International Association for Cryptologic Research

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for Cryptologic Research

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07 July 2025

Fuyuki Kitagawa, Takashi Yamakawa
ePrint Report ePrint Report
Single decryptor encryption (SDE) is public key encryption (PKE) where the decryption key is an unclonable quantum state. Coladangelo, Liu, Liu, and Zhandry (CRYPTO 2021) realized the first SDE assuming subexponentially secure indistinguishability obfuscation (iO) and one-way functions (OWFs), along with the polynomial hardness of the learning with errors (LWE) assumption. Since then, SDE has played a pivotal role in recent advances in quantum cryptography. However, despite its central importance in unclonable cryptography, many fundamental questions about SDE remain unanswered. For example, a line of works has proposed various security notions for SDE, but their relationships have hardly been discussed. Moreover, while many subsequent works have adopted the construction methodology of Coladangelo et al., none have explored its improvement, leaving the possibility of a more efficient approach to SDE.

In this work, we address these fundamental questions concerning SDE. Our contributions are threefold.

New security notion: We introduce a strengthened indistinguishability-based security notion for SDE, which we call CPA+ anti-piracy security. We show that CPA+ security unifies the existing security notions for SDE, as detailed in the third item.

New construction: We present an SDE scheme that satisfies CPA+ anti-piracy security, based solely on polynomially secure iO and OWFs. In addition to relying on weaker and more general assumptions, our SDE scheme offers a significant advantage over the scheme of Coladangelo et al., as both the construction and its security proof are much simpler.

Relationships among security notions: We demonstrate that CPA+ anti-piracy security implies all existing security notions for SDE, with the sole exception of identical challenge ciphertext security proposed by Georgiou and Zhandry (EPRINT 2020). Although we do not establish a direct implication from CPA+ anti-piracy security to identical challenge ciphertext security, we provide a generic transformation from an SDE scheme satisfying the former to one achieving the latter in the quantum random oracle model. Additionally, we establish various relationships among different security notions for SDE. By combining these results with our SDE construction, we derive several new feasibility results.
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Anisha Mukherjee, Sujoy Sinha Roy
ePrint Report ePrint Report
Homomorphic encryption (HE) schemes have gained significant popularity in modern privacy-preserving applications across various domains. While research on HE constructions based on learning with errors (LWE) and ring-LWE has received major attention from both cryptographers and software-hardware designers alike, their module-LWE-based counterpart has remained comparatively under-explored in the literature. A recent work provides a module-LWE-based instantiation (MLWE-HE) of the Cheon-Kim-Kim-Song (CKKS) scheme and showcases several of its advantages such as parameter flexibility and improved parallelism. However, a primary limitation of this construction is the quadratic growth in the size of the relinearization keys. Our contribution is two-pronged: first, we present a new relinearization key-generation technique that addresses the issue of quadratic key size expansion by reducing it to linear growth. Second, we extend the application of MLWE-HE in a multi-group homomorphic encryption (MGHE) framework, thereby generalizing the favorable properties of the single-keyed HE to a multi-keyed setting as well as investigating additional flexibility attributes of the MGHE framework.
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Alexandre Adomnicăi, Wonseok Choi, Yeongmin Lee, Kazuhiko Minematsu, Yusuke Naito
ePrint Report ePrint Report
Authenticated encryption (AE) is a fundamental tool in today's secure communication. Numerous designs have been proposed, including well-known standards such as GCM. While their performance for long inputs is excellent, that for short inputs is often problematic due to high overhead in computation, showing a gap between the real need for IoT-like protocols where packets are often very short. Existing dedicated short-input AEs are very scarce, the classical Encode-then-encipher (Bellare and Rogaway, Asiacrypt 2000) and Manx (Adomnic\u{a}i et al., CT-RSA 2023), using up to two block cipher calls. They have superior performance for (very) short inputs, however, security is up to $n/2$ bits, where $n$ is the block size of the underlying block cipher. This paper proposes a new family of short-input AEs, dubbed Cymric, which ensures beyond-birthday-bound (BBB) security. It supports a wider range of input space than EtE and Manx with the help of one additional block cipher call (thus three calls). In terms of the number of block cipher calls, Cymric is the known minimum construction of BBB-secure AEs, and we also prove this is indeed minimal by presenting an impossibility result on BBB-secure AE with two calls. Finally, we show a comprehensive benchmark on microcontrollers to show performance advantage over existing schemes.
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Debadrita Talapatra, Nimish Mishra, Debdeep Mukhopadhyay
ePrint Report ePrint Report
Prior research on ensuring trust in delegated computation through lattice-based zero-knowledge proofs mostly rely on Learning-With-Errors (LWE) assumption. In this work, we propose a zero-knowledge proof of knowledge using the Ring Learning with Rounding (RLWR) assumption for an interesting and useful class of statements: linear relations on polynomials. We begin by proposing, to the best of our knowledge, the first efficient commitment scheme in literature based on the hardness of RLWR assumption. We establish two properties on RLWR that aid in the construction of our commitments: (i) closure under addition with double rounding, and (ii) closure under multiplication with a short polynomial. Building upon our RLWR commitment scheme, we consequently design a RLWR based $\Sigma_2$ protocol for proving knowledge of a single committed message under linear relations with public polynomials.

As an use-case of our proposed $\Sigma_2$ protocol, we showcase a construction of a quantum-safe Searchable Symmetric Encryption (SSE) scheme by plugging a prior LWR based SSE scheme from (EuroS&P 2023) with our $\Sigma_2$ protocol. Concretely, using our $\Sigma_2$ protocol for linear relations, we prove the correctness of an encrypted search result in a zero-knowledge manner. We implement our verifiable SSE framework and show that the overhead of an extra verification round is negligible ($0.0023$ seconds) and retains the asymptotic query execution time complexity of the original SSE scheme.

Our work establishes results on zero-knowledge proof systems that can be of independent interest. By shifting the setting from RLWE to RLWR, we gain significant (i) efficiency improvements in terms of communication complexity by $O(M)$ (since some prior works on RLWE require rejection sampling by a factor of $M$), as well as (ii) very short proof size ($8.4$ KB) and tighter parameters (since RLWR does not explicitly manipulate error polynomials like RLWE).
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Debadrita Talapatra, Sikhar Patranabis, Debdeep Mukhopadhyay
ePrint Report ePrint Report
Searchable symmetric encryption (SSE) enables query execution directly over sym- metrically encrypted databases. To support realistic query executions over encrypted document collections, one needs SSE schemes capable of supporting both conjunctive and disjunctive keyword queries. Unfortunately, existing solutions are either practi- cally inefficient (incur large storage overheads and/or high query processing latency) or are quantum-unsafe. In this paper, we present the first practically efficient SSE scheme with fast con- junctive and disjunctive keyword searches, compact storage, and security based on the (plausible) quantum-hardness of well-studied lattice-based assumptions. We present NTRU-OQXT – a highly compact NTRU lattice-based conjunctive SSE scheme that outperforms all existing conjunctive SSE schemes in terms of search latency. We then present an extension of NTRU-OQXT that additionally supports disjunctive queries, we call it NTRU-TWINSSE. Technically, both schemes rely on a novel oblivious search protocol based on highly optimized Fast-Fourier trapdoor sampling algorithms over NTRU lattices. While such techniques have been used to design other cryptographic primitives (such as digital signatures), they have not been applied before in the context of SSE. We present prototype implementations of both schemes, and experimentally val- idate their practical performance over a large real-world dataset. Our experiments demonstrate that NTRU-OQXT achieves 2× faster conjunctive keyword searches as compared to all other conjunctive SSE schemes (including the best quantum-unsafe conjunctive SSE schemes), and substantially outperforms many of these schemes in terms of storage requirements. These efficiency benefits also translate to NTRU- TWINSSE, which is practically competitive with the best quantum-unsafe SSE schemes capable of supporting both conjunctive and disjunctive queries.
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Christodoulos Pappas, Dimitrios Papadopoulos
ePrint Report ePrint Report
Zero-knowledge succinct non-interactive arguments (zkSNARKs) are notorious for their large prover space requirements, which almost prohibits their use for really large instances. Space-efficient zkSNARKs aim to address this by limiting the prover space usage, without critical sacrifices to its runtime. In this work, we introduce Hobbit, the only existing space-efficient zkSNARK that achieves optimal prover time $O(|C|)$ for an arithmetic circuit $C$. At the same time, Hobbit is the first transparent and plausibly post-quantum secure construction of its kind. Moreover, our experimental evaluation shows that Hobbit outperforms all prior general-purpose space-efficient zkSNARKs in the literature across four different applications (arbitrary arithmetic circuits, inference of pruned Multi-Layer Perceptron, batch AES128 evaluation, and select-and-aggregate SQL query) by $\times$8-$\times$$56$ in terms or prover time while requiring up to $\times$23 less total space.

At a technical level, we introduce two new building blocks that may be of independent interest: (i) the first sumcheck protocol for products of polynomials with optimal prover time in the streaming setting, and (ii) a novel multi-linear plausibly post-quantum polynomial commitment that outperforms all prior works in prover time (and can be tuned to work in a space-efficient manner). We build Hobbit by combining the above with a modified version of HyperPlonk, providing an explicit routine to stream access to the circuit evaluation.
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Yi-Fan Tseng, Yi-Jiin Lu, Tien-Lin Tsai, Zi-Yuan Liu
ePrint Report ePrint Report
We introduce a novel Public Key Encryption with Equality Test supporting Flexible Authorization scheme offering User-Level, Ciphertext-Level, and User-Specific-Ciphertext-Level authorizations. Notably, our construction achieves security under the Decisional Diffie-Hellman assumption with a tight reduction, whereas the existing works are either not tightly secure or rely heavily on the random oracles. By relying solely on the standard DDH assumption, our scheme offers practical implementation without specialized cryptographic structures.
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Hayder Tirmazi
ePrint Report ePrint Report
Speedrunning is a competition that emerged from communities of early video games such as Doom (1993). Speedrunners try to finish a game in minimal time. Provably verifying the authenticity of submitted speedruns is an open problem. Traditionally, best-effort speedrun verification is conducted by on-site human observers, forensic audio analysis, or a rigorous mathematical analysis of the game mechanics1. Such methods are tedious, fallible, and, perhaps worst of all, not cryptographic. Motivated by naivety and the Dunning-Kruger effect, we attempt to build a system that cryptographically proves the authenticity of speedruns. This paper describes our attempted solutions and ways to circumvent them. Through a narration of our failures, we attempt to demonstrate the difficulty of authenticating live and interactive human input in untrusted environments, as well as the limits of signature schemes, game integrity, and provable play.
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03 July 2025

Universitat Oberta de Catalunya (UOC)
Job Posting Job Posting
We’re looking for a motivated individual to join the KISON research group as a temporary research assistant. The role involves working on tasks related to cryptography and privacy-enhancing technologies. If you have an interest in cutting-edge research in security and privacy, we’d love to hear from you! Apply here: https://selection.uoc.edu/web/offersjob/offerdetails.aspx?offerID=7AEF220E729D78B226BA96C7B4C4059A5ECD9AE0846AB024E66E32BE291A123B For questions or more information, feel free to contact m_mahdavi@uoc.edu or Dr. Helena Rifà Pous hrifa@uoc.edu.

Closing date for applications:

Contact: Helena Rifà Pous

More information: https://selection.uoc.edu/web/offersjob/offerdetails.aspx?offerID=7AEF220E729D78B226BA96C7B4C4059A5ECD9AE0846AB024E66E32BE291A123B

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LuxQuantum, Esch-sur-Alzette, Luxembourg
Job Posting Job Posting

Company Overview

We’re LuxQuantum, a dynamic startup tackling the exciting and complex challenges in quantum cybersecurity. Our goal is to build innovative solutions that address interoperability bottlenecks in quantum communications by seamlessly integrating quantum key distribution (QKD) and post-quantum cryptography (PQC). We’re looking for someone to join our small team—not just as a colleague but as a friend—to help lead this mission.

We’re more than a company; we’re a team of innovators, learners, and dreamers. If you want to explore cutting-edge technology with people who genuinely enjoy working together, we’d love to meet you!


Role Overview

As a Quantum Cybersecurity Engineer, you’ll play a key role in developing solutions to tackle interoperability issues in quantum cybersecurity. Think of yourself as both a problem-solver and a collaborator, directly contributing to the creation of leading-edge quantum cybersecurity solutions in an environment where every voice matters.


Closing date for applications:

Contact: contact@luxquantum.lu

More information: https://www.siliconluxembourg.lu/quantum-cybersecurity-engineer-luxquantum/

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University of Sheffield
Job Posting Job Posting
We are offering a Ph.D. Opportunities at the University of Sheffield, UK. The candidates will have opportunities to work in Sheffield (UK). Requirements for Ph.D. Position • Completed Master’s degree (or equivalent) at a top university in information security, computer science, applied mathematics, electrical engineering, or a similar area • Research experience (such as publishing papers as a first author in reputable venues) • Self-motivated, reliable, creative, can work in a team and want to do excellent research on challenging scientific problems with practical relevance. Desire to publish at top venues (CORE rank A*/A) for information security/applied cryptography (e.g., TDSC, TIFS, S&P, CCS, NDSS, USENIX SEC), ideally on security protocols and secure computation How to apply? Please send me your CV with detailed information. For the Postdoc position, please send three of your best papers. Contact: Dr Prosanta Gope (p.gope@sheffield.ac.uk) Closing date for applications: Contact: Dr Prosanta Gope (p.gope@sheffield.ac.uk)

Closing date for applications:

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

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Indraprastha Institute of Information Technology Delhi
Job Posting Job Posting

I am looking for a motivated and curious student to join my group as a PhD student in the area of cryptanalysis of symmetric ciphers. The research will span classical and quantum cryptanalysis, with intersections in machine learning and cipher design. You are expected to have a strong background in Computer Science or related fields, solid programming skills (C, C++, Python, etc.), and basic knowledge of cryptography and algorithms. Familiarity with Cryptographic tools (SageMath, PyCryptodome, etc.) and exposure to ML is desirable.

You should have a B.Tech/M.Tech (Computer Science or IT) from a recognized institution. CSIR/UGC JRF would be preferable. Stipend will be as per institute norms (INR 60,000 per month, including HRA).

How to apply:
Send an email attached with your CV and transcripts/mark sheets to ravi.anand@iiitd.ac.in, with the subject line “Position -- PhD” by July 15, 2025.

Closing date for applications:

Contact: Ravi Anand (ravi.anand@iiitd.ac.in), IIIT Delhi, New Delhi, India

More information: https://docs.google.com/document/d/1c_wEWSDtR0irAz4T29HAl3o2AWqLZmoWjFtzOJETjQQ/edit?tab=t.0

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02 July 2025

Alexandra Boldyreva, Deep Inder Mohan, Tianxin Tang
ePrint Report ePrint Report
The use of biometric-based security protocols is on the steep rise. As biometrics become more popular, we witness more attacks. For example, recent BrutePrint/InfinityGauntlet attacks showed how to brute-force fingerprints stored on an Android phone in about 40 minutes. The attacks are possible because biometrics, like passwords, do not have high entropy. But unlike passwords, brute-force attacks are much more damaging for biometrics, because one cannot easily change biometrics in case of compromise. In this work, we propose a novel provably secure Brute-Force Resistant Biometrics (BFRB) protocol for biometric-based authentication and key reconstruction that protects against brute-force attacks even when the server storing biometric-related data is compromised. Our protocol utilizes a verifiable partially oblivious pseudorandom function, an authenticated encryption scheme, a pseudorandom function, and a hash. We formally define security for a BFRB protocol and reduce the security of our protocol to the security of the building blocks. We implement the protocol and study its performance for the ND-0405 iris dataset.
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Iván Blanco Chacón, Raúl Durán Díaz, Rodrigo Martín Sanchez-Ledesma
ePrint Report ePrint Report
The Polynomial Learning With Errors problem (PLWE) serves as the background of two of the three cryptosystems standardized in August 2024 by the National Institute of Standards and Technology to replace non-quantum resistant current primitives like those based on RSA, Diffie-Hellman or its elliptic curve analogue. Although PLWE is highly believed to be quantum resistant, this fact has not yet been established, contrariwise to other post-quantum proposals like multivariate and some code based ones. Moreover, several vulnerabilities have been encountered for a number of specific instances. In a search for more flexibility, it becomes fully relevant to study the robustness of PLWE based on other polynomials, not necessarily cyclotomic. In 2015, Elias et al found a good number of attacks based on different features of the roots of the polynomial. In the present work we present an overview of the approximations made against PLWE derived from this and subsequent works, along with several new attacks which refine those by Elias et al. exploiting the order of the trace of roots over finite extensions of the finite field under the three scenarios laid out by Elias et al., allowing to generalize the setting in which the attacks can be carried out.
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01 July 2025

Cryptography Theory and Technology Research Laboratory of Institute of Information Engineering, CAS
Job Posting Job Posting

We are seeking excellent researchers to join the Cryptography Theory and Technology Research Laboratory at IIE. Applicants are encouraged to apply to work on one of the following areas:

  • Post-Quantum Cryptography
  • Fully Homomorphic Encryption
  • Zero-Knowledge Proof
  • Symmetric-Key Cryptography
And we are open to considering other areas of cryptography.

Positions at PostDoc, Assistant/Associate/Full Professor levels are available. Initial appointments are normally made on a fixed-term contract. Subsequent contract renewal, promotion and tenure all follow standard practices.

Application Materials Required:

  1. Curriculum Vitae
  2. 1-5 Representative publications
  3. Research statement

Review of applications will begin July 1, 2025 and continue until positions are filled.

Closing date for applications:

Contact: Xianhui Lu (luxianhui@iie.ac.cn); Yi Deng (deng@iie.ac.cn); Song Tian (tiansong@iie.ac.cn)

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30 June 2025

National Sun Yat-sen University, Department of Computer Science and Engineering; Kaohsiung, Taiwan
Job Posting Job Posting
Applications are invited for the MS and PhD positions at the Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan. The successful candidate will work at the Cryptology and Network Security Lab under the guidance of Dr. Arijit Karati on diverse topics in applied cryptology and network security.
  • Candidates for the applied cryptography domain must comprehend formal security analysis, secure coding, and effective security integration in the application domains.
  • Candidates for the ML/AI domain, must comprehend search/optimization algorithms, classification, regression, and other essential aspects, including backdoor attacks/data poisoning, model inversion, adversarial attack, and membership Inference.

    Responsibilities: Apart from academic work, students must be involved in several activities in a group or individually, such as (not limited to):
  • Design and implementation of safety protocol.
  • Assesment of the security and performance metric.
  • Research meeting with the supervisor.

    Requirements: (02 MS and 02 PhD positions)
    Apart from the university's basic admission policies (https://cse.nsysu.edu.tw/?Lang=en), students are desired to have the following key requirements:
  • Strong motivation on cryptography or AI security.
  • Knowledge of modern technology.
  • Knowledge of field-wise basic mathematics.
  • Knowledge of at least two programming languages, such as Python/Java/C/C++.
  • Master's thesis must match respective research fields. (for PhD positions)

    Scholarship:
  • Under the university policy.
  • Project funding (availability based on the performance of master's and Ph.D. students).

    What students can expect:
  • Cooperation from the supervisor and lab mates.
  • The rich culture in research and related activities.
  • Flexibility in communication, e.g., English.

    Submit your detailed CV by August 30, 2025.

    Application Deadline: September 30, 2025

    Closing date for applications:

    Contact: Arijit Karati (arijit.karati@mail.cse.nsysu.edu.tw)

    More information: https://oia.nsysu.edu.tw/static/file/308/1308/img/NSYSUAY2025-2026AdmissionApplicationGuideforInternationalDegreeStudents.pdf

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    National Sun Yat-sen University, Department of Computer Science and Engineering; Kaohsiung, Taiwan
    Job Posting Job Posting
    Applications are invited for the Post-doc position in applied cryptography and network security at the Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan. Applicants with expertise in at least one of the following areas are preferred: post-quantum cryptography, automotive security, developing novel cryptographic primitives and protocols, side-channel analysis, and machine-learning techniques for safety applications. Applicants require knowledge of formal security analysis, secure coding, and the practical integration of domain security into application domains.

    Essential Qualifications:
  • PhD degree in CSE/Mathematics/IT/electrical engineering with a specialization in Information/Network Security from a reputable Institution (preferably ranked within the QS WUR 500).
  • Outstanding track record of publications in Journals (preferably JCR-Q1 or prestigious IEEE journals) and security-related conferences.

    Application Deadline: 15-08-2025

    Closing date for applications:

    Contact: Arijit Karati (arijit.karati@mail.cse.nsysu.edu.tw)

    More information: https://www.canseclab.com/

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    Zhenhua Zou, Zhuotao Liu, Jinyong Shan, Qi Li, Ke Xu, Mingwei Xu
    ePrint Report ePrint Report
    Collaborative graph processing refers to the joint analysis of inter-connected graphs held by multiple graph owners. To honor data privacy and support various graph processing algorithms, existing approaches employ secure multi-party computation (MPC) protocols to express the vertex-centric abstraction. Yet, due to certain computation-intensive cryptography constructions, state-of-the-art (SOTA) approaches are asymptotically suboptimal, imposing significant overheads in terms of computation and communication. In this paper, we present RingSG, the first system to attain optimal communication/computation complexity within the MPC-based vertex-centric abstraction for collaborative graph processing. This optimal complexity is attributed to Ring-ScatterGather, a novel computation paradigm that can avoid exceedingly expensive cryptography operations (e.g., oblivious sort), and simultaneously ensure the overall workload can be optimally decomposed into parallelizable and mutually exclusive MPC tasks. Within Ring-ScatterGather, RingSG improves the concrete runtime efficiency by incorporating 3-party secure computation via share conversion, and optimizing the most cost-heavy part using a novel oblivious group aggregation protocol. Finally, unlike prior approaches, we instantiate RingSG into two end-to-end applications to effectively obtain application-specific results from the protocol outputs in a privacy-preserving manner. We developed a prototype of RingSG and extensively evaluated it across various graph collaboration settings, including different graph sizes, numbers of parties, and average vertex degrees. The results show RingSG reduces the system running time of SOTA approaches by up to 15.34× and per-party communication by up to 10.36×. Notably, RingSG excels in processing sparse global graphs collectively held by more parties, consistent with our theoretical cost analysis.
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    Ya-Nan Li, Yaqing Song, Qiang Tang, Moti Yung
    ePrint Report ePrint Report
    Git services such as GitHub, have been widely used to manage projects and enable collaborations among multiple entities. Just as in messaging and cloud storage, where end-to-end security has been gaining increased attention, such a level of security is also demanded for Git services. Content in the repositories (and the data/code supply-chain facilitated by Git services) could be highly valuable, whereas the threat of system breaches has become routine nowadays. However, existing studies of Git security to date (mostly open source projects) suffer in two ways: they provide only very weak security, and they have a large overhead.

    In this paper, we initiate the needed study of efficient end-to-end encrypted Git services. Specifically, we formally define the syntax and critical security properties, and then propose two constructions that provably meet those properties. Moreover, our constructions have the important property of platform-compatibility: They are compatible with current Git servers and reserve all basic Git operations, thus can be directly tested and deployed on top of existing platforms. Furthermore, the overhead we achieve is only proportional to the actual difference caused by each edit, instead of the whole file (or even the whole repository) as is the case with existing works. We implemented both constructions and tested them directly on several public GitHub repositories. Our evaluations show (1) the effectiveness of platform-compatibility, and (2) the significant efficiency improvement we got (while provably providing much stronger security than prior ad-hoc treatments).
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    Prabhanjan Ananth, Amit Behera, Zikuan Huang
    ePrint Report ePrint Report
    Quantum copy-protection is a foundational notion in quantum cryptography that leverages the governing principles of quantum mechanics to tackle the problem of software anti-piracy. Despite progress in recent years, precisely characterizing the class of functionalities that can be copy-protected is still not well understood. Two recent works, by [Coladangelo and Gunn, STOC 2024] and [Ananth and Behera, CRYPTO 2024, showed that puncturable functionalities can be copy-protected. Both works have significant caveats with regard to the underlying cryptographic assumptions and additionally restrict the output length of the functionalities to be copy-protected. In this work, we make progress towards simultaneously addressing both caveats. We show the following: - Revisiting Unclonable Puncturable Obfuscation (UPO): We revisit the notion of UPO introduced by [Ananth and Behera, CRYPTO 2024]. We present a new approach to construct UPO and a variant of UPO, called independent-secure UPO. Unlike UPO, we show how to base the latter notion on well-studied assumptions. - Copy-Protection from Independent-secure UPO: Assuming independent-secure UPO, we show that any m-bit, for m ≥ 2, puncturable functionality can be copy-protected. - Copy-Protection from UPO: Assuming UPO, we show that any 1-bit puncturable functionality can be copy-protected. The security of copy-protection holds against identical challenge distributions.
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