Large Prime Numbers Research Articles

New Constructions of Existential Unforgeable Aggregate Signature Scheme from CSP

In future, hundreds of years of mathematical problems that the security of public key cryptography algorithms rely on may be defeated by quantum algorithms. How can a digital signature scheme gracefully balance security and efficiency? This study uses the conjugate search problem and the left self-distributive system to combine and uses the RSA-like algorithm as the underlying structure to propose a new aggregated signature scheme. We, through the EUF game, under the random metaphor model, prove that the security of the scheme satisfies the adaptation unforgeability under selective message attack, the scheme can be finally reduced to the discrete logarithm problem or large prime number decomposition problem. In addition, we can achieve antiquantum attack and exhaustive attack by performing matrix calculations on the message, defining and changing the structure of the matrix by encoding, and setting thresholds for the matrix dimension and the length of the private key. In terms of efficiency, the message signature implementation is linear compared with the expansion rate in terms of storage and computing overhead, and the generation and verification of the final signature pair have nothing to do with the number of users. In addition, the length of the signature is fixed and the size is only the length of a single group, which effectively reduces the generation of public and private key pairs and saves a lot of storage space. The storage space and computational complexity are also effectively improved compared with other solutions.

Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers

In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P1 = 1 + 6 * n (n > 0, n is an integer) P5 = 5 + 6 * m (m ≥ 0, m is an integer) Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Twin prime numbers can only be formed between P1 and P5 prime numbers, n must be 1 greater than m. The largest prime number is known at the moment 2^(82,589,933) – 1 is a P1 prime number.

A secure IoT based healthcare framework using modified RSA algorithm using an artificial hummingbird based CNN

AbstractRecently, a wide variety of real‐world problems are solved via cloud computing, deep learning, machine learning, artificial intelligence, and the Internet of Things (IoT). These methodologies are concerned with different areas like smart cities, agriculture, transportation systems, and healthcare systems. The existing researchers focused on the health care monitoring application along with IoT and cloud computing. It met several shortcomings in case of computational complexity, cost, time, and improper health care data storage and so forth. To overcome these challenges, the novel IoT‐enabled secure healthcare monitoring model is proposed in this study. At first, several sensors are deployed in the human body thereby collecting patients' data with respect to vital parameters like body temperature deviation. The patient's health record dataset consists of 10 attributes namely phone numbers, marital status, address, age, name, heartbeat rate, oxygen level, smoking, temperature, and blood pressure. The data size reduction and normalization are performed in the IoT medical sensor dataset which contains the redundant or irrelevant attributes that are eliminated during pre‐processing. The artificial hummingbird (AHB) algorithm‐based convolutional neural network (AHB‐CNN) model performs both feature extraction and classification of cancer disease. The AHB‐CNN model classifies whether the patient is prone to cancer or not based on the sensor input collected. The received results are then sent to the hospital management for analysis. The Rivest‐Shamir‐Adleman (RSA) encryption method is mostly used in this study due to the major benefits it gives in terms of asymmetric encryption, ease of use, simpler deployment, and high security associated with factoring large prime numbers. A modified RSA algorithm is used in this article which uses the double encryption‐decryption process and “n” prime numbers to enhance the security of the conventional RSA algorithm. The data is always encrypted during transit before being stored in the cloud or for further processing. Depending upon the experimental consequences, the proposed method established superior performances compared to other state‐of‐art techniques.

About the theorem that partially solves the Navarrete–Orellana Conjecture

The Navarrete–Orellana Conjecture states that “given a large prime number a sequence is generated, in such a way that all odd prime numbers, except the given prime, are fixed points of that sequence”. In this work, we formulated a theorem that partially confirms the veracity of this conjecture, more specifically, all prime numbers of a given line segment are fixed points of this sequence.

Comment on “An enhanced and secured RSA public cryptosystem algorithm using Chinese remainder theorem (ESRPKC)”

In [1], Kumar et al. proposed an enhanced and secured RSA public key cryptosystem (ESRPKC) algorithm using Chinese remainder theorem. In their scheme, the public key is defined as (N,e,μ) where N is the product of four distinct large prime numbers, e is a public exponent, and μ is a parameter called encryption key to encrypt the message. Compared to the traditional RSA cryptosystem, the authors used the extra parameter μ, and claimed that security increased due to this extra parameter. They claimed that it is required to use a brute force attack to break the system even if the number N is factorized by an adversary to obtain the private parameters k1 and k2 which are components of μ. The authors claimed that ESRPKC is a highly secure and not easily breakable scheme compared to the traditional RSA scheme.In this paper, we do a cryptanalysis on Kumar et al.'s scheme given in [1] and demonstrate some major security weaknesses. We prove that if N is factorized, there is no need to use brute force to break the system. Additionally, choosing four prime numbers instead of two prime numbers decreases the security significantly and only increases the computation time. Therefore, the proposed system (ESRPKC) is not as efficient as the traditional RSA algorithm.

Two-factor-based RSA key generation from fingerprint biometrics and password for secure communication

In an asymmetric-key cryptosystem, the secure storage of private keys is a challenging task. This paper proposes a novel approach for generating the same public and private key pair on a need basis. Hence, the need for secure storage of the private key is done away with. The proposed approach for generating the key pair is based on two factors: fingerprint biometrics and password. A stable binary string is generated from the distances among pairs of minutiae points in a fingerprint using a gray code-based method. Experiments show that gray code representation significantly reduces the number of inconsistencies between the generated bit strings from two instances of the same fingerprint as against the binary code representation. Hence, the Reed–Solomon error correction code successfully corrects errors due to variations in multiple instances of the same fingerprint to induce stability in the generated string. Hash of the stable string generated from the fingerprint and the string generated from hashed password are XORed to derive a stable seed value. The proposed approach uses this seed value to generate two large prime numbers. These prime numbers are used to generate the public and private key pair using the RSA key generation method. This seed value ensures the generation of the same key pair every time. The experimental results show that the proposed approach can ensure a stable generation of the key. It is not required to store either the fingerprint template or the password. Moreover, the generated private key is also not stored. It can be regenerated on a need basis.

Quantum Cryptography and its Application

Present cryptography algorithms are based on mathematical problem such as factoring two large prime numbers and N vs NP problem. But computing power and evolution in mathematics increases day by day so it’s easy to reverse one way function to quickly such as that of factoring large integers. So the solution is to build a cryptosystem which secure our communication even the computing power is to high so we introduce quantum physics in cryptography which lead to evaluation of quantum cryptography. Quantum cryptography one of the emerging topics in field of computer industry. This paper is an attempt to review fundamentals of quantum cryptography and also analyses few application areas of quantum cryptography and it’s limitation.

Enhanced RSA key encryption application for metering data in smart grid

Purpose Digitizing of the electrical power grid promotes the advantages of efficient energy management alongside the possibilities of major vulnerabilities. A typical inadequacy that needs critical attention to ensure the seamless operation of the smart grid system remains in the data transmission between consumer premises smart devices and the utility centres. Many researches aim at establishing security protocols to ensure secure and efficient energy management resulting in perfect demand–supply balance. Design/methodology/approach In this paper, the authentication of the smart meter data has been proposed with enhanced Rivest–Shamir–Adleman (RSA) key encryption using an efficient way of generating large prime numbers. The trapdoor one-way function applied in the RSA algorithm makes it almost impossible for the reverse engineering attempts of cracking the key pair. Findings The algorithm for generating prime numbers has been tested both with the convention method and with the enhanced method of including a low-level primality test with a first few hundred primes. The combination of low-level and high-level primality tests shows an improvement in execution time of the algorithm. Originality/value There is a considerable improvement in the time complexities when using the combination method. This efficient generation of prime numbers can be successfully applied to the smart meter systems, thereby increasing the strength and speed of the key encryption.

Some Results on Integer Solutions of Quadratic Polynomials in Two Variables

Although it is true that there are several articles that study quadratic equations in two variables, they do so in a general way. We focus on the study of natural numbers ending in one, because the other cases can be studied in a similar way. We have given the subject a different approach, that is why our bibliographic citations are few. In this work, using basic tools of functional analysis, we achieve some results in the study of integer solutions of quadratic polynomials in two variables that represent a given natural number. To determine if a natural number ending in one is prime, we must solve equations (i) <img src=image/13422919_01.gif>, (ii) <img src=image/13422919_02.gif>, (iii) <img src=image/13422919_03.gif>. If these equations do not have an integer solution, then the number P is prime. The advantage of this technique is that, to determine if a natural number p is prime, it is not necessary to know the prime numbers less than or equal to the square root of p. The objective of this work was to reduce the number of possibilities assumed by the integer variables <img src=image/13422919_04.gif> in the equation (i), (ii), (iii) respectively. Although it is true that this objective was achieved, we believe that the lower limits for the sums of the solutions of equations (i), (ii), (iii), were not optimal, since in our recent research we have managed to obtain limits lower, which reduce the domain of the integer variables <img src=image/13422919_04.gif> solve equations (i), (ii), (iii), respectively. In a future article we will show the results obtained. The methodology used was deductive and inductive. We would have liked to have a supercomputer, to build or determine prime numbers of many millions of digits, but this is not possible, since we do not have the support of our respective authorities. We believe that the contribution of this work to number theory is the creation of linear functionals for the study of integer solutions of quadratic polynomials in two variables, which represent a given natural number. The utility of large prime numbers can be used to encode any type of information safely, and the scheme shown in this article could be useful for this process.

Universal Evolutionary Model for Periodical Species

Real-world examples of periodical species range from cicadas, whose life cycles are large prime numbers, like 13 or 17, to bamboos, whose periods are large multiples of small primes, like 40 or even 120. The periodicity is caused by interaction of species, be it a predator-prey relationship, symbiosis, commensalism, or competition exclusion principle. We propose a simple mathematical model, which explains and models all those principles, including listed extremal cases. This rather universal, qualitative model is based on the concept of a local fitness function, where a randomly chosen new period is selected if the value of the global fitness function of the species increases. Arithmetically speaking, the different interactions are related to only four principles: given a couple of integer periods either (1) their greatest common divisor is one, (2) one of the periods is prime, (3) both periods are equal, or (4) one period is an integer multiple of the other.

Collective Signature Protocols for Signing Groups based on Problem of Finding Roots Modulo Large Prime Number

Generally, digital signature algorithms are based on a single difficult computational problem like prime factorization problem, discrete logarithm problem, elliptic curve problem. There are also many other algorithms which are based on the hybrid combination of prime factorization problem and discrete logarithm problem. Both are true for different types of digital signatures like single digital signature, group digital signature, collective digital signature etc. In this paper we propose collective signature protocols for signing groups based on difficulty of problem of finding roots modulo large prime number. The proposed collective signatures protocols have significant merits one of which is connected with possibility of their practical using on the base of the existing public key infrastructures.

COLLECTIVE SIGNATURE PROTOCOLS FOR SIGNING GROUPS BASED ON PROBLEM OF FINDING ROOTS MODULO LARGE PRIME NUMBER

Generally, digital signature algorithms are based on a single difficult computational problem like prime factorization problem, discrete logarithm problem, elliptic curve problem. There are also many other algorithms which are based on the hybrid combination of prime factorization problem and discrete logarithm problem. Both are true for different types of digital signatures like single digital signature, group digital signature, collective digital signature etc. In this paper we propose collective signature protocols for signing groups based on difficulty of problem of finding roots modulo large prime number. The proposed collective signatures protocols have significant merits one of which is connected with possibility of their practical using on the base of the existing public key infrastructures.

Public key image encryption algorithm based on pixel information and random number insertion

In this study, a public key image encryption algorithm is proposed based on pixel information and random number insertion. In the first step, two large prime numbers and the public key are selected randomly to produce the private key. According to the pixel information of the plain image and utilizing the Rivest-Shamir-Adleman (RSA) algorithm, the corresponding cipher information can be obtained by using the public key to encrypt the plain information extracted from the plain image. Both the plain and cipher information are then converted into the initial keys of the chaotic system based on a key transformation mapping. Consequently, the keystreams are generated based on iterative calculations. In the second step, different keystreams are selected for pixel arrangement and modulo operation preprocessing for the plain image. Subsequently, random numbers are inserted in the preprocessed image to obtain an extended image. The final cipher image can be got by selecting different keystreams to encrypt the extended image using XOR diffusion and add modulus operations. Combined with public key cryptosystem RSA, the proposed algorithm can achieve the effect of a one-time pad. Finally, experimental test results show that the cipher image obtained using our algorithm is highly random and robust, and can effectively resist violent, statistical, and differential attacks. A series of security analyses are implemented to validate the proposed asymmetric image encryption algorithm.

Implementation of Elliptic Net Scalar Multiplication Computation for NIST P-192 Curve using Python

Elliptic curve cryptography is one of the most efficient public-key cryptosystems compared to the Rivest-Shamir-Addleman scheme. One of the methods to compute elliptic curve scalar multiplication is division polynomials which utilize the non-linear recurrence relation also known as the elliptic net. Previous studies were related to elliptic curve scalar multiplication via elliptic net which implemented the use of C++, GP/PARI or Java programming. This study aims to implement the Python programming in the SageMath compiler for computing elliptic net scalar multiplication based on the National Institute of Standards and Technology curve of the type P-192. The implementation could be seen at three main processes which are the scalar decomposition, the blocks of elliptic net generation namely the initial and final blocks, as well as scalar multiplication computation. The base point and prime field of the short Weierstrass curve with a large scalar are the parameters used in the process. The results showed that the implemented programming was easier while working with a large prime number field, vulnerable to errors and can be easily verified. The implemented programming can also be used to compute scalar multiplication on other standardizations of elliptic curve cryptography such as Brainpool 384-bit prime field or NUMS 256-bit prime field curves.

A Cryptographic based Approach for Data Hiding in Advanced Video Sequences

Data hiding in video streams became more popular in the present world, since there is a high frequency of data communication over the internet. Hiding the data in video streams provides more security as well as increases embedding capacity than hiding inside the images. The quantity of information to be embedded into the video increases, it can badly influence the video excellence make it inappropriate for certain appliances. The main concerns in data hiding in videos are its high visual excellence, increased hiding capacity, video stream size etc. In this paper, a new data hiding technique is proposed in compressed H.264 Video Streams. At first, the information to be embedded is encrypted using Cryptography approach. The Cryptographic approach helps to encrypt the plain information based on the elliptic points produced by choosing the large prime number. The encrypted data is embedded into the transformed DCT coefficients of I, B and P video frames. The experiment is conducted for different set of video sequences. The results shows that the proposed method yields better performance in terms of Peak signal to noise ratio (PSNR), Structural similarity index (SSIM) and Video quality measure (VQM) when compare to existing methods.

Curve25519 based lightweight end-to-end encryption in resource constrained autonomous 8-bit IoT devices

Robust encryption techniques require heavy computational capability and consume large amount of memory which are unaffordable for resource constrained IoT devices and Cyber-Physical Systems with an inclusion of general-purpose data manipulation tasks. Many encryption techniques have been introduced to address the inability of such devices, lacking in robust security provision at low cost. This article presents an encryption technique, implemented on a resource constrained IoT device (AVR ATmega2560) through utilizing fast execution and less memory consumption properties of curve25519 in a novel and efficient lightweight hash function. The hash function utilizes GMP library for multi-precision arithmetic calculations and pre-calculated curve points to devise a good cipher block using ECDH based key exchange protocols and large random prime number generator function.

McGrath, Sarah. Moral Knowledge. Oxford: Oxford University Press, 2020. Pp. 240. $65.00 (cloth).

McGrath, Sarah. <i>Moral Knowledge</i>. Oxford: Oxford University Press, 2020. Pp. 240. $65.00 (cloth).

A Blockchain Framework in Post-Quantum Decentralization

The decentralization and transparency have provided wide acceptance of blockchain technology in various sectors through numerous applications. The claimed security services by blockchain have been proved using various cryptographic techniques, mainly public key infrastructure and digital signatures. However, the use of generic cryptographic primitives using large prime numbers or elliptic curves with logarithms is going to be an issue with quantum computers as those techniques are vulnerable in post-quantum era. Therefore, the paradigm shift from pre-quantum to the post-quantum era has necessitated new cryptographic developments which are robust against quantum attacks and applicable in blockchain for post-quantum decentralization. Therefore, we have presented a solution for post-quantum decentralization in the blockchain. It uses lattices with polynomials for identity-based encryption (IBE) and aggregate signatures for the consensus to ensure efficiency and suitability in post-quantum blockchain applications. We experiment the proposed approach based on delay, throughput, energy consumption and complexity. The comparative results prove that the presented work is efficient.

Cloud Storage Protection Scheme Based on Fully Homomorphic Encryption

Cloud computing allows enterprises and individuals to have a less physical infrastructure of software and hardware. Nevertheless, there are some concerns regarding privacy protection which may turn out to be a strong barrier. Traditional encryption schemes have been used to encrypt the data before sending them to the cloud. However, the private key has to be provided to the server before any calculations on the data. To solve this security problem, this paper proposes a fully homomorphic encryption scheme for securing cloud data at rest. The scheme is based on prime modular operation, its security depends on factoring multiple large prime numbers (p1, p2,...pn) up to n, which is formed from very large prime numbers up to hundreds of digits as this is an open problem in mathematics. In addition, the elements of the secret key are derived from a series of mathematical operations and the calculation of an Euler coefficient within the modular of integers. Furthermore, it adds the complexity of noise to the plaintext using the number of users of the Cloud Service Provider. Moreover, its randomness is evaluated by the National Institute of Standards and Technology statistical tests, and the results demonstrating that the best statistical performance was obtained with this algorithm.

Building an Elliptic Curve Cryptography to Encode and Decode Vietnamese Texts

This article presents building an Elliptic curve cryptography and using it to encode and decode Vietnamese text. Here we have illustrated the prime number p = 151 in the future, which will use a large prime number. We consider an elliptic curve with a total score of 172 points. Encode and decode with standard Vietnamese text and combine with the special characters in ASCII code. The program is designed and installed and on the C# environment to give the correct result of the encryption algorithm.&#x0D; KeywordsData sequence, Decryption, Discrete logarithm, Elliptic curve, Elliptic curve cryptosystem, Encryption, Public key&#x0D; References[1] Koblitz, “Elliptic curve cryptosystems”, Mathematics of Computation”, 203 - 209, 1987.[2] Miller, “Uses of elliptic curves in cryptography, Advances in Cryptology - Crypto”, Lecture Notes in Computer Science, SpringerVerlag, 1986, pp. 417-426.[3] Sugantha Priya, Dr.M. Mohanraj, “A Review on Secure Elliptic Curve Cryptography (ECC) and Dynamic Secure Routing Link Path Detection Algorithm (DSRLP) Under Jamming Attack”, ISSN 68(30) (2020) 0474-9030.[4] Negin Dinarvand, Hamid Barati, “An efficient and secure RFID authentication protocol using ellipticcurvecryptography”,SpringerfScience+Business Media, LLC, 2017[5] Utku Gulen, Selcuk Baktir, “Elliptic Curve Cryptography for Wireless Sensor Networks Using the Number Theoretic Transform”, journal-sensors, Published: 9 March, 2020.[6] Sravana Kumar, C.H. Suneetha, A.R. Chandrasekh, “Encryption of Data Using Elliptic Curve Over Finite Fields”, International Journal of Distributed and Parallel Systems (IJDPS). 3(1) (2012) 301-308.[7] Amounas, E.H. El Kinani, ECC Encryption and Decryption with a Data Sequence, Applied Mathematical Sciences 6(101) (2012) 5039-5047.[8] Vu Thi Hai Ha, Dinh Thi Hang, Bui Dang Binh, “The influence of volume on the formant of vowels and the identification of Vietnamese speakers”, Vietnam Institute of Linguistics, 2015.[9] Enge, “Elliptic curves and their applications to cryptography”, Norwell, MA: Kulwer Academic publishers, 1999.[10] Neil Koblitz, “An Elliptic Curve implementation of the finite field digital signature algorithm”, in Advances in cryptology,(CRYPTO 1998), SpringerLecture Notes in computer science, 1462 (1998) 327-337.[11] S. Sandeep, Kumar, “Elliptic curve cryptography for constrained devices”, PhD thesis, Ruhr-University Bochum, June, 2006.