Large Prime Numbers Research Articles

A New Cryptographic Key Planning Algorithm Based on Blum Blum Shub

The Blum Blum Shub (BBS) algorithm is one of the known powerful pseudo random number generators. This algorithm can be used for key generation. BBS is basically based on the product of two large prime numbers and a seed value. The selection of these values is a critical issue. In this study, a new approach is proposed to overcome this problem. In the proposed approach, a prime number pool is first created. At this point, the user sets a start and end value. The primes in this range are generated and stored in an array. Then, two primes are randomly selected from this prime number pool with chaotic maps. The positions of these prime numbers in the array are recorded. The seed value is taken as the sum of the positions of these two primes. In other words, the parameters to be selected will be randomly selected in the ranges that the user will enter at that moment. In this study, two random bit sequences were obtained in this way. These sequences are 1 million bits long. NIST SP 800-22 tests were applied to these sequences and the sequences successfully completed all tests.

An Approach for Cipher Communication Utilizing the Concepts of Elliptic Curve Cryptography

Message encryption is a process of protection of the original data in transmission. Several cryptosystems were designed in the history for different environments. Public key or Asymmetric cryptosystem has become more popular in recent era where the key space consists both public and private keys. Elliptic curve cryptography dominates public key cryptosystems like Diffie Hellman key exchange and RSA. ECC is more powerful and alternative technique of RSA, with a simple replacement of exponent calculations of large prime numbers to mathematical representation of Elliptic curve points. Besides the computation ECC works with shorter key length than RSA. The present paper describes an encryption algorithm using elliptic curves over finite fields. Here a group of authentic users transmit.

A new bound for A(A + A) for large sets

For a large prime number p and a set A⊂Fp we prove the following:(1)If A(A+A) does not cover all nonzero residues in Fp, then |A|⩽p/8+o(p).(2)If A is both sum-free and satisfies A=A⁎, then |A|⩽p/9+o(p).(3)If |A|≫log⁡log⁡plog⁡pp, then |A+A⁎|⩾(1+o(1))min⁡(2|A|p,p). Here the constants 1/8, 1/9, 2 are the best possible. Proofs make use of wrappers, subsets of a finite abelian group G, which ‘wrap’ popular values in convolutions of dense sets A,B⊆G. These objects carry certain structural features, making them capable of addressing additive-combinatorial and enumerative problems.

Investigating and improving student understanding of the basics of quantum computing

Quantum information science and engineering (QISE) is a rapidly developing field that leverages the skills of experts from many disciplines to utilize the potential of quantum systems in a variety of applications. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of QISE, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a Quantum Interactive Learning Tutorial, on the basics and applications of quantum computing. These include an overview of key quantum mechanical concepts relevant to quantum computation (including ways a quantum computer is different from a classical computer), properties of single- and multiqubit systems, and the basics of single-qubit quantum gates. The tutorial uses guided inquiry-based teaching-learning sequences. Its development and validation involved conducting cognitive task analysis from both expert and student perspectives and using common student difficulties as a guide. For example, before engaging with the tutorial, after traditional lecture-based instruction, one reasoning primitive that was common in student responses is that a major difference between an N-bit classical and N-qubit quantum computer is that various things associated with a number N for a classical computer should be replaced with the number 2N for a quantum computer (e.g., 2N qubits must be initialized and 2N bits of information are obtained as the output of the computation on the quantum computer). This type of reasoning primitive also led many students to incorrectly think that there are only N distinctly different states available when computation takes place on a classical computer. Research suggests that this type of reasoning primitive has its origins in students learning that quantum computers can provide exponential advantage for certain problems, e.g., Shor’s algorithm for factoring products of large prime numbers, and that the quantum state during the computation can be in a superposition of 2N linearly independent states. The inquiry-based learning sequences in the tutorial provide scaffolding support to help students develop a functional understanding. The final version of the validated tutorial was implemented in two distinct courses offered by the physics department with slightly different student populations and broader course goals. Students’ understanding was evaluated after traditional lecture-based instruction on the requisite concepts and again after engaging with the tutorial. We analyze and discuss their improvement in performance on concepts covered in the tutorial. Published by the American Physical Society 2024

METHOD OF TESTING LARGE NUMBERS FOR PRIMALITY

The current stage of scientific and technological development entails ensuring information security across all domains of human activity. Confidential data and wireless channels of remote control systems are particularly sensitive to various types of attacks. In these cases, various encryption systems are most commonly used for information protection, among which large prime numbers are widely utilized. The subject of research involves methods for generating prime numbers, which entail selecting candidates for primality and determining the primality of numbers. The subject of research involves methods for generating prime numbers, which choice selecting candidates for primality and determining the primality of numbers. The objective of the work is the development and theoretical justification of a method for determining the primality of numbers and providing the results of its testing. The aim to address the following main tasks: analyze the most commonly used and latest algorithms, methods, approaches, and tools for primality testing among large numbers; propose and theoretically justify a method for determining primality for large numbers; and conduct its testing. To achieve this aim, general scientific methods have been applied, including analysis of the subject area and mathematical apparatus, utilization of set theory, number theory, fields theory, as well as experimental design for organizing and conducting experimental research. The following results have been obtained: modern methods for selecting candidates for primality testing of large numbers have been analyzed, options for generating large prime numbers have been considered, and the main shortcomings of these methods for practical application of constructed prime numbers have been identified. Methods for determining candidates for primality testing of large numbers and a three-stage method for testing numbers for primality have been proposed and theoretically justified. The testing conducted on the proposed primality determination method has demonstrated the correctness of the theoretical conclusions regarding the feasibility of applying the proposed method to solve the stated problem. Conclusions. The use of a candidate primality testing strategy allows for a significant reduction in the number of tested numbers. For numbers of size 200 digits, the tested numbers is reduced to 8.82%. As the size of the tested numbers increases, their quantity will decrease. The proposed method for primality testing is sufficiently simple and effective. The first two stages allow for filtering out all composite numbers except for Carmichael numbers. In the first stage, using the first ten prime numbers filters out over 80 percent of the tested numbers. In the second stage, composite numbers with factors greater than 29 are sieved out. In the third stage, Carmichael numbers are sieved out. The test is polynomial, deterministic, and unconditional.

Cryptographic Application of Elliptic Curve Generated through Centered Hexadecagonal Numbers

Background/Objectives: Elliptic Curve Cryptography (ECC) is a public-key encryption method that is similar to RSA. ECC uses the mathematical concept of elliptic curves to achieve the same level of security with significantly smaller keys, whereas RSA's security depends on large prime numbers. Elliptic curves and their applications in cryptography will be discussed in this paper. The elliptic curve is formed by the extension of a Diophantine pair of Centered Hexadecagonal numbers to a Diophantine triple with property D(8). Method: The Diffie–Hellman key exchange, named for Whitfield Diffie and Martin Hellman, was developed by Ralph Merkle and is a mathematical technique for safely transferring cryptographic keys over a public channel. Based on the Diffie–Hellman key exchange, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography. The generation of keys, encryption and decryption are the three main operations of the ElGamal cryptosystem. Findings: Given the relative modesty of our objectives, the fundamental algebraic and geometric characteristics of elliptic curves shall be delineated. Then the behaviour of elliptic curves modulo p: ultimately, there is a fairly strong analogy between the structure of the points on an elliptic curve modulo p and the integers modulo n will be studied. In the end, elliptic curve ElGamal encryption analogues of Diffie–Hellman key exchange will be created. Novelty: Elliptic curves are encountered in a multitude of mathematical contexts and have a varied and fascinating history. Elliptic curves are very significant in number theory and are a focus of much recent work. The earlier research works in Elliptic Curve Cryptography has concentrated on computer algorithms and pairing – based algorithms. In this paper, the concept of polygonal numbers and its extension from Diophantine pair to triples is encountered, thus forming an elliptic curve and perform the encryption-decryption process. MSC Classification Number: 11D09, 11D99,11T71,11G05. Keywords: Elliptic curves, Cryptography, Encryption, Decryption, Centered polygonal numbers

A quantum-secure certificateless aggregate signature protocol for vehicular ad hoc networks

Vehicular ad hoc networks (VANETs) have revolutionized communication between vehicles and infrastructure, notably enhancing traffic management and passenger safety. However, VANETs are vulnerable to security threats, especially regarding data authenticity. Aggregate signature is a powerful technique that reduces computational and communication burdens by aggregating multiple signatures from different signers into a single signature. Traditional aggregate signature schemes, based on large prime number decomposition and the discrete logarithm problem, cannot effectively resist quantum attacks. This paper introduces a novel quantum secure certificateless aggregate signature (QSCLAS) scheme designed to enhance data security and privacy in VANETs. Our proposed scheme employs the number theory research unit (NTRU) algorithm. As a lattice-based cryptographic algorithm, NTRU is renowned for its security against quantum computer attacks, making it an essential component of our quantum-secure solution. By eliminating the need for expensive bilinear pairing operations, our proposed scheme achieves high efficiency and practicality in resource-limited VANETs environments. The security analysis demonstrates our scheme's resilience against both Type-I and Type-II adversaries in the random oracle model under the small integer solution (SIS) problem on the NTRU lattice. Furthermore, compared with existing approaches, the results illustrate that our proposed scheme offers significant advantages in signature generation and verification cost, as well as lower transmission overhead than other lattice-based schemes, thereby making it highly suitable for VANETs environments.

A survey of fermat factorization algorithms for factoring RSA composite numbers

In this research, we delve into the challenges associated with factorization in the context of the RSA cryptosystem. The RSA cryptosystem's security relies on the computational complexity of breaking down its modulus, which is essentially a product of two prime numbers denoted as 'p' and 'q.' Notably, the RSA modulus differs from the typical factorization problem, as it involves exceptionally large prime numbers, each roughly half the size of the modulus itself. If one manages to determine the factors of this modulus, it compromises the security of the entire RSA cryptosystem, as these factors can be exploited to derive the private key.Within this article, we provide an overview of the advancements made in the realm of factoring RSA moduli. Recent efforts in RSA factorization have successfully tackled 250-digit decimal RSA moduli. We also explore the common algorithms employed in factorization and delve into approaches based on the Fermat factorization method. Additionally, we discuss the various programming tools utilized to handle the challenge of factoring large numbers. Furthermore, we delve into the enhancement of the simple Fermat Primality test as a means to expedite the Primality Testing algorithm. Lastly, we offer a concise summary outlining the steps involved in the Number Field Sieve algorithm.

Realizing quantum-safe information sharing: Implementation and adoption challenges and policy recommendations for quantum-safe transitions

By utilizing the properties of quantum mechanics, quantum computers have the potential to factor a key pair of a large prime number and break some of the core cryptographic primitives that most information infrastructures depend on. This means that today's widely used cryptographic algorithms can soon become unsafe and need to be modified with quantum-safe (QS) cryptography. While much work is still needed in developing QS cryptographic algorithms, the institutional, organizational, and policy aspects of transitioning the current infrastructures have received less attention. This paper provides an empirical analysis of QS transition challenges and policy recommendations for moving to a QS situation. We analyzed the data collected through interviews with experts and practitioners from the Dutch government. The results reveal that institutional, organizational and policy aspects of QS transitions are interconnected, and solutions for QS transitions are scattered. Consequently, organizations may face a Catch-22 loop without further actionable approaches and planning for QS transitions.

A New Idea for RSA Backdoors

This article proposes a new method to inject backdoors in RSA (the public-key cryptosystem invented by Rivest, Shamir, and Adleman) and other cryptographic primitives based on the integer factorization problem for balanced semi-primes. The method relies on mathematical congruences among the factors of the semi-primes based on a large prime number, which acts as a “designer key” or “escrow key”. In particular, two different backdoors are proposed, one targeting a single semi-prime and the other one a pair of semi-primes. This article also describes the results of tests performed on a SageMath implementation of the backdoors.

A Novel and Secure Fake-Modulus Based Rabin-Ӡ Cryptosystem

Electronic commerce (E-commerce) transactions require secure communication to protect sensitive information such as credit card numbers, personal identification, and financial data from unauthorized access and fraud. Encryption using public key cryptography is essential to ensure secure electronic commerce transactions. RSA and Rabin cryptosystem algorithms are widely used public key cryptography techniques, and their security is based on the assumption that it is computationally infeasible to factorize the product of two large prime numbers into its constituent primes. However, existing variants of RSA and Rabin cryptosystems suffer from issues like high computational complexity, low speed, and vulnerability to factorization attacks. To overcome the issue, this article proposes a new method that introduces the concept of fake-modulus during encryption. The proposed method aims to increase the security of the Rabin cryptosystem by introducing a fake-modulus during encryption, which is used to confuse attackers who attempt to factorize the public key. The fake-modulus is added to the original modulus during encryption, and the attacker is unable to distinguish between the two. As a result, the attacker is unable to factorize the public key and cannot access the sensitive information transmitted during electronic commerce transactions. The proposed method’s performance is evaluated using qualitative and quantitative measures. Qualitative measures such as visual analysis and histogram analysis are used to evaluate the proposed system’s quality. To quantify the performance of the proposed method, the entropy of a number of occurrences for the pixels of cipher text and differential analysis of plaintext and cipher text is used. When the proposed method’s complexity is compared to a recent variant of the Rabin cryptosystem, it can be seen that it is more complex to break the proposed method—represented as O(ɲ× τ) which is higher than Rabin-P (O(ɲ)) algorithms.

Penggabungan Metode Vigènere Cipher dan ElGamal Pada Pengamanan Pesan Rahasia

Vigènere Cipher is a symmetric cryptographic algorithm that uses the same type of key in the encryption and decryption process. The security of the Vigènere Cipher method lies in the modulo calculation used. ElGamal is an asymmetric cryptographic algorithm that uses two different types of keys in the encryption and decryption process. The security of the ElGamal algorithm lies in the complexity of calculating large prime numbers. The Vigènere Cipher and ElGamal have their advantages and disadvantages. The researchers are interested in combining the advantages of the two methods. In this study, two locks were carried out in the encryption and decryption process. The encryption process uses a public key and the decryption process uses a public key and a secret key . The key used comes from key formation using the ElGamal algorithm. The key security formed from the ElGamal algorithm lies in the large prime , the primitive root of the prime number , and the random integer which comes from the last three digits of Student Number. The conclusion of this study is that the combination of the Vigènere Cipher and ElGamal methods can increase the security of secret messages because it produces a ciphertext with twice the size ) of the original message.

Implementasi Penyisipan Pesan Teks Terenkripsi Menggunakan Kriptografi ElGamal pada Citra Digital Menggunakan Steganogarafi LSB

Message security is very important in maintaining the confidentiality of information, especially those containing important information whose contents should only be known by the entitled party. Therefore, security is needed for messages that are confidential so that messages cannot be intercepted by people who do not want them and the information content of the message becomes more secure. One way is to apply ElGamal and LSB cryptography methods to be able to provide security to text messages by inserting into digital images, the ElGamal algorithm focuses on the strength of the key in solving discrete logarithm problems. By utilizing large prime numbers and discrete logarithm problems which are quite difficult and also by applying the LSB method for text insertion, the security of the key is more guaranteed. The test results on five image objects with the .BMP extension on the application that was built were running well and correctly. The image that is inserted in the encryption process is the insertion of text in the image after it is decrypted and the image produced after the encryption and decryption process is no different from the original image and the same thing for the image size does not change

Vulnerabilities of \(ex^2\) - \(y^2\)\(\phi\) (N)= z Using Modulus of the Form N= \(p^r\)\(q^s\)

The technical details of RSA works on the idea that it is easy to generate the modulus by multiplying two sufficiently large prime numbers together, but factorizing that number back into the original prime numbers is extremely difficult. Suppose that \(N=p^r q^s\) are RSA modulus, where \(p\) and \(q\) are product of two large unknown of unbalance primes for \(2 \leq s<r\). The paper proves that using an approximation of \(\phi(N) \approx\) \(N-N^{\frac{r+8-1}{2 r}}\left(\lambda^{\frac{1-8}{2 r}}+\lambda^{\frac{-8}{2 r}}\right)+N^{\frac{r+8-2}{2 T}} \lambda^{\frac{1-8}{2 r}}\), private keys \(\frac{x^2}{y^2}\) can be found from the convergents of the continued fractions expansion of\[\left|\frac{e}{N-N^{\frac{r+8-1}{2 r}}\left(\lambda^{\frac{1-8}{2 r}}+\lambda^{\frac{-8}{2 r}}\right)+N^{\frac{r+8-2}{2 r}} \lambda^{\frac{1-8}{2 r}}}-\frac{y^2}{x^2}\right|<\frac{1}{2 x^4}\] which leads to the factorization of the moduli \(N=p^r q^s\) into unbalance prime factors p and q in polynomial time. The second part of this reseach report further, how to generalized two system of equations of the form \(e_ux^2\) - \(y^2_u\phi(N_u)\) = \(z_u\) and \(e_ux^2_u\) - \(y^2\phi(N_u)\) = \(z_u\) using simultaneous Diophantine approximation method and LLL algorithm to and the values of the unknown integers \(x,y_u\),\(\phi(N_u)\) and \(x_u\),y,\(\phi(N_u)\) respectively, which yeild to successful factorization of k moduli \(N_u=p^r_uq^s_u\) for u = 1,2, ... k in polynomial time.

Enhancing satellite‐to‐ground communication using quantum key distribution

AbstractClassical Cryptography has been in use for a long time. It has been the only way of securing people's communication. However, there are some flaws observed during the execution of classical cryptography. One of them being the staunch belief that the number composed of multiplication of two large prime numbers cannot be factorised easily, which is under a threat thanks to the computational power of the quantum computers. The next flaw is the non‐detection of the hacker, both of which can be eliminated by using quantum mechanical principles for encryption purposes, which is known as quantum cryptography. Quantum Key Distribution, which provides an information‐theoretically safe solution to the key exchange problem, is the most well‐known example of quantum cryptography. The benefit of quantum cryptography is that it makes it possible to perform a variety of cryptographic operations that have been demonstrated or are hypothesised to be impractical when using solely traditional (i.e., non‐quantum) communication. Free‐space quantum communication has been successfully demonstrated across 300 m by the Indian Space Research Organization (ISRO) in March 2021. With this, ISRO is trying to achieve the same using a satellite‐based communication mechanism, which would revolutionise the mode of modern communication. It is justified that the key generation rate depends on factors like the aperture diameter of the sender and receiver, distance between them, the quantum bit error rate, and many more. The results vary with the parameters in the discussion as explained in the upcoming sections. The avenue of different types of losses that occur while transmitting at large distances, such as Atmospheric Loss, Pointing Loss and Geometric Loss, is explored.

Enriching Information Security via Hybrid of New Expand Rivest Shamir Adleman and Data Encryption Standard Cryptosystem

Cryptography system gives security services ability to protect information from people who are authorized to use it. This research introduces elevated information security system via hybrid of new expand Rivest Shamir Adleman (RSA) and data encryption standard (DES) cryptosystem using appropriate standard lgorithms. An Expand Algorithm (ERSA) algorithm Key cryptosystem based on nth large prime number system was introduced. Then a hybrid between the Expand Algorithm (ERSA) with Data Encryption Standard (DES) was applied using four randomly selected variables, each generated from large factor of “N” prime numbers. The information undergoes a cypher text-decryption process which provides strong security and uphold high information confidentiality and integrity of data. Results showed that the use of ERSA and DES algorithm reduced the key generation period and its complexity analysis of encryption and decryption are stronger, unlike the application of traditional RSA algorithm.

Constructing Representative Collective Signature Protocols Using The GOST R34.10-1994 Standard

The representative collective digital signature, which was suggested by us, is built based on combining the advantages of group digital signature and collective digital signature. This collective digital signature schema helps to create a unique digital signature that deputizes a collective of people representing different groups of signers and may also include personal signers. The advantage of the proposed collective signature is that it can be built based on most of the well-known difficult problems such as the factor analysis, the discrete logarithm and finding modulo roots of large prime numbers and the current digital signature standards of the United States and Russian Federation. In this paper, we use the discrete logarithmic problem on prime finite fields, which has been implemented in the GOST R34.10-1994 digital signature standard, to build the proposed collective signature protocols. These protocols help to create collective signatures: Guaranteed internal integrity and fixed size, independent of the number of members involved in forming the signature. The signature built in this study, consisting of 3 components (U, R, S), stores the information of all relevant signers in the U components, thus tracking the signer and against the “disclaim of liability” of the signer later is possible. The idea of hiding the signer’s public key is also applied in the proposed protocols. This makes it easy for the signing group representative to specify which members are authorized to participate in the signature creation process.

Algorithms for Quantum Computation: The Derivatives of Discontinuous Functions

We hope this work allows one to calculate large prime numbers directly, not by trial-and-error, but following a physical law. We report—contrary to conventional assumptions—that differentiation of discontinuous functions (DDF) exists in the set Q, which becomes central to algorithms for quantum computation. DDF have been thought to exist not in the classical sense, but using distributions. However, DDF using distributions still is defined in terms of mathematical real-numbers (MRN), and do not address the Problem of Closure, here investigated. These facts lead to contradictions using MRN, solved by this work, providing a new unbounded class of physical solutions using physical numbers in quantum mechanics (QM), that were always there (just hidden), allowing DDF without distributions, or MRN. It is worthwhile to see this only in mathematics, to avoid the prejudices found in physics, as this reforms both general relativity and QM. This confirms the opinions of Nicolas Gisin that MRN are non-computable with probability 1, and Niels Bohr that physics is not reality, it is a fitting story about reality. Mathematics can get closer to reality, surprisingly. We just have to base mathematics on nature, not on how it defines nature.

Implementation of Boneh - Lynn - Shacham short digital signature scheme using Weil bilinear pairing based on supersingular elliptic curves

One option for a digital signature solution for devices with low memory and low bandwidth transmission over channels uses a short digital signature scheme based on Weil bilinear pairing aimed at short processing times, fast computation, and convenient deployment on applications. The computational technique of non-degenerate bilinear pairings uses supersingular elliptic curves over a finite field Fpl (where p is a sufficiently large prime number) and has the advantage of being able to avoid Weil-descent, Menezes-Okamoto-Vanstone (MOV) attacks, and attacks by the Number Field Sieve algorithm. Compared to Elliptic Curve Digital Signature Algorithm (ECDSA) digital signature schemes, generating a digital signature for a Boneh-Lynn-Shacham (BLS) scheme using Weil bilinear pairing on a supersingular elliptic curve is simple. In this study, the authors replace non-degenerate bilinear pairing calculations on a supersingular elliptic curve with a Weil pairing with PεE(Fp ), QεE(Fp1) and a higher security multiplier α=12 in the BLS short digital signature scheme. The execution time of the BLS short digital signature program showed improvement compared to the commercial ECDSA digital signature scheme.

An Improved RSA Algorithm for Enhanced Security

The security of the Rivest-Shamir-Adelman (RSA) public key algorithm depends on the difficulty of factoring the modulus calculated by multiplying two large prime numbers. The usefulness of the RSA public key algorithm lies in using one key for encryption and another key for decryption. However, a poor choice of the keys used in encryption and decryption can affect the security of the RSA cryptosystem. Many proposals have been made to modify the RSA cryptosystem in such a way that the attacks on the RSA cryptosystem can be overcome. In this article, we propose concealing the publicly disclosed parameters, the encryption key and the common modulus of the RSA cryptosystem by altering the values sent publicly. The values sent publicly are different from the one used in the algorithm which conceals the actual parameters to others. The implementation of this idea uses two different algorithms and randomly choosing between the algorithms. The choice of the algorithm is done using a random number generator and this choice of the algorithm has to be communicated so the decryptor uses the correct algorithm to decrypt the encrypted data. Finally we explore a faster way to implement the modular exponentiation algorithm used in the RSA encryption and decryption.