International Association
for Cryptologic Research

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)

AIT is Austria's largest non-universitary research institute. Its Cyber Security team focuses on various aspects of security, including anomaly detection, cyber ranges, penetration testing, and cryptography. The cryptography group is conducting research in various directions, including secure communication, privacy-enhancing technologies, and long-term and post-quantum security. The group seeks to grow and thus has a vacancy for a researcher in related areas.

Requirements:

• PhD degree in Computer Science, Cyber Security, or a related field, with a specialization on cryptology
• Profound knowledge in (public key) cryptography, including, e.g., federated computation, long-term and post-quantum secure communication, privacy-enhancing technologies, real-world crypto, zero-knowledge proofs and zkSNARKs
• Strong track record with publications at competitive academic conferences or journals (e.g., Crypto, Eurocrypt, Asiacrypt, TCC, PKC, CCS, S&P, USENIX, ESORICS, ...)
• Experience in the acquisition and execution of national and transnational research projects (e.g., H2020) is a plus
• Good knowledge of a programming language (e.g., C/C++, Rust, Java, Python) and software development is a plus
• Very good written and oral English skills; knowledge of German is not a requirement but willingness to learn German is expected

The salary starts from ~61k€/year, depending on experience. The review process will begin immediately and will continue open until the position has been filled.

Please submit your application including CV, cover letter, full list of publications, and contact details of at least 2 references via email to: stephan.krenn[at]ait.ac.at

Closing date for applications:

Contact: Stephan Krenn; stephan.krenn[at]ait.ac.at

This is an exciting opportunity to join the University of Birmingham’s Centre for Cyber Security and Privacy on the EPSRC funded project "IOTEE: Securing and analysing trusted execution beyond the CPU", led by Prof Oswald and Prof Ryan.

Trusted Execution Environments (TEEs) allow users to run their software in a secure enclave while assuring the integrity and confidentiality of data and applications. However, cloud computing these days relies heavily on peripherals (connected through PCIe) such as GPUs and FPGAs. In this project, together with researchers at the University of Southampton, we will thoroughly evaluate the security guarantees of the new TEE support in the PCIe standard. This could involve the use of formal modelling, as well as researching various software and hardware attacks and countermeasures against them.

We are looking for a person with a PhD in cyber security/computer science/electrical engineering. The candidate must have experience areas such as embedded security, binary analysis, physical attacks such as side-channel analysis and fault injection, and/or formal modelling. This needs to be evidenced through publications in highly ranked conferences/journals in the field. We also welcome experience in writing system level or low-level code in programming languages such as C, C++, or Rust.

The successful candidate will be employed on a full-time, fixed-term contract up to August 2026. Full-time starting salary is normally in the range £33,348 to £43,155. (Some) remote work is possible, depending on the circumstances. The University provides a range of employee benefits, as well as opportunities for career development and training. The project includes substantial funding for conference travel and equipment.

The post-doc will be working in the Centre for Cyber Security and Privacy, which currently has 14 permanent academics as well as 21 postdocs/PhD students.

The application deadline is 12 Oct 2023. Applications have to be made online at: https://edzz.fa.em3.oraclecloud.com/hcmUI/CandidateExperience/en/sites/CX_6001/job/2681/

Closing date for applications:

Contact: Informal enquiries can be made to David Oswald d.f.oswald@bham.ac.uk.

More information: https://edzz.fa.em3.oraclecloud.com/hcmUI/CandidateExperience/en/sites/CX_6001/job/2681/

With the Institute for Photonic Quantum Systems (PhoQS), Paderborn University wants to establish an international research centre in the field of photonic quantum technologies. The aim is to develop new technologies for photon-based quantum applications as well as new theoretical and experimental concepts and research approaches. Ultimately, the focus is on understanding and controlling photonic quantum simulators and quantum computers.

Postdoc (f/m/d) (salary is according to E13 TV-L)

A position with 100 % of the regular working hours is available as of the next possible date. The employment is initially limited to three years and is based on the legal regulations of the Wissen-schaftszeitvertragsgesetzes (WissZeitVG).

Your duties and responsibilities:
• Establishment and expansion of an infrastructure for the integration of quantum computing in high-performance computing.
• Interface of PhoQS to the Paderborn Center for Parallel Computing (PC2) of the Paderborn University
• Supporting users, especially in the natural sciences, in the development and implementation of quantum algorithms
• Optimisation of quantum software platforms for photonic and gate-based quantum computing such as Strawberry Fields, Parceval or Qiskit in collaboration with HPC experts of the PC2
• Organisation and delivery of tutorials and workshops on the use of quantum software plat-forms (basic to advanced)
• Leading a team for the technical integration of quantum computing and high-performance computing

Hiring requirements:
• Completed PhD in computer science, mathematics or physics or comparable qualification
• Solid understanding of many-body quantum mechanics
• Practical experience in high-performance computing and/or in the use of quantum software platforms
• High motivation and willingness for interdisciplinary cooperation between computer science and physics
• Good knowledge of German and English, both written and spoken
• Friendliness, flexibility, ability to work in a team, initiative and willingness to work independently

Closing date for applications:

Contact: Please send your application including a CV (preferably in a single pdf file) using the Ref. No. 6105 by 30th September, 2023 to: bloemer@upb.de

More information: https://cs.uni-paderborn.de/en/cuk

Non-interactive delegation schemes enable producing succinct proofs (that can be efficiently verified) that a machine $M$ transitions from $c_1$ to $c_2$ in a certain number of deterministic steps. We here consider the problem of efficiently \emph{merging} such proofs: given a proof $\Pi_1$ that $M$ transitions from $c_1$ to $c_2$, and a proof $\Pi_2$ that $M$ transitions from $c_2$ to $c_3$, can these proofs be efficiently merged into a single short proof (of roughly the same size as the original proofs) that $M$ transitions from $c_1$ to $c_3$? To date, the only known constructions of such a mergeable delegation scheme rely on strong non-falsifiable ``knowledge extraction" assumptions. In this work, we present a provably secure construction based on the standard LWE assumption.

As an application of mergeable delegation, we obtain a construction of incrementally verifiable computation (IVC) (with polylogarithmic length proofs) for any (unbounded) polynomial number of steps based on LWE; as far as we know, this is the first such construction based on any falsifiable (as opposed to knowledge-extraction) assumption. The central building block that we rely on, and construct based on LWE, is a rate-1 batch argument (BARG): this is a non-interactive argument for NP that enables proving $k$ NP statements $x_1,..., x_k$ with communication/verifier complexity $m+o(m)$, where $m$ is the length of one witness. Rate-1 BARGs are particularly useful as they can be recursively composed a super-constant number of times.

In this paper we address the problem of recovery from failures without re-running entire elections when elections fail to verify. We consider the setting of $\textit{dual voting}$ protocols, where the cryptographic guarantees of end-to-end verifiable voting (E2E-V) are combined with the simplicity of audit using voter-verified paper records (VVPR). We first consider the design requirements of such a system and then suggest a protocol called $\textit{OpenVoting}$, which identifies a verifiable subset of error-free votes consistent with the VVPRs, and the polling booths corresponding to the votes that fail to verify with possible reasons for the failures.

Protocols with \emph{publicly verifiable covert (PVC) security} offer high efficiency and an appealing feature: a covert party may deviate from the protocol, but with a probability (\eg $90\%$, referred to as the \emph{deterrence factor}), the honest party can identify this deviation and expose it using a publicly verifiable certificate. These protocols are particularly suitable for practical applications involving reputation-conscious parties.

However, in the cases where misbehavior goes undetected (\eg with a probability of $10\%$), \emph{no security guarantee is provided for the honest party}, potentially resulting in a complete loss of input privacy and output correctness.

In this paper, we tackle this critical problem by presenting a highly effective solution. We introduce and formally define an enhanced notion called \emph{robust PVC security}, such that even if the misbehavior remains undetected, the malicious party can only gain an additional $1$-bit of information about the honest party's input while maintaining the correctness of the output. We propose a novel approach leveraging \emph{dual execution} and \emph{time-lock puzzles} to design a robust PVC-secure two-party protocol with \emph{low overhead} (depending on the deterrence factor). For instance, with a deterrence factor of $90\%$, our robust PVC-secure protocol incurs \emph{only additional ${\sim}10\%$ overhead} compared to the state-of-the-art PVC-secure protocol.

Given the stronger security guarantees with low overhead, our protocol is highly suitable for practical applications of secure two-party computation.

In CRYPTO 2019, Gohr showed that well-trained neural networks could perform cryptanalytic distinguishing tasks superior to differential distribution table (DDT)-based distinguishers. This suggests that the differential-neural distinguisher (ND) may use additional information besides pure ciphertext differences. However, the explicit knowledge beyond differential distribution is still unclear. In this work, we provide explicit rules that can be used alongside DDTs to enhance the effectiveness of distinguishers compared to pure DDT-based distinguishers. These rules are based on strong correlations between bit values in right pairs of XOR-differential propagation through addition modulo $2^n$. Interestingly, they can be closely linked to the earlier study of the multi-bit constraints and the recent study of the fixed-key differential probability. In contrast, combining these rules does not improve the NDs' performance. This suggests that these rules or their equivalent form have already been exploited by NDs, highlighting the power of neural networks in cryptanalysis. In addition, we find that to enhance the differential-neural distinguisher's accuracy and the number of rounds, regulating the differential propagation is imperative. Introducing differences into the keys is typically believed to help eliminate differences in encryption states, resulting in stronger differential propagations. However, differential-neural attacks differ from traditional ones as they don't specify output differences or follow a single differential trail. This questions the usefulness of introducing differences in a key in differential-neural attacks and the resistance of Speck against such attacks in the related-key setting. This work shows that the power of differential-neural cryptanalysis in the related-key setting can exceed that in the single-key setting by successfully conducting a 14-round key recovery attack on Speck32/64.

Data formats used for cryptographic inputs have historically been the source of many attacks on cryptographic protocols, but their security guarantees remain poorly studied. One reason is that, due to their low-level nature, formats often fall outside of the security model. Another reason is that studying all of the uses of all of the formats within one protocol is too difficult to do by hand, and requires a comprehensive, automated framework.

We propose a new framework, “Comparse”, that specifically tackles the security analysis of data formats in cryptographic protocols. Comparse forces the protocol analyst to systematically think about data formats, formalize them precisely, and show that they enjoy strong enough properties to guarantee the security of the protocol.

Our methodology is developed in three steps. First, we introduce a high-level cryptographic API that lifts the traditional game-based cryptographic assumptions over bitstrings to work over high-level messages, using formats. This allows us to derive the conditions that secure formats must obey in order for their usage to be secure. Second, equipped with these security criteria, we implement a framework for specifying and verifying secure formats in the F* proof assistant. Our approach is based on format combinators, which enable compositional and modular proofs. In many cases, we relieve the user of having to write those combinators by hand, using compile-time term synthesis via Meta-F*. Finally, we show that our F* implementation can replace the symbolic notion of message formats previously implemented in the DY* protocol analysis framework. Our newer, bit-level precise accounting of formats closes the modeling gap, and allows DY* to reason about concrete messages and identify protocol flaws that it was previously oblivious to.

We evaluate Comparse over several classic and real-world protocols. Our largest case studies use Comparse to formalize and provide security proofs for the formats used in TLS 1.3, as well as upcoming protocols like MLS and Compact TLS 1.3 (cTLS), providing confidence and feedback in the design of these protocols.

Registration-Based Encryption (RBE) [Garg et al. TCC'18] is a public-key encryption mechanism in which users generate their own public and secret keys, and register their public keys with a central authority called the key curator. Similarly to Identity-Based Encryption (IBE), in RBE users can encrypt by only knowing the public parameters and the public identity of the recipient. Unlike IBE, though, RBE does not suffer the key escrow problem — one of the main obstacles of IBE's adoption in practice — since the key curator holds no secret.

In this work, we put forward a new methodology to construct RBE schemes that support large users identities (i.e., arbitrary strings). Our main result is the first efficient pairing-based RBE for large identities. Prior to our work, the most efficient RBE is that of [Glaeser et al. ePrint'22] which only supports small identities. The only known RBE schemes with large identities are realized either through expensive non-black-box techniques (ciphertexts of 3.6 TB for 1000 users), or via a specialized lattice-based construction [Döttling et al. Eurocrypt'23] (ciphertexts of 2.4 GB). By unlocking the use of pairings for RBE with large identity space, we enable a further improvement of three orders of magnitude, as our ciphertexts for a system with 1000 users are $1.7$ MB.

The core technique of our approach is a novel use of cuckoo hashing in cryptography that can be of independent interest. We give two main applications. The first one is the aforementioned RBE methodology, where we use cuckoo hashing to compile an RBE with small identities into one for large identities. The second one is a way to convert any vector commitment scheme into a key-value map commitment. For instance, this leads to the first algebraic pairing-based key-value map commitments.

Sigma protocols are one of the most common and efficient zero-knowledge proofs (ZKPs). Over the decades, a large number of efficient Sigma protocols are proposed, yet few works pay attention to the common design principal. In this work, we propose a generic framework of Sigma protocols for algebraic statements from verifiable secret sharing (VSS) schemes. Our framework provides a general and unified approach to understanding Sigma protocols for proving knowledge of openings of algebraic commitments. It not only neatly explains the classic protocols such as Schnorr, Guillou–Quisquater and Okamoto protocols, but also leads to new Sigma protocols that were not previously known.

Furthermore, we show an application of our framework in designing ZKPs for composite statements, which contain both algebraic and non-algebraic statements. We give a generic construction of ZKPs for composite statements by combining Sigma protocols from VSS and ZKPs following MPC-in-the-head paradigm seamlessly via a technique of witness sharing reusing. Our construction has advantages of requiring no trusted setup, being public-coin and having a fast prover runtime. By instantiating our construction using Ligero++ (Bhadauria et al., CCS 2020), we obtain a new ZK protocol for composite statements, which achieves a new balance between running time and the proof size, thus resolving the open problem left by Backes et al. (PKC 2019). Concretely, the proof size is polylogarithmic to the circuit size and the number of public-key operations that both the prover and the verifier require is independent to the circuit size.

In this paper, we initiate the study of the Rank Decoding (RD) problem and LRPC codes with blockwise structures in rank-based cryptosystems. First, we introduce the blockwise errors ($\ell$-errors) where each error consists of $\ell$ blocks of coordinates with disjoint supports, and define the blockwise RD ($\ell$-RD) problem as a natural generalization of the RD problem whose solutions are $\ell$-errors (note that the standard RD problem is actually a special $\ell$-RD problem with $\ell=1$). We adapt the typical attacks on the RD problem to the $\ell$-RD problem, and find that the blockwise structures do not ease the problem too much: the $\ell$-RD problem is still exponentially hard for appropriate choices of $\ell>1$. Second, we introduce blockwise LRPC ($\ell$-LRPC) codes as generalizations of the standard LPRC codes whose parity-check matrices can be divided into $\ell$ sub-matrices with disjoint supports, i.e., the intersection of two subspaces generated by the entries of any two sub-matrices is a null space, and investigate the decoding algorithms for $\ell$-errors. We find that the gain of using $\ell$-errors in decoding capacity outweighs the complexity loss in solving the $\ell$-RD problem, which makes it possible to design more efficient rank-based cryptosystems with flexible choices of parameters.

As an application, we show that the two rank-based cryptosystems submitted to the NIST PQC competition, namely, RQC and ROLLO, can be greatly improved by using the ideal variants of the $\ell$-RD problem and $\ell$-LRPC codes. Concretely, for 128-bit security, our RQC has total public key and ciphertext sizes of 2.5 KB, which is not only about 50% more compact than the original RQC, but also smaller than the NIST Round 4 code-based submissions HQC, BIKE, and Classic McEliece.

The proof of stake (PoS) mechanism, which allows stakeholders to issue a block with a probability proportional to their wealth instead of computational power, is believed to be an energy-efficient alternative to the proof of work (PoW). The privacy concern of PoS, however, is more subtle than that of PoW. Recent research has shown that current anonymous PoS (APoS) protocols do not suffice to protect the stakeholder's identity and stake, and the loss of privacy is theoretically inherent for any (deterministic) PoS protocol that provides liveness guarantees. In this paper, we consider the concrete stake privacy of PoS when considering the limitations of attacks in practice. To quantify the concrete stake privacy of PoS, we introduce the notion of $(T, \delta, \epsilon)$-privacy. Our analysis of $(T, \delta, \epsilon)$-privacy on Cardano shows to what extent the stake privacy can be broken in practice, which also implies possible parameters setting of rational $(T, \delta, \epsilon)$-privacy for PoS in the real world. The data analysis of Cardano demonstrates that the $(T, \delta, \epsilon)$-privacy of current APoS is not satisfactory, mainly due to the deterministic leader election predicate in current PoS constructions. Inspired by the differential privacy technique, we propose an efficient non-deterministic leader election predicate, which can be used as a plugin to APoS protocols to protect stakes against frequency analysis. Based on our leader election predicate, we construct anonymous PoS with noise (APoS-N), which can offer better $(T, \delta, \epsilon)$-privacy than state-of-the-art works. Furthermore, we propose a method of proving the basic security properties of PoS in the noise setting, which can minimize the impact of the noise on the security threshold. This method can also be applied to the setting of PoS with variable stakes, which is of independent interest.