Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.26258

CRT-Decomposed $Σ$-Protocols for CSIDH

I. Dey·I. Cherkaoui
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct a zero-knowledge proof of knowledge for the CSIDH group action that exploits the Chinese Remainder Theorem (CRT) structure of the ideal class group, available whenever the group structure is known exactly, as for CSIDH-512. We prove perfect completeness, perfect special honest-verifier zero-knowledge, and 2-special soundness, in which the secret is recovered from two accepting transcripts by one subtraction and one modular inversion per CRT component; no rewinding loop, lattice reduction, or heuristic sampling appears in the extractor. Because each round admits exactly two responses, Unruh's transform yields a non-interactive proof with straight-line extraction in the quantum random oracle model (QROM), removing the multiplicative forking-lemma loss. We instantiate the scheme on the CSIDH-512 class group, verify the algebraic layer by machine over the exact 258-bit modulus ($10^4$ random instances, all passing), and report protocol-level simulations in the exponent model: Monte Carlo soundness rates matching the proven $2^{-t}$ bound within 95\% confidence at every tested $t$, serialized signature sizes within 1.6\% of the formulas, instrumented action counts, and a scaled meet-in-the-middle attack whose measured cost follows the predicted $\sqrt{q}$ law. We further prove two delimiting results: CRT decomposition cannot enlarge the per-round challenge space, and publishing the CRT hop curves lowers classical key-recovery cost from about $2^{128.6}$ to about $2^{67.3}$ group action evaluations. The construction is therefore correct and structurally complete today, but quantitatively secure only on future parameters whose class number has large prime factors. We compare against CSI-FiSh, CSI-Otter, and Tanuki; a tightly secure blind signature from this proof of knowledge is left to future work.

Explore related subjects

cs.CRcs.ITmath.ITquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

I. Dey, I. Cherkaoui. 2026-08-13. CRT-Decomposed $Σ$-Protocols for CSIDH. https://arxiv.org/abs/2609.26258

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

UPPRESSO: Untraceable and Unlinkable Privacy-PREserving Single Sign-On Services

Single sign-on (SSO) allows a user to maintain only the credential for an identity provider (IdP) to log into multiple relying parties (RPs). However, SSO introduces privacy threats, as (a) a curious IdP could track a user's all visits to RPs, and (b) colluding RPs could learn a user's online profile by linking her identities across these RPs. This paper presents a privacypreserving SSO scheme, called UPPRESSO, to protect an honest user's online profile against (a) an honest-but-curious IdP and (b) malicious RPs colluding with other users. UPPRESSO proposes an identity-transformation approach to generate untraceable ephemeral pseudo-identities for an RP and a user from which the target RP derives a permanent account for the user, while the transformations also provide unlinkability. This approach protects the identities of the user and the target RPs in a login flow, while working compatibly with widely-deployed SSO protocols and providing services accessed from a commercial-off-the-shelf browser without plug-ins or extensions. We built a prototype of UPPRESSO on top of MITREid Connect, an open-source SSO system. The extensive evaluations show that it fulfills the security and privacy requirements of SSO with reasonable overheads.

cs.CR↗
arXiv2026

Attack Tree Distance: a practical examination of tree difference measurement within cyber security

Attack trees are a popular threat modeling method. In practice, there is often a need to compare attack tree models produced by human experts, based on both the structure of the tree and the meaning of the node labels. In this work, we investigate the problem of comparing attack trees and measuring their similarity. We define five different measures for measuring the distance between two attack trees: Label Distance (LD), Tree Edit Distance (TED), Radical Distance (RD), Multiset Distance (MSD) and Weighted Sum Distance (WSD). We further propose a repeatable method of both theoretical and experimental attack tree distance measures validation. Our theoretical validation consists of a series of basic transformations to evaluate the behavior of distance measures with respect to specific types of transformations that may appear between two attack trees. To experimentally validate our distance measures, we designed and executed a human study ($n=39$) to collect a dataset of attack trees to be used for evaluation and comparison of the measures. From our theoretical and experimental results, we find that applying semantic similarity as a means of comparing node labels is a valid approach. Further, we find four of the five attack tree distance measures are valid approaches in certain, varying circumstances. Our results suggest that these methods can already be used to identify similar real-world attack trees. Overall, this work lays the groundwork for improved threat model analysis, validation of AI-generated attack trees, and future research into threat similarity measurement in cybersecurity.

cs.CR↗
arXiv2026

Probabilistic Modeling of Jailbreak on Multimodal LLMs: From Quantification to Application

Recently, Multimodal Large Language Models (MLLMs) have demonstrated their superior ability in understanding multimodal content. However, they remain vulnerable to jailbreak attacks, which exploit weaknesses in their safety alignment to generate harmful responses. Previous studies categorize jailbreaks as successful or failed based on whether responses contain malicious content. However, given the stochastic nature of MLLM responses, this binary classification of an input's ability to jailbreak MLLMs is inappropriate. Derived from this viewpoint, we introduce jailbreak probability to quantify the jailbreak potential of an input, which represents the likelihood that MLLMs generated a malicious response when prompted with this input. We approximate this probability through multiple queries to MLLMs. After modeling the relationship between input hidden states and their corresponding jailbreak probability using Jailbreak Probability Prediction Network (JPPN), we use continuous jailbreak probability for optimization. Specifically, we propose Jailbreak-Probability-based Attack (JPA) that optimizes adversarial perturbations on input image to maximize jailbreak probability, and further enhance it as Multimodal JPA (MJPA) by including monotonic text rephrasing. To counteract attacks, we also propose Jailbreak-Probability-based Finetuning (JPF), which minimizes jailbreak probability through MLLM parameter updates. Extensive experiments show that (1) (M)JPA yields significant improvements when attacking a wide range of models under both white and black box settings. (2) JPF vastly reduces jailbreaks by at most over 60\%. Both of the above results demonstrate the significance of introducing jailbreak probability to make nuanced distinctions among input jailbreak abilities.

cs.CR↗