Biblio

Found 19604 results

2020-08-10
Kim, Byoungchul, Jung, Jaemin, Han, Sangchul, Jeon, Soyeon, Cho, Seong-je, Choi, Jongmoo.  2019.  A New Technique for Detecting Android App Clones Using Implicit Intent and Method Information. 2019 Eleventh International Conference on Ubiquitous and Future Networks (ICUFN). :478–483.
Detecting repackaged apps is one of the important issues in the Android ecosystem. Many attackers usually reverse engineer a legitimate app, modify or embed malicious codes into the app, repackage and distribute it in the online markets. They also employ code obfuscation techniques to hide app cloning or repackaging. In this paper, we propose a new technique for detecting repackaged Android apps, which is robust to code obfuscation. The technique analyzes the similarity of Android apps based on the method call information of component classes that receive implicit intents. We developed a tool Calldroid that implemented the proposed technique, and evaluated it on apps transformed using well-known obfuscators. The evaluation results showed that the proposed technique can effectively detect repackaged apps.
2020-05-11
Liu, Weiyou, Liu, Xu, Di, Xiaoqiang, Qi, Hui.  2019.  A novel network intrusion detection algorithm based on Fast Fourier Transformation. 2019 1st International Conference on Industrial Artificial Intelligence (IAI). :1–6.
Deep learning techniques have been widely used in intrusion detection, but their application on convolutional neural networks (CNN) is still immature. The main challenge is how to represent the network traffic to improve performance of the CNN model. In this paper, we propose a network intrusion detection algorithm based on representation learning using Fast Fourier Transformation (FFT), which is first exploration that converts traffic to image by FFT to the best of our knowledge. Each traffic is converted to an image and then the intrusion detection problem is turned to image classification. The experiment results on NSL-KDD dataset show that the classification performence of the algorithm in the CNN model has obvious advantages compared with other algorithms.
2020-02-18
Griffioen, Paul, Weerakkody, Sean, Sinopoli, Bruno.  2019.  An Optimal Design of a Moving Target Defense for Attack Detection in Control Systems. 2019 American Control Conference (ACC). :4527–4534.
In this paper, we consider the problem of designing system parameters to improve detection of attacks in control systems. Specifically, we study control systems which are vulnerable to integrity attacks on sensors and actuators. We aim to defend against strong model aware adversaries that can read and modify all sensors and actuators. Previous work has proposed a moving target defense for detecting integrity attacks on control systems. Here, an authenticating subsystem with time-varying dynamics coupled to the original plant is introduced. Due to this coupling, an attack on the original system will affect the authenticating subsystem and in turn be revealed by a set of sensors measuring the extended plant. Moreover, the time-varying dynamics of the extended plant act as a moving target, preventing an adversary from developing an effective adaptive attack strategy. Previous work has failed to consider the design of the time-varying system matrices and as such provides little in terms of guidelines for implementation in real systems. This paper proposes two optimization problems for designing these matrices. The first designs the auxiliary actuators to maximize detection performance while the second designs the coupling matrices to maximize system estimation performance. Numerical examples are presented that validate our approach.
2020-09-04
Saad, Muhammad, Cook, Victor, Nguyen, Lan, Thai, My T., Mohaisen, Aziz.  2019.  Partitioning Attacks on Bitcoin: Colliding Space, Time, and Logic. 2019 IEEE 39th International Conference on Distributed Computing Systems (ICDCS). :1175—1187.
Bitcoin is the leading example of a blockchain application that facilitates peer-to-peer transactions without the need for a trusted intermediary. This paper considers possible attacks related to the decentralized network architecture of Bitcoin. We perform a data driven study of Bitcoin and present possible attacks based on spatial and temporal characteristics of its network. Towards that, we revisit the prior work, dedicated to the study of centralization of Bitcoin nodes over the Internet, through a fine-grained analysis of network distribution, and highlight the increasing centralization of the Bitcoin network over time. As a result, we show that Bitcoin is vulnerable to spatial, temporal, spatio-temporal, and logical partitioning attacks with an increased attack feasibility due to network dynamics. We verify our observations by simulating attack scenarios and the implications of each attack on the Bitcoin . We conclude with suggested countermeasures.
2020-03-02
Wang, Qing, Wang, Zengfu, Guo, Jun, Tahchi, Elias, Wang, Xinyu, Moran, Bill, Zukerman, Moshe.  2019.  Path Planning of Submarine Cables. 2019 21st International Conference on Transparent Optical Networks (ICTON). :1–4.
Submarine optical-fiber cables are key components in the conveying of Internet data, and their failures have costly consequences. Currently, there are over a million km of such cables empowering the Internet. To carry the ever-growing Internet traffic, additional 100,000s of km of cables will be needed in the next few years. At an average cost of \$28,000 per km, this entails investments of billions of dollars. In current industry practice, cable paths are planned manually by experts. This paper surveys our recent work on cable path planning algorithms, where we use several methods to plan cable paths taking account of a range of cable risk factors in addition to cable costs. Two methods, namely, the fast marching method (FMM) and the Dijkstra's algorithm are applied here to long-haul cable path design in a new geographical region. A specific example is given to demonstrate the benefit of the FMM-based method in terms of the better path planning solutions over the Dijkstra's algorithm.
Ullah, Rehmat, Ur Rehman, Muhammad Atif, Kim, Byung-Seo, Sonkoly, Balázs, Tapolcai, János.  2019.  On Pending Interest Table in Named Data Networking based Edge Computing: The Case of Mobile Augmented Reality. 2019 Eleventh International Conference on Ubiquitous and Future Networks (ICUFN). :263–265.
Future networks require fast information response time, scalable content distribution, security and mobility. In order to enable future Internet many key enabling technologies have been proposed such as Edge computing (EC) and Named Data Networking (NDN). In EC substantial compute and storage resources are placed at the edge of the network, in close proximity to end users. Similarly, NDN provides an alternative to traditional host centric IP architecture which seems a perfect candidate for distributed computation. Although NDN with EC seems a promising approach for enabling future Internet, it can cause various challenges such as expiry time of the Pending Interest Table (PIT) and non-trivial computation of the edge node. In this paper we discuss the expiry time and non-trivial computation in NDN based EC. We argue that if NDN is integrated in EC, then the PIT expiry time will be affected in relation with the processing time on the edge node. Our analysis shows that integrating NDN in EC without considering PIT expiry time may result in the degradation of network performance in terms of Interest Satisfaction Rate.
2020-04-03
Gerl, Armin, Becher, Stefan.  2019.  Policy-Based De-Identification Test Framework. 2019 IEEE World Congress on Services (SERVICES). 2642-939X:356—357.
Protecting privacy of individuals is a basic right, which has to be considered in our data-centered society in which new technologies emerge rapidly. To preserve the privacy of individuals de-identifying technologies have been developed including pseudonymization, personal privacy anonymization, and privacy models. Each having several variations with different properties and contexts which poses the challenge for the proper selection and application of de-identification methods. We tackle this challenge proposing a policy-based de-identification test framework for a systematic approach to experimenting and evaluation of various combinations of methods and their interplay. Evaluation of the experimental results regarding performance and utility is considered within the framework. We propose a domain-specific language, expressing the required complex configuration options, including data-set, policy generator, and various de-identification methods.
2020-06-22
Adesuyi, Tosin A., Kim, Byeong Man.  2019.  Preserving Privacy in Convolutional Neural Network: An ∊-tuple Differential Privacy Approach. 2019 IEEE 2nd International Conference on Knowledge Innovation and Invention (ICKII). :570–573.
Recent breakthrough in neural network has led to the birth of Convolutional neural network (CNN) which has been found to be very efficient especially in the areas of image recognition and classification. This success is traceable to the availability of large datasets and its capability to learn salient and complex data features which subsequently produce a reusable output model (Fθ). The Fθ are often made available (e.g. on cloud as-a-service) for others (client) to train their data or do transfer learning, however, an adversary can perpetrate a model inversion attack on the model Fθ to recover training data, hence compromising the sensitivity of the model buildup data. This is possible because CNN as a variant of deep neural network does memorize most of its training data during learning. Consequently, this has pose a privacy concern especially when a medical or financial data are used as model buildup data. Existing researches that proffers privacy preserving approach however suffer from significant accuracy degradation and this has left privacy preserving model on a theoretical desk. In this paper, we proposed an ϵ-tuple differential privacy approach that is based on neuron impact factor estimation to preserve privacy of CNN model without significant accuracy degradation. We experiment our approach on two large datasets and the result shows no significant accuracy degradation.
2020-09-04
Osia, Seyed Ali, Rassouli, Borzoo, Haddadi, Hamed, Rabiee, Hamid R., Gündüz, Deniz.  2019.  Privacy Against Brute-Force Inference Attacks. 2019 IEEE International Symposium on Information Theory (ISIT). :637—641.
Privacy-preserving data release is about disclosing information about useful data while retaining the privacy of sensitive data. Assuming that the sensitive data is threatened by a brute-force adversary, we define Guessing Leakage as a measure of privacy, based on the concept of guessing. After investigating the properties of this measure, we derive the optimal utility-privacy trade-off via a linear program with any f-information adopted as the utility measure, and show that the optimal utility is a concave and piece-wise linear function of the privacy-leakage budget.
2020-06-02
Coiteux-Roy, Xavier, Wolf, Stefan.  2019.  Proving Erasure. 2019 IEEE International Symposium on Information Theory (ISIT). :832—836.

It seems impossible to certify that a remote hosting service does not leak its users' data - or does quantum mechanics make it possible? We investigate if a server hosting data can information-theoretically prove its definite deletion using a "BB84-like" protocol. To do so, we first rigorously introduce an alternative to privacy by encryption: privacy delegation. We then apply this novel concept to provable deletion and remote data storage. For both tasks, we present a protocol, sketch its partial security, and display its vulnerability to eavesdropping attacks targeting only a few bits.

2020-03-30
Mao, Huajian, Chi, Chenyang, Yu, Jinghui, Yang, Peixiang, Qian, Cheng, Zhao, Dongsheng.  2019.  QRStream: A Secure and Convenient Method for Text Healthcare Data Transferring. 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC). :3458–3462.
With the increasing of health awareness, the users become more and more interested in their daily health information and healthcare activities results from healthcare organizations. They always try to collect them together for better usage. Traditionally, the healthcare data is always delivered by paper format from the healthcare organizations, and it is not easy and convenient for data usage and management. They would have to translate these data on paper to digital version which would probably introduce mistakes into the data. It would be necessary if there is a secure and convenient method for electronic health data transferring between the users and the healthcare organizations. However, for the security and privacy problems, almost no healthcare organization provides a stable and full service for health data delivery. In this paper, we propose a secure and convenient method, QRStream, which splits original health data and loads them onto QR code frame streaming for the data transferring. The results shows that QRStream can transfer text health data smoothly with an acceptable performance, for example, transferring 10K data in 10 seconds.
Brito, J. P., López, D. R., Aguado, A., Abellán, C., López, V., Pastor-Perales, A., la Iglesia, F. de, Martín, V..  2019.  Quantum Services Architecture in Softwarized Infrastructures. 2019 21st International Conference on Transparent Optical Networks (ICTON). :1–4.
Quantum computing is posing new threats on our security infrastructure. This has triggered a new research field on quantum-safe methods, and those that rely on the application of quantum principles are commonly referred as quantum cryptography. The most mature development in the field of quantum cryptography is called Quantum Key Distribution (QKD). QKD is a key exchange primitive that can replace existing mechanisms that can become obsolete in the near future. Although QKD has reached a high level of maturity, there is still a long path for a mass market implementation. QKD shall overcome issues such as miniaturization, network integration and the reduction of production costs to make the technology affordable. In this direction, we foresee that QKD systems will evolve following the same path as other networking technologies, where systems will run on specific network cards, integrable in commodity chassis. This work describes part of our activity in the EU H2020 project CiViQ in which quantum technologies, as QKD systems or quantum random number generators (QRNG), will become a single network element that we define as Quantum Switch. This allows for quantum resources (keys or random numbers) to be provided as a service, while the different components are integrated to cooperate for providing the most random and secure bit streams. Furthermore, with the purpose of making our proposal closer to current networking technology, this work also proposes an abstraction logic for making our Quantum Switch suitable to become part of software-defined networking (SDN) architectures. The model fits in the architecture of the SDN quantum node architecture, that is being under standardization by the European Telecommunications Standards Institute. It permits to operate an entire quantum network using a logically centralized SDN controller, and quantum switches to generate and to forward key material and random numbers across the entire network. This scheme, demonstrated for the first time at the Madrid Quantum Network, will allow for a faster and seamless integration of quantum technologies in the telecommunications infrastructure.
2020-05-08
Wang, Dongqi, Shuai, Xuanyue, Hu, Xueqiong, Zhu, Li.  2019.  Research on Computer Network Security Evaluation Method Based on Levenberg-Marquardt Algorithms. 2019 International Conference on Communications, Information System and Computer Engineering (CISCE). :399—402.
As we all know, computer network security evaluation is an important link in the field of network security. Traditional computer network security evaluation methods use BP neural network combined with network security standards to train and simulate. However, because BP neural network is easy to fall into local minimum point in the training process, the evalu-ation results are often inaccurate. In this paper, the LM (Levenberg-Marquard) algorithm is used to optimize the BP neural network. The LM-BP algorithm is constructed and applied to the computer network security evaluation. The results show that compared with the traditional evaluation algorithm, the optimized neural network has the advantages of fast running speed and accurate evaluation results.
2020-08-24
Liang, Dai, Pan, Peisheng.  2019.  Research on Intrusion Detection Based on Improved DBN-ELM. 2019 International Conference on Communications, Information System and Computer Engineering (CISCE). :495–499.
To leverage the feature extraction of DBN and the fast classification and good generalization of ELM, an improved method of DBN-ELM is proposed for intrusion detection. The improved model uses deep belief network (DBN) to train NSL-KDD dataset and feed them back to the extreme learning machine (ELM) for classification. A classifier is connected at each intermediate level of the DBN-ELM. By majority voting on the output of classifier and ELM, the final output is calculated by integration. Experiments show that the improved model increases the classification confidence and accuracy of the classifier. The model has been benchmarked on the NSL-KDD dataset, and the accuracy of the model has been improved to 97.82%, while the false alarm rate has been reduced to 1.81%. Proposed improved model has been also compared with DBN, ELM, DBN-ELM and achieves competitive accuracy.
2020-01-27
Xuefeng, He, Chi, Zhang, Yuewu, Jing, Xingzheng, Ai.  2019.  Risk Evaluation of Agricultural Product Supply Chain Based on BP Neural Network. 2019 16th International Conference on Service Systems and Service Management (ICSSSM). :1–8.

The potential risk of agricultural product supply chain is huge because of the complex attributes specific to it. Actually the safety incidents of edible agricultural product emerge frequently in recent years, which expose the fragility of the agricultural product supply chain. In this paper the possible risk factors in agricultural product supply chain is analyzed in detail, the agricultural product supply chain risk evaluation index system and evaluation model are established, and an empirical analysis is made using BP neural network method. The results show that the risk ranking of the simulated evaluation is consistent with the target value ranking, and the risk assessment model has a good generalization and extension ability, and the model has a good reference value for preventing agricultural product supply chain risk.

2020-06-12
Gu, Feng, Zhang, Hong, Wang, Chao, Wu, Fan.  2019.  SAR Image Super-Resolution Based on Noise-Free Generative Adversarial Network. IGARSS 2019 - 2019 IEEE International Geoscience and Remote Sensing Symposium. :2575—2578.

Deep learning has been successfully applied to the ordinary image super-resolution (SR). However, since the synthetic aperture radar (SAR) images are often disturbed by multiplicative noise known as speckle and more blurry than ordinary images, there are few deep learning methods for the SAR image SR. In this paper, a deep generative adversarial network (DGAN) is proposed to reconstruct the pseudo high-resolution (HR) SAR images. First, a generator network is constructed to remove the noise of low-resolution SAR image and generate HR SAR image. Second, a discriminator network is used to differentiate between the pseudo super-resolution images and the realistic HR images. The adversarial objective function is introduced to make the pseudo HR SAR images closer to real SAR images. The experimental results show that our method can maintain the SAR image content with high-level noise suppression. The performance evaluation based on peak signal-to-noise-ratio and structural similarity index shows the superiority of the proposed method to the conventional CNN baselines.

2020-06-29
Blazek, Petr, Gerlich, Tomas, Martinasek, Zdenek.  2019.  Scalable DDoS Mitigation System. 2019 42nd International Conference on Telecommunications and Signal Processing (TSP). :617–620.
Distributed Denial of Service attacks (DDoS) are used by attackers for their effectiveness. This type of attack is one of the most devastating attacks in the Internet. Every year, the intensity of DDoS attacks increases and attackers use sophisticated multi-target DDoS attacks. In this paper, a modular system that allows to increase the filtering capacity linearly and allows to protect against the combination of DDoS attacks is designed and implemented. The main motivation for development of the modular filtering system was to find a cheap solution for filtering DDoS attacks with possibility to increase filtering capacity. The proposed system is based on open-source detection and filtration tools.
2020-01-20
Wu, Di, Chen, Tianen, Chen, Chienfu, Ahia, Oghenefego, Miguel, Joshua San, Lipasti, Mikko, Kim, Younghyun.  2019.  SECO: A Scalable Accuracy Approximate Exponential Function Via Cross-Layer Optimization. 2019 IEEE/ACM International Symposium on Low Power Electronics and Design (ISLPED). :1–6.

From signal processing to emerging deep neural networks, a range of applications exhibit intrinsic error resilience. For such applications, approximate computing opens up new possibilities for energy-efficient computing by producing slightly inaccurate results using greatly simplified hardware. Adopting this approach, a variety of basic arithmetic units, such as adders and multipliers, have been effectively redesigned to generate approximate results for many error-resilient applications.In this work, we propose SECO, an approximate exponential function unit (EFU). Exponentiation is a key operation in many signal processing applications and more importantly in spiking neuron models, but its energy-efficient implementation has been inadequately explored. We also introduce a cross-layer design method for SECO to optimize the energy-accuracy trade-off. At the algorithm level, SECO offers runtime scaling between energy efficiency and accuracy based on approximate Taylor expansion, where the error is minimized by optimizing parameters using discrete gradient descent at design time. At the circuit level, our error analysis method efficiently explores the design space to select the energy-accuracy-optimal approximate multiplier at design time. In tandem, the cross-layer design and runtime optimization method are able to generate energy-efficient and accurate approximate EFU designs that are up to 99.7% accurate at a power consumption of 3.73 pJ per exponential operation. SECO is also evaluated on the adaptive exponential integrate-and-fire neuron model, yielding only 0.002% timing error and 0.067% value error compared to the precise neuron model.

2020-09-18
Kleckler, Michelle, Mohajer, Soheil.  2019.  Secure Determinant Codes: A Class of Secure Exact-Repair Regenerating Codes. 2019 IEEE International Symposium on Information Theory (ISIT). :211—215.
{1 We present a construction for exact-repair regenerating codes with an information-theoretic secrecy guarantee against an eavesdropper with access to the content of (up to) ℓ nodes. The proposed construction works for the entire range of per-node storage and repair bandwidth for any distributed storage system with parameters (n
2020-03-09
Xiaoxin, LOU, Xiulan, SONG, Defeng, HE, Liming, MENG.  2019.  Secure estimation for intelligent connected vehicle systems against sensor attacks. 2019 Chinese Control Conference (CCC). :6658–6662.
Intelligent connected vehicle system tightly integrates computing, communication, and control strategy. It can increase the traffic throughput, minimize the risk of accidents and reduce the energy consumption. However, because of the openness of the vehicular ad hoc network, the system is vulnerable to cyber-attacks and may result in disastrous consequences. Hence, it is interesting in design of the connected vehicular systems to be resilient to the sensor attacks. The paper focuses on the estimation and control of the intelligent connected vehicle systems when the sensors or the wireless channels of the system are attacked by attackers. We give the upper bound of the corrupted sensors that can be corrected and design the state estimator to reconstruct the initial state by designing a closed-loop controller. Finally, we verify the algorithm for the connected vehicle system by some classical simulations.
2020-03-30
Ahamed, Md. Salahuddin, Asiful Mustafa, Hossen.  2019.  A Secure QR Code System for Sharing Personal Confidential Information. 2019 International Conference on Computer, Communication, Chemical, Materials and Electronic Engineering (IC4ME2). :1–4.
Securing and hiding personal confidential information has become a challenge in these modern days. Due to the lack of security and confidentiality, forgery of confidential information can cause a big margin loss to a person. Personal confidential information needs to be securely shared and hidden with the expected recipient and he should be able to verify the information by checking its authenticity. QR codes are being used increasingly to share data for different purposes. In information communication, QR code is important because of its high data capacity. However, most existing QR code systems use insecure data format and encryption is rarely used. A user can use Secure QR Code (SQRC) technology to keep information secured and hidden. In this paper, we propose a novel SQRC system which will allow sharing authentic personal confidential information by means of QR code verification using RSA digital signature algorithm and also allow authorizing the information by means of QR code validation using RSA public key cryptographic algorithm. We implemented the proposed SQRC system and showed that the system is effective for sharing personal confidential information securely.
2020-08-17
Garg, Hittu, Dave, Mayank.  2019.  Securing User Access at IoT Middleware Using Attribute Based Access Control. 2019 10th International Conference on Computing, Communication and Networking Technologies (ICCCNT). :1–6.
IoT middleware is an additional layer between IoT devices and the cloud applications that reduces computation and data handling on the cloud. In a typical IoT system model, middleware primarily connects to different IoT devices via IoT gateway. Device data stored on middleware is sensitive and private to a user. Middleware must have built-in mechanisms to address these issues, as well as the implementation of user authentication and access control. This paper presents the current methods used for access control on middleware and introduces Attribute-based encryption (ABE) on middleware for access control. ABE combines access control with data encryption for ensuring the integrity of data. In this paper, we propose Ciphertext-policy attribute-based encryption, abbreviated CP-ABE scheme on the middleware layer in the IoT system architecture for user access control. The proposed scheme is aimed to provide security and efficiency while reducing complexity on middleware. We have used the AVISPA tool to strengthen the proposed scheme.
2020-09-28
Homoliak, Ivan, Venugopalan, Sarad, Hum, Qingze, Szalachowski, Pawel.  2019.  A Security Reference Architecture for Blockchains. 2019 IEEE International Conference on Blockchain (Blockchain). :390–397.
Due to their specific features, blockchains have become popular in recent years. Blockchains are layered systems where security is a critical factor for their success. The main focus of this work is to systematize knowledge about security and privacy issues of blockchains. To this end, we propose a security reference architecture based on models that demonstrate the stacked hierarchy of various threats as well as threat-risk assessment using ISO/IEC 15408. In contrast to the previous surveys [23], [88], [11], we focus on the categorization of security vulnerabilities based on their origins and using the proposed architecture we present existing prevention and mitigation techniques. The scope of our work mainly covers aspects related to the nature of blockchains, while we mention operational security issues and countermeasures only tangentially.
2020-04-24
Rodriguez, Manuel, Fathy, Hosam.  2019.  Self-Synchronization of Connected Vehicles in Traffic Networks: What Happens When We Think of Vehicles as Waves? 2019 American Control Conference (ACC). :2651—2657.

In this paper we consider connected and autonomous vehicles (CAV) in a traffic network as moving waves defined by their frequency and phase. This outlook allows us to develop a multi-layer decentralized control strategy that achieves the following desirable behaviors: (1) safe spacing between vehicles traveling down the same road, (2) coordinated safe crossing at intersections of conflicting flows, (3) smooth velocity profiles when traversing adjacent intersections. The approach consist of using the Kuramoto equation to synchronize the phase and frequency of agents in the network. The output of this layer serves as the reference trajectory for a back-stepping controller that interfaces the first-order dynamics of the phase-domain layer and the second order dynamics of the vehicle. We show the performance of the strategy for a single intersection and a small urban grid network. The literature has focused on solving the intersection coordination problem in both a centralized and decentralized manner. Some authors have even used the Kuramoto equation to achieve synchronization of traffic lights. Our proposed strategy falls in the rubric of a decentralized approach, but unlike previous work, it defines the vehicles as the oscillating agents, and leverages their inter-connectivity to achieve network-wide synchronization. In this way, it combines the benefits of coordinating the crossing of vehicles at individual intersections and synchronizing flow from adjacent junctions.

2020-03-23
Hayashi, Masahito.  2019.  Semi-Finite Length Analysis for Secure Random Number Generation. 2019 IEEE International Symposium on Information Theory (ISIT). :952–956.
To discuss secure key generation from imperfect random numbers, we address the secure key generation length. There are several studies for its asymptotic expansion up to the order √n or log n. However, these expansions have errors of the order o(√n) or o(log n), which does not go to zero asymptotically. To resolve this problem, we derive the asymptotic expansion up to the constant order for upper and lower bounds of these optimal values. While the expansions of upper and lower bonds do not match, they clarify the ranges of these optimal values, whose errors go to zero asymptotically.