Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1705.01380

Linear complexity of Legendre-polynomial quotients

Zhixiong Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We continue to investigate binary sequence $(f_u)$ over $\{0,1\}$ defined by $(-1)^{f_u}=\left(\frac{(u^w-u^{wp})/p}{p}\right)$ for integers $u\ge 0$, where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and we restrict $\left(\frac{0}{p}\right)=1$. In an earlier work, the linear complexity of $(f_u)$ was determined for $w=p-1$ under the assumption of $2^{p-1}\not\equiv 1 \pmod {p^2}$. In this work, we give possible values on the linear complexity of $(f_u)$ for all $1\le w<p-1$ under the same conditions. We also state that the case of larger $w(\geq p)$ can be reduced to that of $0\leq w\leq p-1$.

Explore related subjects

cs.CRmath.NT

Keep this discovery

BibTeXRIS

Zhixiong Chen. 2017-05-03. Linear complexity of Legendre-polynomial quotients. https://doi.org/10.1049/iet-ifs.2017.0307

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

KEEP EXPLORING

Related papers

arXiv2026

Empirical Evaluation of Membership Inference Attacks on NLP Text Classifiers: A Baseline Study on SST-2

Membership inference attacks (MIAs) try to determine whether a specific record was used to train a model, a privacy risk that matters in natural language processing (NLP), where training data can contain sensitive user text. This paper presents a controlled benchmark of membership inference vulnerability for text classification on the GLUE SST-2 sentiment dataset. A TF-IDF + Logistic Regression pipeline and a fine-tuned DistilBERT classifier are compared under a loss-threshold MIA, with utility measured by development accuracy and macro F1. DistilBERT reached 0.9466 accuracy and 0.9460 macro F1 against 0.8756 and 0.8727 for Logistic Regression, yet both models leaked membership signal (Attack AUC 0.5615 and 0.5800, respectively). Two mitigations were tested. Stronger regularization reduced leakage for Logistic Regression at a visible utility cost, whereas fine-tuning DistilBERT for 2 epochs instead of 3 reduced leakage with negligible accuracy loss. Lightweight training adjustments can improve the privacy-utility trade-off without complex defenses.

cs.CR↗
arXiv2026

Empirical Evaluation of Data Poisoning Attacks in Supervised Learning

Data poisoning corrupts training data to degrade a model or to plant attacker-controlled behavior. This study evaluates two representative training-time attacks, label flipping and backdoor poisoning, on MNIST and Fashion-MNIST with three baseline classifiers: Logistic Regression, Linear SVM, and Random Forest. Clean training is compared with poisoning rates of 5%, 10%, and 20% using clean-test accuracy, macro-precision, macro-recall, macro-F1, and, for backdoors, attack success rate. Label flipping caused clear degradation, largest for Logistic Regression and Linear SVM, while Random Forest stayed comparatively stable. Backdoor poisoning reached attack success rates from 0.9667 to 1.0000 on both datasets and all three models while often keeping clean-test performance near baseline. The results separate indiscriminate poisoning, which shows up in standard metrics, from targeted backdoor poisoning, which stays comparatively stealthy while embedding highly effective malicious behavior, and they support security-oriented evaluation beyond conventional clean-test metrics.

cs.CR↗
arXiv2026

DeFiFusion: Combining Transaction Events with Smart Contracts to Detect Price Manipulation Attacks

Decentralized Finance (DeFi) has emerged as a rapidly growing blockchain-based financial service, where market transaction dynamics and underlying smart contract logic are intricately intertwined. This autonomous interplay, while eliminating centralized intermediaries, significantly expands the vulnerability surface of DeFi protocols to Price Manipulation Attacks (PMAs), which have already inflicted catastrophic financial losses. Despite their gravity, existing detection paradigms suffer from fundamental limitations. Transaction-centric methods lack awareness of contract execution semantics, making them prone to false positives under legitimate market volatility, while static contract analyses ignore real transaction behaviors and frequently report vulnerabilities that are infeasible to exploit in practice. We present DeFiFusion, a dual-modal PMA detection framework that closes this gap by jointly modeling transaction events and smart contract semantics within a unified pipeline. Our core insight is that PMA maliciousness emerges only from the interaction between transaction behaviors and the contract logic they exploit; neither signal suffices in isolation. Accordingly, we derive price-manipulation-aware event encoding for extracting fine-grained temporal and economic features tailored to manipulation patterns. We further introduce LLM-based contract semantic extraction to supply the execution-logic context that prior behavioral methods lack. To fuse these modalities, we propose a Dual-Modal Projection-Fusion Transformer with T5-style relative positional encoding, capturing the cyclic multi-stage execution structures that distinguish PMAs from benign market activity. Extensive experiments demonstrate that DeFiFusion consistently achieves state-of-the-art detection performance, effectively recalling 222 of the 225 PMA cases while maintaining a precision of 96.10%.

cs.CR↗