Related papers: On Polynomial Modular Number Systems over $\mathbb…

We consider an application to the discrete log problem using completely regular semigroups which may provide a more secure symmetric cryptosystem than the classic system based on groups. In particular we describe a scheme that would appear…

In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in…

In this paper I consider all possible properties from commutative algebra for polynomial composites and monoid domains. The aim is full characterization of these structures. I start with the examination of group, ring, modules properties,…

Computation biology helps to understand all processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a…

Counting the number of nodes in Anonymous Dynamic Networks is enticing from an algorithmic perspective: an important computation in a restricted platform with promising applications. Starting with Michail, Chatzigiannakis, and Spirakis…

We introduce a class of digital machines we name Digital Memcomputing Machines (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space and energy). An abstract…

Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…

Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…

Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time…

We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…

These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…

This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a…

Many variants of RSA cryptosystem exist in the literature. One of them is RSA over polynomials based on Galois approach. In standard RSA modulus is product of two large primes whereas in the Galois approach author considered the modulus as…

As society becomes more reliant on computers, cryptographic security becomes increasingly important. Current encryption schemes include the ElGamal signature scheme, which depends on the complexity of the discrete logarithm problem. It is…

This thesis aims to use intelligent systems to extend and improve performance and security of cryptographic techniques. Genetic algorithms framework for cryptanalysis problem is addressed. A novel extension to the differential cryptanalysis…

This paper presents the Quanta Prime Sequence (QPS) and its foundational theorem, showcasing a unique class of polynomials with substantial implications. The study uncovers profound connections between Quanta Prime numbers and essential…

Ensuring data integrity is a critical requirement in complex systems, especially in financial platforms where vast amounts of data must be consistently accurate and reliable. This paper presents a robust approach using polynomial…

Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer…