An approximating k-ary GCD algorithm

© 2016, Pleiades Publishing, Ltd.In our paper we elaborate a new version of the k-ary GCD algorithm. Our algorithm is based on the Farey Series and surpasses all existing realizations of the k-ary algorithm. It can have practical applications inMathematics and Cryptography

An algorithm for counting smooth integers

© 2016, Pleiades Publishing, Ltd.An integer number n > 0 is called y-smooth for y > 0 if any prime factor p of n satisfies p ≤ y. Let ψ(x, y) be the number of all y-smooth integers less or equal to x. In this paper we elaborate a new algorithm for approximate calculation of ψ(x, y) at large x and relatively small y < log x

On distribution of semiprime numbers

A semiprime is a natural number which is the product of two (possibly equal) prime numbers. Let y be a natural number and g(y) be the probability for a number y to be semiprime. In this paper we derive an asymptotic formula to count g(y) for large y and evaluate its correctness for different y. We also introduce strongly semiprimes, i.e., numbers each of which is a product of two primes of large dimension, and investigate distribution of strongly semiprimes. © 2014 Allerton Press, Inc

Analysis of the k-ary Euclid for tuples of integers

In our paper we discuss the k-ary Euclid Algorithm for counting the great common divisor (GCD) of two or more integers and suggest some improvements. This gives us a possibility to parallelize and speed up the calculating of GCD, which has a variety of applications in the Number Theory, Modular Arithmetic and the Cryptography Algorithms such as RSA, ElGamal encryption system and others.1-

One approach to factorization of positive integers

Factorization of positive integers into primes is a hard computational task. Its complexity lies in the base of the most popular method of cryptography, the RSA method. In this paper we propose a new technique in a factorization procedure which combines ideas of the Number Field Sieve (NFS) and the Quadratic Sieve (QS) in a special manner. © Allerton Press, Inc., 2011

The Error Probability of the Miller–Rabin Primality Test

© 2018, Pleiades Publishing, Ltd. In our paper we give theoretical and practical estimations of the error probability in the well-known Miller–Rabin probabilistic primality test. We show that a theoretical probability of error 0.25 for a single round of the test is very overestimated and, in fact, error is diminishing with the growth of length of numbers involved by a rate limited with ln n/n

Calculation of Bezout’s coefficients for the k-ary algorithm of finding GCD

© 2017, Allerton Press, Inc. Bezout’s equation is a representation of the greatest common divisor d of integers A and B as a linear combination Ax + By = d, where x and y are integers called Bezout’s coefficients. The task of finding Bezout’s coefficients has numerous applications in the number theory and cryptography, for example, for calculation of multiplicative inverse elements in modular arithmetic. Usually Bezout’s coefficients are caclulated using the extended version of the classical Euclidian algorithm. We elaborate a new algorithm for calculating Bezout’s coefficients based on the k-ary GCD algorithm

Uptake of halloysite clay nanotubes by human cells: Colourimetric viability tests and microscopy study

© 2018 This study is a systemic investigation of the uptake and toxicity of halloysite nanotubes using human adenocarcinoma epithelial cells (A549). A549 cells were chosen as a popular model of cancer cells extensively studied in nanotoxicity and drug delivery research. The adverse effects of a range of halloysite concentrations were evaluated. The viability of A549 cells was determined using several colourimetric assays. Dark-field microscopy was used to visualize the uptake and distribution of halloysite nanotubes in cells. The morphology of the cells was evaluated using dark-field, transmission electron and atomic force microscopies. The results showed that halloysite had a dose-dependent effect on human cells at concentrations of 5–900 μg per 105 cells in the MTT assay. The reduced toxicity of halloysite nanotubes at lower concentrations (5–75 μg per 105 cells) was additionally supported by the results of other colorimetric assays. Microscopy assays have demonstrated that the nanotubes, though affecting the biochemical processes, do not alter the morphology of the cells and do not penetrate into the nuclei

Towards effective COVID\u201119 vaccines: Updates, perspectives and challenges (Review)

In the current context of the pandemic triggered by SARS-COV-2, the immunization of the population through vaccination is recognized as a public health priority. In the case of SARS\u2011COV\u20112, the genetic sequencing was done quickly, in one month. Since then, worldwide research has focused on obtaining a vaccine. This has a major economic impact because new technological platforms and advanced genetic engineering procedures are required to obtain a COVID\u201119 vaccine. The most difficult scientific challenge for this future vaccine obtained in the laboratory is the proof of clinical safety and efficacy. The biggest challenge of manufacturing is the construction and validation of production platforms capable of making the vaccine on a large scale