Computationally Efficient Approach to Implementation of the Chinese Remainder Theorem Algorithm in Minimally Redundant Residue Number System

In this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. Throughout the article, an efficient approach for implementing the CRT algorithm is described. The structure of the rank of an RNS number, a principal positional characteristic of the residue code, is investigated. It is shown that the rank of a number can be represented by a sum of an inexact rank and a two-valued correction to it. We propose a new variant of minimally redundant RNS, which provides low computational complexity for the rank calculation, and its effectiveness analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the excess residue modulo 2, the complexity of the rank calculation goes down from $O\left (k^{2}\right )$ O k 2 to $O\left (k\right )$ O k with respect to required modular addition operations and lookup tables, where k equals the number of non-redundant RNS moduli.

On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime p to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.

Symmetries of abelian Chern-Simons theories and arithmetic

We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including U(1)k Chern-Simons theory and (ℤk)ℓ gauge theories. For example, we prove that U(1)k Chern-Simons theory is time-reversal invariant if and only if −1 is a quadratic residue modulo k, which happens if and only if all the prime factors of k are Pythagorean (i.e., of the form 4n + 1), or Pythagorean with a single additional factor of 2. Many distinct non-abelian finite symmetry groups are found.

The least quadratic non-residue

For a prime number p p , we say a a is a quadratic non-residue modulo p p if there is no integer x x such that x 2 ≡ a mod p x^2\equiv a\bmod {p} . The problem of bounding the least quadratic non-residue modulo p p has a rich mathematical history. Moreover, there have been recent results, especially concerning explicit estimates. In this survey paper we give the history of the problem and explain many of the main achievements, giving explicit versions of these results in most cases. The paper is intended as a self-contained collection of the main ideas that have been used to attack the problem.

An efficient implementation of the Chinese Remainder Theorem in minimally redundant Residue Number System

The Chinese Remainder Theorem (CRT) widely used in many modern computer applications. This paper presents an efficient approach to the calculation of the rank of a number, a principal positional characteristic used in the Residue Number System (RNS). The proposed method does not use large modulo addition operations compared to a straightforward implementation of the CRT algorithm. The rank of a number is equal to a sum of an inexact rank and a two-valued correction factor that only takes on the values 0 or 1. We propose a minimally redundant RNS, which provides low computational complexity of the rank calculation. The effectiveness of the novel method is analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the extra residue modulo 2, the complexity of rank calculation goes down from \(O(k^2)\) to \(O(k)\), where \(k\) equals the number of residues in non-redundant RNS.

Magic and Modulo Magicness of Paley Digraph

A special digraph arises in round robin tournaments. More exactly, a tournament Tq with q players 1, 2, ... , q in which there are no draws. This gives rise to a digraph in which either (u, v) or (v, u) is an arc for each pair u, v. Graham and Spencer defined the tournament as, The nodes of digraph Dp are {0, 1, ... , p -1} and Dp contains the arc (u, v) if and only if u - v is a quadratic residue modulo p where p  3(mod 4) be a prime. This digraph is referred as the Paley tournament. Raymond Paley was a person raised Hadamard matrices by using this quadratic residues. So to honor him this tournament was named as Paley tournament. These results were extended by Bollobas for prime powers. Modular super edge trimagic labeling and modular super vertex magic total labeling has been investigated in this paper.

ON THE RELATIVE FREQUENCY OF RESIDUE CLASSES IN PASCAL’S TRIANGLE MODULO A PRIME

When the entries of Pascal’s triangle which are congruent to a given nonzero residue modulo a fixed prime are mapped to corresponding locations of the unit square, a fractal-like structure emerges. In a previous publication, Bradley, Khalil, Niemeyer and Ossanna [The box-counting dimension of Pascal’s triangle [Formula: see text] mod [Formula: see text], Fractals 26(5) (2018) 1850071] showed that this mapping yields a nonempty compact set which can be realized as a limit of a sequence of sets representing incrementally refined approximations. Moreover, it was shown therein that for any fixed prime, the sequence converges to the same set, regardless of the nonzero residue or combination of nonzero residues considered. Consequently, the fractal (box-counting) dimension of the limiting set is independent of the residue. To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first [Formula: see text] rows of Pascal’s triangle which are congruent to a given nonzero residue [Formula: see text] modulo the prime [Formula: see text]. Unfortunately, the numerical evidence presented in this paper supports the contention that there is no such formula. Nevertheless, the evidence indicates that for sufficiently large primes [Formula: see text], the number of entries congruent to [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] is well approximated by the respective linear functions [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, for large primes [Formula: see text] there are approximately six times as many occurrences of the residue [Formula: see text] in the first [Formula: see text] rows of Pascal’s triangle reduced modulo [Formula: see text] than there are of any other residue [Formula: see text] in the range [Formula: see text], and three times as many as [Formula: see text]. On the other hand, if we let the nonnegative integer [Formula: see text] vary while keeping the prime [Formula: see text] fixed, and look at the relative frequency of various residue classes that occur in the first [Formula: see text] rows, the seemingly substantial differences in frequency between [Formula: see text], [Formula: see text], and [Formula: see text] when [Formula: see text] are increasingly dissipated as [Formula: see text] grows without bound. We show that in the limit as [Formula: see text] tends to infinity, all nonzero residues are equally represented with asymptotic proportion [Formula: see text].

Quadratic residue modulo a power of 2

Quadratic residue modulo an odd prime power has been studied for centuries. Many mathematical tools have been devised to deal with those odd prime power. The left moduli of powers of 2 are thus naturally the subject of this article. We set the objectives of showing some intriguing properties of quadratic residues modulo an even prime power. Though humble in its significance, our results are achieved after years of reading Prasolov’s and I. F. Sharygin’s maths books.

Performance analysis of arithmetic algorithms implemented in C++ and Python programming languages

This paper presents the results of the numerical experiment, which aims to clarify the actual performance of arithmetic algorithms implemented in C ++ and Python programming languages using arbitrary precision arithmetic. "Addition machine" has been chosen as a mathematical model for integer arithmetic algorithms. "Addition machine" is a mathematical abstraction, introduced by R. Floyd and D. Knuth. The essence of "addition machine" is the following: using only operations of addition, subtraction, comparison, assignment and a limited number of registers it is possible to calculate more complex operations such as finding the residue modulo, multiplication, finding the greatest common divisor, exponentiation modulo with reasonable computational efficiency. One of the features of this implementation is the use of arbitrary precision arithmetic, which may be useful in cryptographic algorithms.