On the Design and Security of Post-Quantum Aggregate Signatures

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On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Recently, Blanco-Chac\'on proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number

RLWE and PLWE over cyclotomic fields are not equivalent

We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field $\mathbb{Q}(\zeta_n)$ are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in $n$. We do so by providing a lower bound, holding for infinitely many positive integers $n$, for the condition number of the Vandermonde matrix of the $n$th cyclotomic polynomial

History-Free Sequential Aggregate Signatures from Generic Trapdoor Functions

A sequential aggregate signature (SAS) scheme allows multiple users to sequentially combine their respective signatures in order to reduce communication costs. Historically, early proposals required the use of trapdoor permutation (e.g., RSA). In recent years, a number of attempts have been made to extend SAS schemes to post-quantum assumptions. Many post-quantum signatures have been proposed in the hash-and-sign paradigm, which requires the use of trapdoor functions and appears to be an ideal candidate for sequential aggregation attempts. However, the hardness in achieving post-quantum one-way permutations makes it difficult to obtain similarly general constructions. Direct attempts at generalizing permutation-based schemes have been proposed, but they either lack formal security or require additional properties on the trapdoor function, which are typically not available for multivariate or code-based functions. In this paper, we propose a history-free sequential aggregate signature based on generic trapdoor functions, generalizing existing techniques. We prove the security of our scheme in the random oracle model by adopting the probabilistic hash-and-sign with retry paradigm, and we instantiate our construction with three post-quantum schemes, comparing their compression capabilities. Finally, we discuss how direct extensions of permutation-based SAS schemes are not possible without additional properties, showing the insecurity of two existing multivariate schemes when instantiated with Unbalanced Oil and Vinegar

A deep learning mixed-data type approach for the classification of FHR signals

The Cardiotocography (CTG) is a widely diffused monitoring practice, used in Ob-Gyn Clinic to assess the fetal well-being through the analysis of the Fetal Heart Rate (FHR) and the Uterine contraction signals. Due to the complex dynamics regulating the Fetal Heart Rate, a reliable visual interpretation of the signal is almost impossible and results in significant subjective inter and intra-observer variability. Also, the introduction of few parameters obtained from computer analysis did not solve the problem of a robust antenatal diagnosis. Hence, during the last decade, computer aided diagnosis systems, based on artificial intelligence (AI) machine learning techniques have been developed to assist medical decisions. The present work proposes a hybrid approach based on a neural architecture that receives heterogeneous data in input (a set of quantitative parameters and images) for classifying healthy and pathological fetuses. The quantitative regressors, which are known to represent different aspects of the correct development of the fetus, and thus are related to the fetal healthy status, are combined with features implicitly extracted from various representations of the FHR signal (images), in order to improve the classification performance. This is achieved by setting a neural model with two connected branches, consisting respectively of a Multi-Layer Perceptron (MLP) and a Convolutional Neural Network (CNN). The neural architecture was trained on a huge and balanced set of clinical data (14.000 CTG tracings, 7000 healthy and 7000 pathological) recorded during ambulatory non stress tests at the University Hospital Federico II, Napoli, Italy. After hyperparameters tuning and training, the neural network proposed has reached an overall accuracy of 80.1%, which is a promising result, as it has been obtained on a huge dataset

Fetal states identification in cardiotocographic tracings through discrete emissions multivariate hidden markov models

Background and objectives: Computerized Cardiotocography (cCTG) allows to analyze the Fetal Heart Rate (FHR) objectively and thoroughly, providing valuable insights on fetal condition. A challenging but crucial task in this context is the automatic identification of fetal activity and quiet periods within the tracings. Different neural mechanisms are involved in the regulation of the fetal heart, depending on the behavioral states. Thereby, their correct identification has the potential to increase the interpretability and diagnostic capabilities of FHR quantitative analysis. Moreover, the most common pathologies in pregnancy have been associated with variations in the alternation between quiet and activity states. Methods: We address the problem of fetal states clustering by means of an unsupervised approach, resorting to the use of a multivariate Hidden Markov Models (HMM) with discrete emissions. A fixed length sliding window is shifted on the CTG traces and a small set of features is extracted at each slide. After an encoding procedure, these features become the emissions of a multivariate HMM in which quiet and activity are the hidden states. After an unsupervised training procedure, the model is used to automatically segment signals. Results: The achieved results indicate that our developed model exhibits a high degree of reliability in identifying quiet and activity states within FHR signals. A set of 35 CTG signals belonging to different pregnancies were independently annotated by an expert gynecologist and segmented using the proposed HMM. To avoid any bias, the physician was blinded to the results provided by the algorithm. The overall agreement between the HMM's predictions and the clinician's interpretations was 90%.Conclusions: The proposed method reliably identified fetal behavioral states, the alternance of which is an important factor in the fetal development. One key strength of our approach lies in the ease of interpreting the obtained results. By utilizing a small set of parameters that are already used in cCTG and possess clear intrinsic meanings, our method provides a high level of explainability. Another significant advantage of our approach is its fully unsupervised learning process. The states identified by our model using the Baum-Welch algorithm are associated with the "Active" and "Quiet" states only after the clustering process, removing the reliance on expert annotations. By autonomously identifying the clusters based solely on the intrinsic characteristics of the signal, our method achieves a more objective evaluation that overcomes the limitations of subjective interpretations. Indeed, we believe it could be integrated in cCTG systems to obtain a more complete signal analysis

Influence of Gestational Diabetes on Fetal Heart Rate in Antepartum Cardiotocographic Recordings

In pregnancy, diabetes is known to increase the risk of adverse maternal and neonatal outcomes. It would be beneficial to find techniques that allow early investigation of the physio-pathological mechanisms involved to provide clinicians with tools for prevention and therapies. For that, cardiotocography (CTG) is a promising tool. However, the evidence is still scarce and the impact on clinical practice little. In this study, we aim at characterizing the changes induced by gestational diabetes (GDM) on the fetal heart rate series. To do so, we performed a retrospective cohort study on a CTG dataset containing more than 20000 recordings of which 852 belong to 301 GDM-diagnosed patients. We divided the recordings by gestational age (G.A.) into 4 groups (weeks: 31-35, 36, 37, 38 to delivery) and for each we identified a control population of equal size matched by comorbidities. We analyzed a comprehensive set of parameters from the time domain, frequency domain and non-linear analysis and assessed variations in median values on each feature. For all G.A. below the 38th week, we found a significant increase in the power in the movement frequency band (p<0.01) and an increase in the absolute value of Deceleration Reserve (p<0.01) in GDM vs control. Other significant values were also identified and are discussed in more detail in the paper

Prediction of IUGR condition at birth by means of CTG recordings and a ResNet model

Objective: Sub-optimal uterine-placental perfusion and fetal nutrition can lead to intrauterine growth restriction (IUGR), also called fetal growth restriction (FGR). Antenatal cardiotocography (CTG) can aid in the early detection of IUGR. Reliably diagnosing IUGR before delivery remains challenging, and deep learning (DL) techniques offer potential solutions. This paper describes the development of a DL approach to predict an IUGR condition at birth by using CTG signals collected during antenatal monitoring. Materials and methods: Our method is encapsulated in the concept of a two-step training process of a ResNet architecture. The primary focus is on the minimization of data loss, which motivates the division into “presumed” and “confirmed” datasets, which is employed to distinguish based on the presence of information at birth. The method involves fine-tuning: the initial training utilizes “presumed” data to train the network, and the subsequent training employs data representing certain knowledge to refine its performance. Results: The DL model reaches a balanced accuracy of 80% on a hold-out test set of confirmed cases, which is better than what obtained by using standard clinical guidelines. Discussion: The results of our work are compared to the results of similar papers dealing with the prediction of IUGR condition at birth and in general with the prediction of fetal pathological conditions. Our final results are obtained using a very large dataset compared to other papers reported in the literature. Conclusion: The inclusion of DL methods on CTG signals may complement imaging technologies and improve the early detection of IUGR