Mersenne Prime Research Articles

ON INVERSES AND EIGENPAIRS OF PERIODIC TRIDIAGONAL TOEPLITZ MATRICES WITH PERTURBED CORNERS

In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type $I$. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type $II$ matrix can be also obtained, which benefits from the relation between Type $I$ and $II$ matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.

HMNT: Hash Function Based on New Mersenne Number Transform

In the field of information security, hash functions are considered important as they are used to ensure message integrity and authentication. Despite various available methods to design hash functions, the methods have been proven to time inefficient and have security flaws (such as a lack of collision resistance or susceptibility to birthday attacks). In the current study, we propose a novel hash function scheme based on a new Mersenne number transform. The suggested hash function called Hash Mersenne Number Transform (HMNT) takes an arbitrary length as input to generate a hash value with variable lengths (128, 256 and 512-bits or longer). The proposed scheme is evaluated in terms of the sensitivity of the hash value to the message, secret key and image, distribution of hashes, confusion and diffusion, robustness against collision and birthday attacks, alongside flexibility. Based on the simulation outcomes, the suggested scheme possess high sensitivity to the original message, the secret key and images, along with strong collision resistance. In conclusion, the proposed hash scheme is simple and efficient compared with the existing hash functions, making it viable for practical implementation.

A primality test for $$4Kp^n-1$$ numbers

We present a Lucasian type primality test, not explicitly based on Lucas sequences, for numbers written in the form $$N=4Kp^n-1$$. This test is a generalization of the classical Lucas–Lehmer test for Mersenne numbers using as underlying group $$\mathcal {G}_N:=\{z \in (\mathbb {Z}/N\mathbb {Z})[i]: z\overline{z} \equiv 1 \pmod N\}$$.

Improved cryptanalysis of the AJPS Mersenne based cryptosystem

AbstractAt Crypto 2018, Aggarwal, Joux, Prakash and Santha (AJPS) described a new public-key encryption scheme based on Mersenne numbers. Shortly after the publication of the cryptosystem, Beunardeau et al. described an attack with complexity 𝓞(22h). In this paper, we describe an improved attack with complexity 𝓞(21.75h).

ON THE PRIME DECOMPOSITION OF INTEGERS OF THE FORM zn-yn/z-y

In this work, the author shows a sufficient and necessary condition for an integer of the form $(zn-y^n)/(z-y)$ to be divisible by some perfect $mth$ power $p$, where $p$ is an odd prime and $m$ is a positive integer. A constructive method of this type of integers is explained with details and examples. Links between the main result and known ideas such as Termat's last theorem, Goormaghtigh conjecture and Mersenne numbers are discussed. other related ideas, examples and applications are provides.

Algebraic Construction of Near-Bent and APN Functions

Boolean functions play an important role in symmetric cryptosystems. In this paper, we have constructed near-bent Boolean functions algebraically with the help of Niho power function exponent in the trace form over a finite field. Specific cryptographic properties which may gain attention with suitable modifications are observed using these functions. If exponents are either Mersenne numbers or Fermat numbers, then the resulting functions are found to be APN (almost perfect nonlinear).

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

Solutions to Non-Linear Diophantine Equation Pχ+ (P+5)y =z2 with P is Mersenne Prime

This research seeks for a solution (if any) to non-linear Diophantine equation 2 ( 5) x y p p z    with p is Mersenne prime. There are 3 possibilities of solution to the non-linear Diophantine equation, which are single solution, many solutions, or no solution. The research methodology is conducted in two stages, which are using simulation to seek for solution (if any) to non-linear Diophantine equation 2 ( 5) x y p p z    with p is Mersenne prime and using Catalan’s conjecture and characteristics of congruency theory. it is proven that the non-linear Diophantine equation has no solution for p  3.

Development of modified RSA algorithm using fixed mersenne prime numbers for medical ultrasound imaging instrumentation

Purpose: Encryption of patient information has become an important issue in medical ultrasound instrumentation to secure information when images are accessed off-site. The proposed algorithm is used to encrypt private medical images and transfer the encrypted images to improve the encryption capability and elapsed time.Materials and methods: We generate a public key using three prime numbers, including a fixed Mersenne prime number, in the modified Rivest-Shamir-Adelman (RSA) algorithm to compare the encryption capability. We calculated and compared the elapsed time using the modified RSA algorithm with a breast phantom in the medical ultrasound imaging instrumentation.Results: The encryption capability is improved because the elapsed time when using three prime numbers is longer (1.2337 s) than that when using two prime numbers (1.0712 s). However, the elapsed time using fixed Mersenne prime numbers (0.8360 s) is a similar to that using two prime numbers (0.8389 s).Conclusions: Our proposed cryptographic algorithm provides improved encryption in medical ultrasound imaging compared to algorithms that use two prime numbers that are not Mersenne prime numbers, while transmitting images with adequate elapsed times.

The necessary security requirements for the values used by the AJPS cryptosystem

D. Aggarwal and others have recently proposed a new quantum-resistant asymmetric cryptosystem AJPS [1] which uses operations modulo a Mersenne number. This paper imposes the restrictions on the public key of this cryptosystem and ciphertext requirements, and presents the active attack on AJPS and the AJPS cryptosystem usage recommendations.

Design and implementation in VHDL code of the discrete cosine transform over finite fields for symmetric convolution operation

The discrete cosine transform is a very important mathematical tool for image processing operations, such as filtering, recognition, segmentation, compression, etc. We present the design and implementation of the description and synthesis in VHDL code of the discrete cosine transform over the finite field with an order given by the Mersenne prime number 127, in order to perform the symmetric convolution operation. The development of the VHDL code for the discrete cosine transform and the symmetric convolution operations are performed using the HDL Coder in Simulink. The discrete cosine transform over finite field is defined using a new trigonometric transform proposed recently. The main aim for the development of the discrete cosine transform and the symmetric convolution hardware architectures over finite field is to integrate these architectures in applications of filtering, encryption and recognition of images.

IoT Light Weight (LWT) Crypto Functions

We are in the era of IoT and 5G technologies. IoT has wide range of applications in Smart Home, Smart cities, Agriculture, Health etc. Due to that, the number of connected sensor devices become increased. Along with that security of these devices become a challenging issue. By the next year there would be a great increase in the number of connected sensor devices. For the power constrained devices like sensors and actuators, they requires lightweight security mechanism. There are several Lightweight (LW) energy efficient Hashing techniques are available. They are photon, quark, spongent, Lesamnta-LW etc. These all are fixed length block sized and key sized LW hashing techniques. All transformation methods used today in LW hash function only support fixed block size and key size and requires high hardware requirements too. In this paper, we compare different types of LW hash families in terms of their design and introduce the possibility of variable length hash function using Mersenne number based transform.

Leading Digits of Mersenne Numbers

It has long been known that sequences such as the powers of 2 and the factorials satisfy Benford’s Law; that is, leading digits in these sequences occur with frequencies given by , . In this article, we consider the leading digits of the Mersenne numbers , where pn is the n-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of has worse distribution properties than “smooth” sequences with similar rates of growth, such as . Surprisingly, the opposite seems to be true; indeed, we present data, based on the first billion terms of the sequence , showing that leading digits of Mersenne numbers behave in many respects more regularly than those in the above smooth sequences. We state several conjectures to this effect, and we provide an heuristic explanation for the observed phenomena based on classic models for the distribution of primes.

A fast realization of new Mersenne number transformation and its applications

SummaryThe new Mersenne number transform (NMNT) can be realized by the fast Fourier transform (FFT) with power‐of‐two length, which results in great flexibility in real‐world fixed‐point computations, such as the convolution‐based signal processing in the embedded device. Yet the FFT realization exists the truncation errors and in order to further reduce the computations, this paper puts forward a novel realization structure for the NMNT, where the Walsh‐Hadamard transform (WHT) is employed to accelerate the NMNT computation. Moreover, we also propose the refined computing structure using the matrix decomposition and multiple constant multiplication (MCM), and then all multiplications can be replaced by the shift‐addition operations without precision loss. Besides, the use of lookup table (LUT) to reduce the complexity is also discussed in our study. A typical convolution application is tested by computer simulations, while the result demonstrates that the proposed scheme produces precise computing results with reduced complexity.

A congruence relation of the Catalan-Mersenne numbers

The Catalan-Mersenne numbers cn are double Mersenne numbers defined by c0 = 2 and $${c_n} = {2^{{c_{n - 1}}}} - 1$$ for positive integers n. We prove a certain congruence relation of the Catalan- Mersenne numbers.

FIR Filter Realization via Deferred End-Around Carry Modular Addition

Hardware realization of FIR filters that are based on residue number systems leads to increased speed and reduced power, where besides the popular Mersenne numbers, several moduli of the form ${2}^{n} \pm \delta (\delta \ge 3)$ are commonly used. However, additional weighted ${2}^{i}(i>1)$ end-around carries (EACs) slow down and complicate the required modular adders in comparison to modulo-( ${2}^{n}-1$ ) adders. For example, for $\delta =3$ , the modular sum is obtained via $A+B\mp 3{c}_{\mathrm {out}}$ , where ${A}$ and ${B}$ are modulo-( ${2}^{n}\pm 3$ ) operands and ${c}_{\mathrm {out}}$ is the carryout of binary addition $A+B$ . In this paper, a new multioperand modular adder is proposed, where the key improvement is that all the required EAC additions (e.g., $+ 3c_{\mathrm {out}}$ ) are postponed until after the last filter tap, whereby tens of addition operations take place without the EAC secondary addition; hence considerable savings of time, area consumption, and power dissipation. The proposed deferred EAC addition scheme has been applied to three previous relevant works. The corresponding synthesis results showed over 11%–32%, 27%–29%, and 21%–37%, reductions in delay, area, and power measures, respectively. This is achieved despite area and power overhead of the few appended stages into the pipelined architecture of the filter, which are nevertheless shown to become less significant as the number of filter taps grows.

Faster Residue Multiplication Modulo 521-bit Mersenne Prime and an Application to ECC

We present faster algorithms for the residue multiplication modulo 521-bit Mersenne prime on 32- and 64-bit platforms by using Toeplitz matrix-vector product. The total arithmetic cost of our proposed algorithms is less than that of existing algorithms, with algorithms for 64- and 32-bit residue multiplication giving the best timing results on our test machine. The transition from 64- to 32-bit implementation is full of challenges because the number of limbs doubles and the limbs’ bitlengths are cut in half. Without using any intrinsics or SIMD/assembly instructions in our implementation on an Intel(R) Core i5 – 6402P CPU @ 2.80GHz, we find 136 and 550 cycles for our 64-and 32-bit residue multiplications, respectively. In addition, we implement constant-time variable- and fixed-base scalar multiplication for the standard NIST curve P-521 and Edwards curve E-521.

On Some Identities and Generating Functions for Mersenne Numbers and Polynomials

In this paper, we introduce a new operator in order to derive some properties of homogeneous symmetric functions. By making use of the proposed operator, we give some new generating functions for Mersenne numbers, Mersenne numbers and product of sequences and Chebychev polynomials of second kind.

On generalized Mersenne numbers, their interpretations and matrix generators

In this paper we introduce generalized Mersenne numbers. We shall present some of their interpretations and matrix generators which are very useful for determining identities.

There Are Infinitely Many Mersnne Composite Numbers with Prime Exponents

By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.