signature scheme with sizes comparable to SQIsignHD, slightly slower signing than SQIsignHD but still much faster than SQIsign, and the fastest verification of any known variant of SQIsign. We achieve this without compromising on the security proof: the assumptions behind SQIsign2D-West are similar to those of the eight-dimensional variant of SQIsignHD. Additionally, like SQIsignHD, SQIsign2D-West favourably scales to high levels of security Concretely, for NIST level I we achieve signing times of 80 ms and verifying times of 4.5 ms, using optimised arithmetic based on intrinsics available to the Ice Lake architecture. For NIST level V, we achieve 470 ms for signing and 31 ms for verifying.

A zero-bit watermarked language model produces text that is indistinguishable from that of the underlying model, but which can be detected as machine-generated using a secret key. Unfortunately, merely detecting AI-generated spam, say, as watermarked may not prevent future abuses. If we could additionally trace the text to a spammer's API token or account, we could then cut off their access or pursue legal action.

We introduce multi-user watermarks, which allow tracing model-generated text to individual users or to groups of colluding users. We construct multi-user watermarking schemes from undetectable zero-bit watermarking schemes. Importantly, our schemes provide both zero-bit and multi-user assurances at the same time: detecting shorter snippets just as well as the original scheme, and tracing longer excerpts to individuals. Along the way, we give a generic construction of a watermarking scheme that embeds long messages into generated text.

Ours are the first black-box reductions between watermarking schemes for language models. A major challenge for black-box reductions is the lack of a unified abstraction for robustness — that marked text is detectable even after edits. Existing works give incomparable robustness guarantees, based on bespoke requirements on the language model's outputs and the users' edits. We introduce a new abstraction to overcome this challenge, called AEB-robustness. AEB-robustness provides that the watermark is detectable whenever the edited text "approximates enough blocks" of model-generated output. Specifying the robustness condition amounts to defining approximates, enough, and blocks. Using our new abstraction, we relate the robustness properties of our message-embedding and multi-user schemes to that of the underlying zero-bit scheme, in a black-box way. Whereas prior works only guarantee robustness for a single text generated in response to a single prompt, our schemes are robust against adaptive prompting, a stronger and more natural adversarial model.

Multivariate public key cryptography (MPKC) is one of the most promising alternatives to build quantum-resistant signature schemes, as evidenced in NIST's call for additional post-quantum signature schemes. The main assumption in MPKC is the hardness of the Multivariate Quadratic (MQ) problem, which seeks for a common root to a system of quadratic polynomials over a finite field. Although the Crossbred algorithm is among the most efficient algorithm to solve MQ over small fields, its complexity analysis stands on shaky ground. In particular, it is not clear for what parameters it works and under what assumptions. In this work, we provide a rigorous analysis of the Crossbred algorithm over any finite field. We provide a complete explanation of the series of admissible parameters proposed in previous literature and explicitly state the regularity assumptions required for its validity. Moreover, we show that the series does not tell the whole story, hence we propose an additional condition for Crossbred to work. Additionally, we define and characterize a notion of regularity for systems over a small field, which is one of the main building blocks in the series of admissible parameters.

In our highly digitalized world, an adversary is not constrained to purely digital attacks but can monitor or influence the physical execution environment of a target computing device. Such side-channel or fault-injection analysis poses a significant threat to otherwise secure cryptographic implementations. Hence, it is important to consider additional adversarial capabilities when analyzing the security of cryptographic implementations besides the default black-box model. For side-channel analysis, this is done by providing the adversary with knowledge of some internal values, while for fault-injection analysis the capabilities of the adversaries include manipulation of some internal values.

In this work, we extend probabilistic security models for physical attacks, by introducing a general random probing model and a general random fault model to capture arbitrary leakage and fault distributions, as well as the combination of these models. Our aim is to enable a more accurate modeling of low-level physical effects. We then analyze important properties, such as the impact of adversarial knowledge on faults and compositions, and provide tool-based formal verification methods that allow the security assessment of design components. These methods are introduced as extension of previous tools VERICA and IronMask which are implemented, evaluated and compared.

Asymmetric Password-Authenticated Key Exchange (aPAKE) protocols, particularly Strong aPAKE (saPAKE) have enjoyed significant attention, both from academia and industry, with the well-known OPAQUE protocol currently undergoing standardization. In (s)aPAKE, a client and a server collaboratively establish a high-entropy key, relying on a previously exchanged password for authentication. A main feature is its resilience against offline and precomputation (for saPAKE) attacks. OPAQUE, as well as most other aPAKE protocols, have been designed and analyzed in a single-user setting, i.e., modelling that only a single user interacts with the server. By the composition framework of UC, security for the actual multi-user setting is then conjectured. As any real-world (s)aPAKE instantiation will need to cater multiple users, this introduces a dangerous gap in which developers are tasked to extend the single-user protocol securely and in a UC-compliant manner. In this work, we extend the (s)aPAKE definition to directly model the multi-user setting, and explicitly capture the impact that a server compromise has across user accounts. We show that the currently standardized multi-user version of OPAQUE might not provide the expected security, as it is insecure against offline attacks as soon as the file for one user in the system is compromised. This is due to using shared state among different users, which violates the UC composition framework. However, we show that another change introduced in the standardization draft which also involves a shared state does not compromise security. When extending the aPAKE security in the multi-client setting, we notice that the widely used security definition captures significantly weaker security guarantees than what is offered by many protocols. Essentially, the aPAKE definition assumes that the server stores unsalted password-hashes, whereas several protocols explicitly use a salt to protect against precomputation attacks. We therefore propose a definitional framework that captures different salting approaches -- thus showing that the security gap between aPAKE and saPAKE can be smaller than expected.

In this paper, we present efficient protected software implementations of the authenticated cipher Ascon, the recently announced winner of the NIST standardization process for lightweight cryptography. Our implementations target theoretical and practical security against second-order power analysis attacks. First, we propose an efficient second-order extension of a previously presented first-order masking of the Keccak S-box that does not require online randomness. The extension itself is inspired by a previously presented second-order masking of an AND-XOR construction. We then discuss implementation tricks that further improve performance and reduce the chance of unintended combination of shares during the execution of masked software on microprocessors. This allows us to retain the theoretic protection orders of masking in practice with low performance overhead, which we also confirm via TVLA on ARM microprocessors. The formal correctness of our designs is additionally verified using Coco on the netlist of a RISC-V IBEX core. We benchmark our masked software designs on 32-bit ARM and RISC-V microprocessor platforms. On both platforms, we can perform Ascon-128 authenticated encryption with a throughput of about 300 or 550 cycles/byte when operating on 2 or 3 shares. When utilizing a leveled implementation technique, the throughput of our masked implementations generally increases to about 90 cycles/byte. We publish our masked software implementations together with a generic software framework for evaluating performance and side-channel resistance of various masked cryptographic implementations.

Creating an adversary resilient Learned Bloom filter with provable guarantees is an open problem. We define a strong adversarial model for the Learned Bloom Filter. We also construct two adversary resilient variants of the Learned Bloom Filter called the Uptown Bodega Filter and the Downtown Bodega Filter. Our adversarial model extends an existing adversarial model designed for the classical (i.e not ``learned'') Bloom Filter by Naor and Yogev and considers computationally bounded adversaries that run in probabilistic polynomial time (PPT). We show that if pseudo-random permutations exist, then a secure Learned Bloom Filter may be constructed with $\lambda$ extra bits of memory and at most one extra pseudo-random permutation in the critical path. We further show that, if pseudo-random permutations exist, then a high utility Learned Bloom Filter may be constructed with $2\lambda$ extra bits of memory and at most one extra pseudo-random permutation in the critical path. Finally, we construct a hybrid adversarial model for the case where a fraction of the workload is chosen by an adversary. We show realistic scenarios where using the Downtown Bodega Filter gives better performance guarantees compared to alternative approaches in this model.

In many real-world scenarios, there are cases where a client wishes to check if a data element they hold is included in a set segmented across a large number of data holders. To protect user privacy, the client’s query and the data holders’ sets should remain encrypted throughout the whole process. Prior work on Private Set Intersection (PSI), Multi-Party PSI (MPSI), Private Membership Test (PMT), and Oblivious RAM (ORAM) falls short in this scenario in many ways. They either require data holders to possess the sets in plain- text, incur prohibitively high latency for aggregating results from a large number of data holders, leak the information about the party holding the intersection element, or induce a high false positive. This paper introduces the primitive of a Private Segmented Mem- bership Test (PSMT). We give a basic construction of a protocol to solve PSMT using a threshold variant of approximate-arithmetic homomorphic encryption and show how to overcome existing challenges to construct a PSMT protocol without leaking information about the party holding the intersection element or false positives for a large number of data holders ensuring IND-CPA^? security. Our novel approach is superior to existing state-of-the-art approaches in scalability with regard to the number of supported data holders. This is enabled by a novel summation-based homo- morphic membership check rather than a product-based one, as well as various novel ideas addressing technical challenges. Our PSMT protocol supports many more parties (up to 4096 in experiments) compared to prior related work that supports only around 100 parties efficiently. Our experimental evaluation shows that our method’s aggregation of results from data holders can run in 92.5s for 1024 data holders and a set size of 2^25, and our method’s over- head increases very slowly with the increasing number of senders. We also compare our PSMT protocol to other state-of-the-art PSI and MPSI protocols and discuss our improvements in usability with a better privacy model and a larger number of parties

Embedded curves are elliptic curves defined over a prime field whose order (characteristic) is the prime subgroup order (the scalar field) of a pairing-friendly curve. Embedded curves have a large prime-order subgroup of cryptographic size but are not pairing-friendly themselves. Sanso and El Housni published families of embedded curves for BLS pairing-friendly curves. Their families are parameterized by polynomials, like families of pairing-friendly curves are. However their work did not found embedded families for KSS pairing-friendly curves. In this note we show how the problem of finding families of embedded curves is related to the problem of finding optimal formulas for $\mathbb{G}_1$ subgroup membership testing on the pairing-friendly curve side. Then we apply Smith's technique and Dai, Lin, Zhao, and Zhou criteria to obtain the formulas of embedded curves with KSS, and outline a generic algorithm for solving this problem in all cases. We provide two families of embedded curves for KSS18 and give examples of cryptographic size. We also suggest alternative embedded curves for BLS that have a seed of much lower Hamming weight than Sanso et al.~and much higher 2-valuation for fast FFT. In particular we highlight BLS12 curves which have a prime-order embedded curve that form a plain cycle (no pairing), and a second (plain) embedded curve in Montgomery form. A Brezing-Weng outer curve to have a pairing-friendly 2-chain is also possible like in the BLS12-377-BW6-761 construction. All curves have $j$-invariant 0 and an endomorphism for a faster arithmetic on the curve side.

Unclonable cryptography is concerned with leveraging the no-cloning principle to build cryptographic primitives that are otherwise impossible to achieve classically. Understanding the feasibility of unclonable encryption, one of the key unclonable primitives, satisfying indistinguishability security in the plain model has been a major open question in the area. So far, the existing constructions of unclonable encryption are either in the quantum random oracle model or are based on new conjectures.

We present a new approach to unclonable encryption via a reduction to a novel question about nonlocal quantum state discrimination: how well can non-communicating -- but entangled -- players distinguish between different distributions over quantum states? We call this task simultaneous state indistinguishability. Our main technical result is showing that the players cannot distinguish between each player receiving independently-chosen Haar random states versus all players receiving the same Haar random state.

We leverage this result to present the first construction of unclonable encryption satisfying indistinguishability security, with quantum decryption keys, in the plain model. We also show other implications to single-decryptor encryption and leakage-resilient secret sharing.

Multi-scalar multiplication (MSM) is one of the core components of many zero-knowledge proof systems, and a primary performance bottleneck for proof generation in these schemes. One major strategy to accelerate MSM is utilizing precomputation. Several algorithms (e.g., Pippenger and BGMW) and their variants have been proposed in this direction. In this paper, we revisit the recent precomputation-based MSM calculation method proposed by Luo, Fu and Gong at CHES 2023 and generalize their approach. In particular, we presented a general construction of optimal buckets. This improvement leads to significant performance improvements, which are verified by both theoretical analysis and experiments.

Attribute-based encryption (ABE) is a generalization of public-key encryption that enables fine-grained access control to encrypted data. In (ciphertext-policy) ABE, a central trusted authority issues decryption keys for attributes $x$ to users. In turn, ciphertexts are associated with a decryption policy $\mathcal{P}$. Decryption succeeds and recovers the encrypted message whenever $\mathcal{P}(x) = 1$. Recently, Hohenberger, Lu, Waters, and Wu (Eurocrypt 2023) introduced the notion of registered ABE, which is an ABE scheme without a trusted central authority. Instead, users generate their own public/secret keys (just like in public-key encryption) and then register their keys (and attributes) with a key curator. The key curator is a transparent and untrusted entity.

Currently, the best pairing-based registered ABE schemes support monotone Boolean formulas and an a priori bounded number of users $L$. A major limitation of existing schemes is that they require a (structured) common reference string (CRS) of size $L^2 \cdot |\mathcal{U}|$ where $|\mathcal{U}|$ is the size of the attribute universe. In other words, the size of the CRS scales quadratically with the number of users and multiplicatively with the size of the attribute universe. The large CRS makes these schemes expensive in practice and limited to a small number of users and a small universe of attributes.

In this work, we give two ways to reduce the CRS size in pairing-based registered ABE schemes. First, we introduce a combinatoric technique based on progression-free sets that enables registered ABE for the same class of policies but with a CRS whose size is sub-quadratic in the number of users. Asymptotically, we obtain a scheme where the CRS size is nearly linear in the number of users $L$ (i.e., $L^{1 + o(1)}$). If we take a more concrete-efficiency-oriented focus, we can instantiate our framework to obtain a construction with a CRS of size $L^{\log_2 3} \approx L^{1.6}$. For instance, in a scheme for 100,000 users, our approach reduces the CRS by a factor of over $115\times$ compared to previous approaches (and without incurring any overhead in encryption/decryption time). Our second approach for reducing the CRS size is to rely on a partitioning-based argument when arguing security of the registered ABE scheme. Previous approaches took a dual-system approach. Using a partitioning-based argument yields a registered ABE scheme where the size of the CRS is independent of the size of the attribute universe. The cost is the resulting scheme satisfies a weaker notion of static security. Our techniques for reducing the CRS size can be combined, and taken together, we obtain a pairing-based registered ABE scheme that supports monotone Boolean formulas with a CRS size of $L^{1 + o(1)}$. Notably, this is the first pairing-based registered ABE scheme that does not require imposing a bound on the size of the attribute universe during setup time.

As an additional application, we also show how to apply our techniques based on progression-free sets to the batch argument (BARG) for $\mathsf{NP}$ scheme of Waters and Wu (Crypto 2022) to obtain a scheme with a nearly-linear CRS without needing to rely on non-black-box bootstrapping techniques.