Large Prime Numbers Research Articles

On Distribution of Elements of Subgroups in Arithmetic Progressions Modulo a Prime

Let $$\mathbb{F}_p$$ be the field of residue classes modulo a large prime number p. We prove that if $$\mathcal{G}$$ is a subgroup of the multiplicative group $$\mathbb{F}_p^*$$ and if $$\mathcal{I} \subset \mathbb{F}_p$$ is an arithmetic progression, then $$|\mathcal{G} \cap \mathcal{I}| = (1 + o(1))|\mathcal{G}|\mathcal{I}|/p + R$$ , where $$|R| < (|\mathcal{I}|^{1/2} + |\mathcal{G}|^{1/2} + |\mathcal{I}|^{1/2}|\mathcal{G}|^{3/8}p^{-1/8})p^{o(1)}$$ . We use this bound to show that the number of solutions to the congruence xn ≡ λ (mod p), x ∈ ℕ, L < x < L + p/n, is at most p1/3−1/390+o(1) uniformly over positive integers n, λ and L. The proofs are based on results and arguments of Cilleruelo and the author (2014), Murphy, Rudnev, Shkredov and Shteinikov (2017) and Bourgain, Konyagin, Shparlinski and the author (2013).

The finite Littlewood problem in $${{\mathbb {F}}}_p$$ F p

Let p be a large prime number and f(x) be an integer-valued function defined in $${\mathbb F}_p$$ . The Littlewood problem in $${{\mathbb {F}}}_p$$ is to establish non-trivial lower bounds for the $$\ell _1$$ norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of $$\mathbb {F}_p^*.$$

Distributing of point division on 4 results of Edwards curve points group to adjacement classes

Elliptic curves in Edwards form are perspective for usage in modern asymmetric cryptosystems. Such curves have a series of advantages in compare with elliptic curves in canonical form, such as speed of addition, universality of addition law, existence of affine coordinates for neutral element of group of points. The fact that Edwards curves are symmetric in both variables involves some properties of such curves that are used in cryptogogy. These days Edwards curves are actively investigates all over the world, for instance, the possibility is investigated to design new digital signature standards on Edwards curves. The most interesting for practical usage are Edwards curves which orders are equal to 4n, where n is large prime number. The security of digital signature on Edwards curves is based on complication of DLP (Discrete logarithm problem) in subgroup of Edwards curve points. The usage of Edwards curve for new digital signature standards stipulates the actuality of cryptanalysis of such cryptosystems. The important place among attacks on DLP-based cryptosystems take special attacks that use the features of the cyclic group in which the DLP problem is considered. Because of this it is necessary to investigate the structure of the cyclic group and its features for cryptanalysis of such systems. One of the algebraic tasks which may be useful in cryptanalysis is representation of Edwards curve points by the pair of left (right) adjacent classes by subgroups of the order 4 and of the maximal prime order n. One of the algorithms for cryptographic analysis of the Edwards curve cryptosystems is the division of point of Edwards curve by four. Division results are tightly connected with the split of point groups of Edward curve by adjacent classes of subgoups of maximum prime order and of the order 4. Structure of the Edwards curve points group allows to determine definitively position of any point of this group, simultaneously in two adjacent classes of subgroups of maximum prime order or fourth order. Example is given of discrete logarithmic problem solution using division of point by four and classification of results of division by adjacent classes for point groups of Edwards curve of order twenty eight and seventy six.

A New Cryptographic Scheme Utilizing the Difficulty of Big Boolean Satisfiability

A search problem may be identified as one, which requires an actual “search” for an answer or a solution. Such a problem may have no obvious method, which could be followed to determine a solution, other than to intelligently search through all candidate or potential solutions, which constitute the search space, until a satisfactory one is found. Typically, we may have an efficient way of determining whether one of the possible solutions is actually correct, but no efficient way of determining how to find a correct solution. There are many such search problems, both theoretically and practically motivated, but they all have these difficulties in common. Consider the example of RSA cryptanalysis, where we are given an integer n which is the product of two large prime numbers a and b, and we need to factor n into its factors a and b. This can be achieved by attempting (according to the sieve of Eratosthenes) to divide n by every prime integer between 2 and √n, and hence it is a special kind of a search problem in itself. Several efforts in the past aimed to translate various encryption and hashing schemes into Boolean satisfiability (SAT). The SAT problem is a computationally intractable (NP-hard) problem but relatively efficient SAT-Solvers are built having computational complexity of 2^k (1-∈), where 0&lt; ∈ &lt;1 and thus can prune the search space significantly. Guided by the above concepts, we propose herein a scheme that can encrypt a message by using a ‘big’ Boolean function, which produces an equation that cannot be solved by the conventional SAT-Solvers and leads to a dramatic increase in the search space from 2^n to 〖〖(2〗^(2^m ))〗^n in the worst case. Logical cryptanalysis shows that the proposed scheme is very hard to break, indeed. To the best of our knowledge, the adversary cannot reduce or prune the search space (except for shortening the task needed at every node), and is forced to traverse the whole search space. He might arrive at several candidate solutions, and has to search for clues as to which of them is the correct solution.

Using random search and brute force algorithm in factoring the RSA modulus

Abstract. The security of the RSA cryptosystem is directly proportional to the size of its modulus, n. The modulus n is a multiplication of two very large prime numbers, notated as p and q. Since modulus n is public, a cryptanalyst can use factorization algorithms such as Euler’s and Pollard’s algorithms to derive the private keys, p and q. Brute force is an algorithm that searches a solution to a problem by generating all the possible candidate solutions and testing those candidates one by one in order to get the most relevant solution. Random search is a numerical optimization algorithm that starts its search by generating one candidate solution randomly and iteratively compares it with other random candidate solution in order to get the most suitable solution. This work aims to compare the performance of brute force algorithm and random search in factoring the RSA modulus into its two prime factors by experimental means in Python programming language. The primality test is done by Fermat algorithm and the sieve of Eratosthenes.

On factoring RSA modulus using random-restart hill-climbing algorithm and Pollard’s rho algorithm

The security of the widely-used RSA public key cryptography algorithm depends on the difficulty of factoring a big integer into two large prime numbers. For many years, the integer factorization problem has been intensively and extensively studied in the field of number theory. As a result, a lot of deterministic algorithms such as Euler’s algorithm, Kraitchik’s, and variants of Pollard’s algorithms have been researched comprehensively. Our study takes a rather uncommon approach: rather than making use of intensive number theories, we attempt to factorize RSA modulus n by using random-restart hill-climbing algorithm, which belongs the class of metaheuristic algorithms. The factorization time of RSA moduli with different lengths is recorded and compared with the factorization time of Pollard’s rho algorithm, which is a deterministic algorithm. Our experimental results indicates that while random-restart hill-climbing algorithm is an acceptable candidate to factorize smaller RSA moduli, the factorization speed is much slower than that of Pollard’s rho algorithm.

Chaotic synchronization cryptosystems combined with RSA encryption algorithm

Chaotic synchronization has suffered from demodulating information by hackers via the public channel. Different methods have been developed to hide the contents of a message using chaotic signals. Nevertheless, many studies have shown that most of the existing methods are unreliable in the aspect of security. According to the above, in order to enhance the strength of the cryptosystem and provide greater security, we carry out double encryption which combines chaotic synchronization with RSA algorithm. In this study, a design methodology for neural-network (NN)-based secure communications in multiple time-delay chaotic (MTDC) systems is proposed to obtain double encryption via RSA algorithm and chaotic synchronization. RSA algorithm is an asymmetric encryption and its strength is based on the tremendous difficulty of factorizing two large prime numbers. Accordingly, it takes much more time to process RSA algorithm if the length of the key is much longer. A shorter key is used to decrease the processing time of RSA algorithm in this work. Nevertheless, it will result in a lower security of the cryptosystem. In order to enhance the strength of the cryptosystem, we carry out double encryption which combines chaotic synchronization with RSA algorithm. Moreover, an IGA is proposed in this study to effectively overcome the weakness of the traditional GA. On the basis of the IGA, a fuzzy controller is synthesized to not only realize the exponential synchronization, but also achieve optimal H∞ performance by minimizing the disturbances attenuation level. Finally, a numerical example with simulations is given to illustrate the effectiveness of our approach.

Generating Mersenne Prime Number Using Rabin Miller Primality Probability Test to Get Big Prime Number in RSA Cryptography

Cryptography RSA method (Rivest - Shamir - Adelman) require large-scale primes to obtain high security that is in greater than or equal to 512, in the process to getting the securities is done to generation or generate prime numbers greater than or equal to 512. Using the Sieve of Eratosthenes is needed to bring up a list of small prime numbers to use as a large prime numbers, the numbers from the result would be combined, so the prime numbers are more produced by the combination Eratosthenes. In this case the prime numbers that are in the range 1500 < prime <2000, for the next step the result of the generation it processed by using the Rabin - Miller Primarily Test. Cryptography RSA method (Rivest - Shamir - Adleman) with the large-scale prime numbers would got securities or data security is better because the difficulty to describe the RSA code gain if it has no RSA Key same with data sender.

Enhanced Hybrid Algorithm of Secure and Fast Chaos-based, AES, RSA and ElGamal Cryptosystems

Objectives: The principal objective of this paper is to develop an enhanced algorithm of cryptography through the hybridization of existing symmetric and asymmetric encryption. Methods/Analysis: The strength of several prevailing symmetric and asymmetric encryption were utilized in the proposed methodology. ElGamal’s key security based on difficulty of solving discrete logarithm, Chaos-based’s avalanche effect of using multiple keys, RSA’s security based on the difficulty of factorization of large numbers and AES’s S-Box mapping, which makes the encryption complex and unpredictable were adapted. Findings: The strength of security of the proposed hybrid algorithm is tested by the provided examples. It was found out that with the use of different keys in encryption, various ciphertexts were generated. The published public keys using such algorithm also make it infeasible to intrude because of the difficulty of solving discrete logarithm with large prime numbers. Furthermore, the value of the ciphertext of the proposed method is significantly different and, in some cases, larger than that of RSA’s. Applications: With this newly enhanced hybrid algorithm of data encryption, transferring and conveying of data will be much more secured.

A new generalization of the KMOV cryptosystem

The KMOV scheme is a public key cryptosystem based on an RSA modulus \(n=pq\) where p and q are large prime numbers with \(p\equiv q\equiv 2\pmod 3\). It uses the points of an elliptic curve with equation \(y^2\equiv x^3+b\pmod n\). In this paper, we propose a generalization of the KMOV cryptosystem with a prime power modulus of the form \(n=p^{r}q^{s}\) and study its resistance to the known attacks.

Fully homomorphic encryption with matrix based digital signature standard

In this work, a novel mechanism is considered for Asymmetric mode encrypting data. A generator matrix is used to generate a field with a large prime number. The generator matrix, prime number and ternary vector are used as global variables. Those global variables are used to calculate public key and sub keys. Using the Global Variables and Private key, a Digital signature algorithm is proposed which supports the feature like Authenticity of users. The mechanism can also be used in Homomorphic Encryption where computations to be carried out on cipher text and generate an encrypted result which, when decrypted, matches the result of operations performed on the plaintext.

A novel approach with matrix based public key crypto systems

Here in this model, a new mechanism is used for Public Key Cryptography. A generator matrix is used to generate a field with a large prime number. The generator matrix, prime number and quaternary vector are used as global variables. The Generator Matrix is powered by a private key to generate Public Key. Since the model is based on Discrete Logarithm Problem, which is Hard problem, the proposed algorithm supports the features like Authenticity of users, Security & Confidentiality of data transmitted.Going by the construction of the algorithm, Encryption is being done on blocks of data for which it consumes less computing resources. Going by complexity of the algorithm, the key length needed is about 72 bit lengths to provide sufficient strengths against crypto analysis.

The Universal Block Lanczos–Padé Method for Linear Systems Over Large Prime Fields

In this paper, we propose a universal algorithm designed for solving large sparse linear systems over finite fields with a large prime number of elements. Such systems arise in the solution of the discrete logarithm problem modulo a prime number. The algorithm has been developed for parallel computing systems with various parallel architectures and properties. The new method inherits the structural properties of the Lanczos method. However, it provides flexible control over the complexity of parallel computations and the intensity of exchanges.

Comment on “An Enhanced and Secured RSA Key Generation Scheme (ESRKGS)”

In this article, we do the cryptanalysis of the Enhanced and Secured RSA Key Generation Scheme (ESRKGS) given in Thangavel et al. (2015). The authors claimed that ESRKGS is a highly secure and not easily breakable scheme compared to the traditional RSA scheme. In their scheme, the public key n is a product of two large prime numbers, but the values of Encryption E and Decryption D keys are based on the product of four large prime numbers, N, which is a multiple of n. They claimed that even if someone factorizes n, in order to obtain the private key D, brute force must be used to obtain the other two primes.We prove that we can use an alternative private key D1 to break the system. Someone does not need to employ brute force to break the system when n is factored. Hence, ESRKGS has the same security level as the traditional RSA.

Étude probabiliste des quotients de Fermat

For a fixed integer a>1, we suggest that the probability of nullity of the p-Fermat quotient q(p,a) is much lower than 1/p for any arbitrary large prime number p. For this we use various heuristics, justified by means of numerical computations and analytical results, which may imply the finiteness of the q(p,a) equal to 0 and the existence of integers a such that q(p,a) is different from 0 for all p. However no proofs are obtained concerning these heuristics.

Evaluation of Key Dependent S-Box Based Data Security Algorithm using Hamming Distance and Balanced Output

Data security is a major issue because of rapid evolution of data communication over unsecured internetwork. Here the proposed system is concerned with the problem of randomly generated S-box. The generation of S-box depends on Pseudo-Random-Number-Generators and shared-secret-key. The process of Pseudo-Random-Number-Generator depends on large prime numbers. All Pseudo-Random-Numbers are scrambled according to shared-secret-key. After scrambling, the S-box is generated. In this research, large prime numbers are the inputs to the Pseudo-Random-Number-Generator. The proposed S-box will reduce the complexity of S-box generation. Based on S-box parameters, it experimentally investigates the quality and robustness of the proposed algorithm which was tested. It yields better results with the S-box parameters like Hamming Distance, Balanced Output and Avalanche Effect and can be embedded to popular cryptography algorithms

Algebraic points on Shimura curves of $$\varGamma _0(p)$$-type (III)

In previous works, we proved that under a certain assumption, the set of rational points over a number field on the Shimura curve of \(\varGamma _0(p)\)-type consists of at most elliptic points for every sufficiently large prime number p. In this article, we relax the assumption of the previous result and prove the non-existence of elliptic points under a mild extra assumption.

Proposal of the Multivariate Public Key Cryptosystem Relying on the Difficulty of Factoring a Product of Two Large Prime Numbers

Proposal of the Multivariate Public Key Cryptosystem Relying on the Difficulty of Factoring a Product of Two Large Prime Numbers

CUDA programs for solving the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

In this paper we present new versions of previously published numerical programs for solving the dipolar Gross-Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank-Nicolson method as in the previous version (R. Kishor Kumar et al., Comput. Phys. Commun. 195, 117 (2015)), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12 to 25, depending on the program and input size.

Genefer: Programs for Finding Large Probable Generalized Fermat Primes

n+1 (GFNs) using a Fermat test. Optimised implementations are available for modern CPUs using single instruction, multiple data (SIMD) instructions, as well as for GPUs using CUDA or OpenCL. Genefer has been extensively used by PrimeGrid – a volunteer computing project searching for large prime numbers of various kinds, including GFNs. Genefer’s architecture separates the high level logic such as checkpointing and user interface from the architecture-specific performance-critical parts of the implementation, which are suitable for re-use. Genefer is released under the MIT license. Source and binaries are available from www.assembla.com/ spaces/genefer.