Exponentiation Operations Research Articles

Arithmetic Proficiency of Pre-Service Science Teachers: An Empirical Study

Mastery of arithmetic operations is fundamental for students pursuing science education, as these skills are essential in solving complex problems in physics, chemistry, and biology. However, gaps in students' arithmetic proficiency can hinder their academic and professional development. This study aims to examine the arithmetic skills of pre-service science teachers in solving mathematical problems across six domains: addition, subtraction, multiplication, division, exponentiation, and mixed operations. A total of 37 short-answer questions were administered, and the results were analyzed by domain. The findings indicate that students demonstrate proficiency in basic operations involving whole numbers, particularly in addition and subtraction. However, challenges persist in the areas of fractions, decimals, and mixed operations, where accuracy rates were notably lower. These gaps in understanding may affect students' ability to apply mathematical concepts in their science courses and future teaching roles. The study's limitations include a focus on quantitative results without exploring the cognitive processes behind student errors. Future research should investigate intervention strategies to address these weaknesses, potentially through targeted instructional approaches or the use of technology to enhance learning outcomes

Hamacher Operations on Interval Valued Picture Fuzzy Matrices

Picture Fuzzy Sets (PFS) extend fuzzy sets and intuitionistic fuzzy sets, offering a more robust framework for handling ambiguous, uncertain, and imprecise data. Interval-valued Picture Fuzzy Set represent membership, neutral, and non-membership degrees as intervals, providing a flexible solution for situations where precise values are difficult to determine. This paper introduces novel Hamacher operations for interval-valued picture fuzzy matrices, building upon existing Hamacher operations for Picture Fuzzy matrices. We develop and examine scalar multiplication and exponentiation operations using these new Hamacher operations. This research advances the development of sophisticated medical diagnosis systems, leading to improved patient outcomes and more accurate decision-making.

Constructing a Nondegenerate m-Dimensional Integer-Domain Chaotic Map Model over GF(2n) with Application in PRNG

To address the problem of dynamic degradation over a finite-precision platform of chaotic maps and the reversibility of linear chaotic maps, we propose an improved model over GF([Formula: see text]) that is called the nondegenerate m-Dimensional [Formula: see text] Integer-Domain Chaotic Maps (mD-IDCMs). This model incorporates modular exponentiation operation, and is capable of constructing nondegenerate IDCMs of any dimension. Moreover, we prove the irreversibility of mD-IDCM and analyze its chaotic behaviors in terms of positive Lyapunov Exponents (LEs). The results of theoretical analysis show that the proposed mD-IDCM model can obtain the desired positive LEs by appropriately configuring its coefficient matrix. Then, we present two instances, and analyze their LEs, Kolmogorov entropy, Sample entropy, Correlation dimension, and the dynamic analysis indicates that the chaotic map constructed by mD-IDCM has ergodicity within a sufficiently large chaotic range. Finally, we design a Pseudo-Random Number Generator (PRNG) with a key to verify the practicability of the mD-IDCM.

Cost-Effective Signcryption for Securing IoT: A Novel Signcryption Algorithm Based on Hyperelliptic Curves

Security and efficiency remain a serious concern for Internet of Things (IoT) environments due to the resource-constrained nature and wireless communication. Traditional schemes are based on the main mathematical operations, including pairing, pairing-based scalar multiplication, bilinear pairing, exponential operations, elliptic curve scalar multiplication, and point multiplication operations. These traditional operands are cost-intensive and require high computing power and bandwidth overload, thus affecting efficiency. Due to the cost-intensive nature and high resource requirements, traditional approaches are not feasible and are unsuitable for resource-limited IoT devices. Furthermore, the lack of essential security attributes in traditional schemes, such as unforgeability, public verifiability, non-repudiation, forward secrecy, and resistance to denial-of-service attacks, puts data security at high risk. To overcome these challenges, we have introduced a novel signcryption algorithm based on hyperelliptic curve divisor multiplication, which is much faster than other traditional mathematical operations. Hence, the proposed methodology is based on a hyperelliptic curve, due to which it has enhanced security with smaller key sizes that reduce computational complexity by 38.16% and communication complexity by 62.5%, providing a well-balanced solution by utilizing few resources while meeting the security and efficiency requirements of resource-constrained devices. The proposed strategy also involves formal security validation, which provides confidence for the proposed methodology in practical implementations.

Efficient privacy-preserving Gaussian process via secure multi-party computation

Gaussian processes (GPs), known for their flexibility as non-parametric models, have been widely used in practice involving sensitive data (e.g., healthcare, finance) from multiple sources. With the challenge of data isolation and the need for high-performance models, how to jointly develop privacy-preserving GP for multiple parties has emerged as a crucial topic. In this paper, we propose a new privacy-preserving GP algorithm, namely PP-GP, which employs secret sharing (SS) techniques. Specifically, we introduce a new SS-based exponentiation operation (PP-Exp) through confusion correction and an SS-based matrix inversion operation (PP-MI) based on Cholesky decomposition. However, the advantages of the GP come with a great computational burden and space cost. To further enhance the efficiency, we propose an efficient split learning framework for privacy-preserving GP, named Split-GP, which demonstrably improves performance on large-scale data. We leave the private data-related and SMPC-hostile computations (i.e., random features) on data holders, and delegate the rest of SMPC-friendly computations (i.e., low-rank approximation, model construction, and prediction) to semi-honest servers. The resulting algorithm significantly reduces computational and communication costs compared to PP-GP, making it well-suited for application to large-scale datasets. We provide a theoretical analysis in terms of the correctness and security of the proposed SS-based operations. Extensive experiments show that our methods can achieve competitive performance and efficiency under the premise of preserving privacy.

Vector finite fields of characteristic two as algebraic support of multivariate cryptography

The central issue of the development of the multivariate public key algorithms is the design of reversible non-linear mappings of $n$-dimensional vectors over a finite field, which can be represented in a form of a set of power polynomials. For the first time, finite fields $GF\left((2^d)^m\right)$ of characteristic two, represented in the form of $m$-dimensional finite algebras over the fields $GF(2^d)$ are introduced for implementing the said mappings as exponentiation operation. This technique allows one to eliminate the use of masking linear mappings, usually used in the known approaches to the design of multivariate cryptography algorithms and causing the sufficiently large size of the public key. The issues of using the fields $GF\left((2^d)^m\right)$ as algebraic support of non-linear mappings are considered, including selection of appropriate values of $m$ and $d$. In the proposed approach to development of the multivariate cryptography algorithms, a superposition of two non-linear mappings is used to define resultant hard-to-reverse mapping with a secret trap door. The used two non-linear mappings provide mutual masking of the corresponding reverse maps, due to which the size of the public key significantly reduces as compared with the known algorithms-analogues at a given security level.

PENGGABUNGAN KRIPTOGRAFI RIVEST SHAMIR ADLEMAN (RSA) DAN ADVANCED ENCRYPTION STANDARD (AES) PADA APLIKASI PENGIRIM E-MAIL

One of the most commonly used communication tools is e-mail. Messages sent via e-mail can contain confidential information, security measures are needed for confidential messages sent via e-mail. Cryptography is one of the security measures that can be taken. Based on the types of keys, there are two types of cryptography namely symmetric and asymmetric. Symmetric cryptography uses the same key in the encryption and decryption process, while asymmetric cryptography uses different keys. Symmetric keys can be encrypted by asymmetric keys to make the transmission of secret keys more secure. The symmetric and asymmetric cryptography chosen are AES (Advanced Encryption Standard) and enhanced RSA (Rivest Shamir Adleman). RSA cryptography is known for its strong security based on the difficulty of exponential and factorization operations, while AES cryptography is a secure cryptography standard that has a relatively fast computation time. An e-mail application to send secret messages using combined cryptography algorithm constructed using Python programming language.

Extended Diffie Hellman Protocol Based on Exponential and Multiplication Operation Over Modular Arithmetic

Extended Diffie Hellman Protocol Based on Exponential and Multiplication Operation Over Modular Arithmetic

Extended Diffie-Hellman protocol based on exponential and multiplication operations over modular arithmetic

Extended Diffie-Hellman protocol based on exponential and multiplication operations over modular arithmetic

A visual approximation of a Fourier transforms operation using GeoGebra

Many engineering fields use the Fourier Transform. The purpose of the Fourier Transform is to decompose periodic signals into fundamental frequencies and harmonic frequencies. The Fourier transform operation in ordinary textbooks is mathematically analytical, this often makes it difficult for students who study it. This paper describes the mathematical operations that occur in the Fourier transform. Discrete Fourier transform operations consist of conversion operations of continuous-time signals into data sequences, exponential complex number multiplication operations, and integration operations of number data sequences. Visualization of Fourier transform operations using the GeoGebra application. This research obtains visualization results of Fourier transform operations. The resulting visualization includes the effect of increasing signal frequency and the integral limit. The influence of the greater integral limit causes the peak of the frequency spectrum to become larger. The integral result of the pulse signal is in the form of a sinc function and corresponds to the illustration in the textbook. The limitation of the results of this research is that the Fourier transform visualization was carried out at low frequencies

Public Integrity Verification for Cloud Storage with Efficient Key-update

<p>To improve the security of the data on cloud storage, numbers of data integrity auditing schemes have been proposed in the past several years. However, there only a few schemes considered the security challenge that the user’s key is exposed unknowingly which is very likely to happen in real-life. To cope with the problem, we propose a public data integrity auditing scheme for cloud storage with efficient key updating. In our scheme, the user’s key is updated periodically to resist the risk of key exposure. Meanwhile, the authentication tags of blocks are updated simultaneously with the key updating so as to guarantee the data integrity can be verified normally. The algorithm of key updating in our scheme is very efficient which only needs a hash operation while previous schemes need two or three exponentiation operations. Moreover, the workload of tag updating is undertaken by cloud servers with a re-tag-key which reduces the burden of users and improves the efficiency of the scheme. The communication cost of the scheme is also reduced greatly, for instance, the information size in ‘re-key’ step is decreased from two group members to one. Furthermore, we give the formal security model of our scheme and prove the security under the CDH assumption. The experimental results show that our proposal is efficient and feasible.</p> <p> </p>

Efficient and Flexible Multiauthority Attribute-Based Authentication for IoT Devices

The correctness and reliability of data sources are the keys to the practicality of data collected by IoT devices. Attribute-based signature(ABS) is a cryptographic primitive for users to sign with their own attributes, which can be applied to the authentication process in IoT scenarios. The attribute authority is responsible for issuing the attribute key to the user in ABS. Multiple authorities can complete attribute management tasks to avoid the threat of a single authority. However, attribute authorities need to execute multiple interactions to collaborate to generate attribute keys for users, which brings a large transmission burden. In addition, a lot of resource-constrained terminals in the IoT mostly play the role of signer or verifier in authentication protocols. The signature generation and verification algorithms often have heavy pairing and exponentiation operations. Currently, no ABS scheme takes into account the efficiency of all participating entities simultaneously. In this paper, we present an aggregated anonymous key issue (AAKI) protocol to reduce the transmission burden between multiple authorities. Meanwhile, the non-interactive zero-knowledge proof aggregate exponentiation (NI-ZKPoKAE) protocol is designed to aggregate the transmitted secret values in AAKI. To reduce the computational burden of signers and verifiers, Blakley secret sharing, where the Hadamard matrix is used more efficiently to handle the (n,n)-threshold, is used to construct an efficient and fine-grained multi-authority ABS (EFMA-ABS) scheme. This brings high efficiency to all three types of parties involved in IoT authentication. Our above-mentioned protocols have been proven to be feasible and effective.

PPS-FPCM: PRIVACY-PRESERVING SEMI-FUZZY POSSIBILISTIC C-MEANS

Applying traditional clustering techniques to big data on the cloud while preserving the privacy of the data is a challenge due to the required division and exponential operations in each iteration, which complicate its implementation on encrypted data. Several existing approaches are based on approximating the formulas of centers, weights, and memberships as three polynomial functions according to the multivariate Taylor formula. However, they usually suffer an increase in complexity and a slight drop in accuracy. In this paper, a novel Privacy-Preserving semi-fuzzy clustering algorithm based on the possibilistic paradigm, termed PPS-FPCM, is presented. Its main feature is that it avoids exponentiation and division operations, at each iteration, without losing accuracy. By restricting the typicality to an ordered set of discrete values between zero and one decided by the data owner (DO), the computation is simplified. The second key idea is the use of this soft typicality to detect outliers and compute the corresponding semi-fuzzy memberships, which is used to increase the in-between cluster distance. However, the initial typicality requires a magnitude relation comparison, which is still difficult to do over encrypted data. In this research study, we show how the existing incomplete re-encryption method can be used to tackle this problem. In each iteration, centers and distances to the new centers are computed on a calculator cloud server (CaCS) which is responsible for storing the cipher texts of the (DO)’s data and processing them. Then, CaCS sends the incompletely re-encrypted difference between these distances and iteratively updated bin values that correspond to the discrete possibilistic memberships that are initially decided by the (DO) to the comparator cloud server (CoCS). CoCS decrypts the difference and returns the results of comparisons. When CaCS receives the results of comparison from CoCS, it decides on an appropriate soft typicality or resends the difference of the same distance to another bin value. The required number of comparisons is O(log the number of bins). CaCS iteratively computes the corresponding semi-fuzzy memberships, computes the refined memberships, and updates the centers. In the end, CaCS sends the final soft memberships and centers to the (DO). The proposed algorithm is applicable to normal data and homomorphically encrypted data, is more effective than several related algorithms, and can produce accurate results using large enough (16 or more) discrete values with a high reduction on runtime as the number of comparisons is much less complex than exponential and division operations with added communication cost between CaCS and CoCS.

Antiquantum Privacy Protection Scheme in Advanced Metering Infrastructure of Smart Grid Based on Consortium Blockchain and RLWE

Smart meters in advanced metering infrastructure (AMI) collect fine-grained, real-time power consumption data may result in user privacy leakage. Aiming at the problem that existing privacy protection schemes cannot resist quantum attacks, an antiquantum privacy protection scheme in AMI of smart grid based on consortium blockchain and ring learning with errors (RLWE) is proposed. The scheme has the following contributions. First, the RLWE-based secure communication scheme is designed in this article, which ensures the resistance of the scheme to quantum attacks. Second, all smart meters in AMI use anonymous identity to communicate with the control center, and the fine-grained power consumption data is also encrypted and digitally signed to prevent user privacy leakage and tampering of power consumption data. And the scheme does not use bilinear pairings and exponential operations and has efficient computational performance. Third, a two-tier consortium blockchain storage system consisted of regional gateways and data aggregators are designed to eliminate the single point of failure problems of the data aggregators. The practical byzantine fault tolerance consensus mechanism is improved to reduce the communication cost in the consensus process. The scheme's security is proven theoretically, and the experimental results show that the scheme has a low computational cost.

A Lightweight, Secure Big Data-Based Authentication and Key-Agreement Scheme for IoT with Revocability

With the rapid development of Internet of Things (IoT), designing a secure two-factor authentication scheme for IoT is becoming increasingly demanding. Two-factor protocols are deployed to achieve a higher security level than single-factor protocols. Given the resource constraints of IoT devices, other factors such as biometrics are ruled out as additional authentication factors due to their large overhead. Smart cards are also prone to side-channel attacks. Therefore, historical big data have gained interest recently as a novel authentication factor in IoT. In this paper, we show that existing big data-based schemes fail to achieve their claimed security properties such as perfect forward secrecy (PFS), key compromise impersonation (KCI) resilience, and server compromise impersonation (SCI) resilience. Assuming a real strong attacker rather than a weak one, we show that previous schemes not only fail to provide KCI and SCI but also do not provide real two-factor security and revocability and suffer inside attack. Then, we propose our novel scheme which can indeed provide real two-factor security, PFS, KCI, and inside attack resilience and revocability of the client. Furthermore, our performance analysis shows that our scheme has reduced modular exponentiation operation and multiplication for both the client and the server compared to Liu et al.’s scheme which reduces the execution time by one third for security levels of λ = 128 . Moreover, in order to cope with the potential threat of quantum computers, we suggest using lightweight XMSS signature schemes which provide the desired security properties with λ = 128 bit postquantum security. Finally, we prove the security of our proposed scheme formally using both the real-or-random model and the ProVerif analysis tool.

Einstein exponential operation laws of spherical fuzzy sets and aggregation operators in decision making.

The spherical fuzzy set (SFS) model is one of the newly developed extensions of fuzzy sets (FS) for the purpose of dealing with uncertainty or vagueness in decision making. The aim of this paper is to define new exponential and Einstein exponential operational laws for spherical fuzzy sets and their corresponding aggregation operators. We introduce the operational laws for exponential and Einstein exponential SFSs in which the base values are crisp numbers and the exponents (weights) are spherical fuzzy numbers. Some of the properties and characteristics of the proposed operations are then discussed. Based on these operational laws, some new aggregation operators for the SFS model, namely Spherical Fuzzy Weighted Exponential Averaging (SFWEA) and Spherical Fuzzy Einstein Weighted Exponential Averaging (SFEWEA) operators are introduced. Finally, a decision-making algorithm based on these newly introduced aggregation operators is proposed and applied to a multi-criteria decision making (MCDM) problem related to ranking different types of psychotherapy.

Efficient Revocable Storage Attribute-based Encryption With Arithmetic Span Programs in Cloud-Assisted Internet of Things

Revocable storage and efficient description of the access policy are necessary to enhance the practicality of the attribute-based encryption (ABE) in real-life scenarios, such as cloud-assisted Internet of Things (IoT). Nevertheless, existing ABE works fail to balance the two vital factors. In this paper, we construct an efficient revocable storage ciphertext-policy attribute-based encryption with arithmetic span programs (RS- CPABE-ASP). The arithmetic span program (ASP) is elegantly utilized as the access structure to reduce the unnecessary cost for defining access policy. Combining the indirect revocation and the ciphertext update mechanism, our work prevents the revoked user unable to access the newly generated data and the old data that can be accessed before. As shown in the outsourced version of RS-CPABE-ASP, the costly part for users to decrypt the data can be outsourced to powerful cloud servers. In this way, users in our RS-CPABE-ASP are able to access their data in a more efficient way by merely one exponential operation. Finally, we carry out detailed theoretical analysis and experimental simulations to evaluate the performance of our work. The results fairly show that our proposed work is efficient and feasible in cloud-assisted IoT.

A new type of digital signature algorithms with a hidden group

The known designs of digital signature schemes with a hidden group, which use finite non-commutative algebras as algebraic support, are based on the computational complexity of the so-called hidden discrete logarithm problem. A similar design, used to develop a signature algorithm based on the difficulty of solving a system of many quadratic equations in many variables, is introduced. The significant advantage of the proposed method compared with multivariate-cryptography signature algorithms is that the said system of equations, which occurs as the result of performing the exponentiation operations in the hidden group, has a random look and is specified in a finite field of a higher order. This provides the ability to develop post-quantum signature schemes with significantly smaller public-key sizes at a given level of security.

Algebraic signature algorithms with a hidden group group, based on hardness of solving systems of quadratic equations

A new-type algebraic digital signature schemes on non-commutative associative algebras are developed using technique of performing exponentiation operations in a hidden group. The signature contains two elements: a randomization integer e and a vector S. The used verification equations are characterized in multiple entries of the signature element S. The post-quantum security of the introduced signature algorithms is provided by the computational difficulty of solving a system of many quadratic equations in many variables, like in the public-key multivariate cryptosystems. However in the former case the quadratic equations are set over the finite fields having the order of significantly larger size.

Efficient and Strong Symmetric Password Authenticated Key Exchange With Identity Privacy for IoT

Password authenticated key exchange (PAKE) allows two parties with a shared password to establish a session key. In order to provide secure and private communication between devices in an Internet of Things (IoT) environment, the PAKE protocol is considered one of the most common and promising security methods. However, many existing PAKE proposals still face challenges in security and efficiency. First, both the terminals of participants may suffer from the compromise attack and precomputation attack, which will lead to the leakage of password. Second, the majority of the existing schemes cannot guarantee participants’ identity privacy. Third, most PAKE protocols are not suitable for IoT devices with limited computing capability because of a large number of exponential and pairing operations. To address these issues, we propose a strong symmetric PAKE protocol for IoT devices, which only requires 3 exponentiations per party. The proposed scheme can protect both parties from compromise and precomputation attacks, in which the password file relies on the identity and random salt. What’s more, our protocol guarantees the identity privacy, that is, the transmitted record and password file will not reveal the identity information. We further present a new security model for our protocol, and prove that the proposed scheme is secure under this model. Finally, we show the practicality of our PAKE via experiments and efficiency analysis.