ON THE DISTRIBUTION OF THE GREATEST COMMON DIVISOR OF GAUSSIAN INTEGERS.

A new method is proposed for estimating an image from two of its distorted versions without the a priori knowledge of the distortion functions. In z-domain, the original image can be regarded as the greatest common polynomial divisor between the distorted versions. With the assumption that the distortion filters are FIR and relatively co-prime, this becomes a problem of taking the greatest common divisor (GCD) of two or more two-dimensional polynomials. Exact GCD is not desirable because even extremely small variations due to quantization error or additive noise will destroy the integrity of the polynomial system and lead to a trivial solution. Our method of blind image deconvolution translates the two-dimensional GCD problem into a robust one-dimensional Sylvester-type GCD algorithm. Experimental results show that it is computationally efficient and moderately noise robust.

P. What is Number Theory? 1. The Integers. Numbers and Sequences. Sums and Products. Mathematical Induction. The Fibonacci Numbers. 2. Integer Representations and Operations. Representations of Integers. Computer Operations with Integers. Complexity of Integer Operations. 3. Primes and Greatest Common Divisors. Prime Numbers. The Distribution of Primes. Greatest Common Divisors. The Euclidean Algorithm. The Fundemental Theorem of Arithmetic. Factorization Methods and Fermat Numbers. Linear Diophantine Equations. 4. Congruences. Introduction to Congruences. Linear Congrences. The Chinese Remainder Theorem. Solving Polynomial Congruences. Systems of Linear Congruences. Factoring Using the Pollard Rho Method. 5. Applications of Congruences. Divisibility Tests. The perpetual Calendar. Round Robin Tournaments. Hashing Functions. Check Digits. 6. Some Special Congruences. Wilson's Theorem and Fermat's Little Theorem. Pseudoprimes. Euler's Theorem. 7. Multiplicative Functions. The Euler Phi-Function. The Sum and Number of Divisors. Perfect Numbers and Mersenne Primes. Mobius Inversion. 8. Cryptology. Character Ciphers. Block and Stream Ciphers. Exponentiation Ciphers. Knapsack Ciphers. Cryptographic Protocols and Applications. 9. Primitive Roots. The Order of an Integer and Primitive Roots. Primitive Roots for Primes. The Existence of Primitive Roots. Index Arithmetic. Primality Tests Using Orders of Integers and Primitive Roots. Universal Exponents. 10. Applications of Primitive Roots and the Order of an Integer. Pseudorandom Numbers. The EIGamal Cryptosystem. An Application to the Splicing of Telephone Cables. 11. Quadratic Residues. Quadratic Residues and nonresidues. The Law of Quadratic Reciprocity. The Jacobi Symbol. Euler Pseudoprimes. Zero-Knowledge Proofs. 12. Decimal Fractions and Continued. Decimal Fractions. Finite Continued Fractions. Infinite Continued Fractions. Periodic Continued Fractions. Factoring Using Continued Fractions. 13. Some Nonlinear Diophantine Equations. Pythagorean Triples. Fermat's Last Theorem. Sums of Squares. Pell's Equation. 14. The Gaussian Integers. Gaussian Primes. Unique Factorization of Gaussian Integers. Gaussian Integers and Sums of Squares.

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relatively coprime, in the absence of noise, this becomes a problem of taking the greatest common divisor (GCD) of two or more two-dimensional (2-D) polynomials. Exact GCD is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution. Our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2-D GCD method based on a one-dimensional (1-D) Sylvester-type GCD algorithm. Experimental results with both synthetically blurred images and real motion-blurred pictures show that it is computationally efficient and moderately noise robust.

Efficient calculation of the greatest common divisor (GCD) for big integers each whose number of bits is greater than or equal to 1024 has drawn a considerable amount of attention because it can be used to detect a weakness of the RSA security infrastructure. This paper presents a parallel binary GCD algorithm and its implementation for big integers on the Intel Xeon Phi coprocessor. This algorithm is capable of computing GCDs efficiently on many pairs of big integers in parallel by utilizing all cores on a Xeon Phi coprocessor as well as taking advantage of all vector processing units of the coprocessor to speed up critical integer operations within the algorithm. Using 240 threads on a Xeon Phi coprocessor to carry out GCD calculations for a large amount of 2048-bit integers, the implementation achieves the speedup of 30 times over a sequential binary GCD algorithm implementation on a single CPU core, and it delivers twice amount of performance in comparison to the same sequential binary GCD implementation running on 240 threads of the Xeon Phi.

We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.

(1912). Notes on Greatest Common Divisor and Least Common Multiple of Integers. The American Mathematical Monthly: Vol. 19, No. 1, pp. 4-6.

A simple linear transformation of the biquaternionic parameters α and β of the group SL(2, C), which is two-to-one homomorphic to the restricted Lorentz group ℒ, is used to express each element of SL(2, C) and ℒ in terms of the first column of an element of ℒ and the quaternionic parameters α. This parametrization is shown to bring the determination of the integral realizations of the restricted Lorentz group under the purview of classical Diophantine analysis, involving the expression of a given pair of integers, respectively, as a sum of three squares and a sum of four squares. These solutions are further constrained by the condition that a set of four integers, which is linear in the solution of the three squares problem and in the solution of the four squares problem, have a common factor. These are all classical problems addressed and solved by Euclid, Euler, Fermat, Gauss, Jacobi, and others. The corresponding realizations of SL(2, C) fall into three distinct classes: those in which the elements of the matrix are Gaussian integers, Gaussian integers divided by √, and Gaussian half-odd integers. These results apply also to the principal subgroups of SL(2, C).

We present an asymptotically fast algorithm for the computation of the greatest common divisor (GCD) of two Gaussian integers. Our algorithm is based on a controlled Euclidean descent in that the operands are not reduced too much in each Euclidean step. To compute a descent of n bits, the algorithm recursively calculates two descents of approximately n/2 bits each with short operands and transfers the thereby calculated cofactors to the original operands. Overall, this algorithm achieves a time bound of \(O(n(\rm log ~\it n)^{2} \rm log~ log~ \it n)\) bit operations for operands bounded by 2 n in absolute value.

The object of this paper is to give asymptotic estimates for some number theoretic sums over Gaussian integers. As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of Gaussian integers with only “large” prime factors and for the number of Gaussian integers with only “small” prime factors are given.

In the paper we give an introduction to a new algorithm counting the greatest common divisor (GCD) of natural integers called the approximating GCD algorithm introduced by S.Ishmukhametov in 2016. We compare it with the classical Euclidean GCD algorithm and the kary GCD algorithm in spirit of J. Sorenson and K. Weber and outline their advantages and disadvantages.

A matrix Euclidean algorithm induced by state space realization

We start with a number of fairly elementary results and techniques, mainly about greatest common divisors. You have probably met some of this material already, though it may not have been treated as formally as here. There are several good reasons for giving very precise definitions and proofs, even when there is general agreement about the validity of the mathematics involved. The first is that ‘general agreement’ is not the same as convincing proof: it is not unknown for majority opinion to be seriously mistaken about some point. A second reason is that, if we know exactly what assumptions are required in order to deduce certain conclusions, then we may be able to deduce similar conclusions in other areas where the same assumptions hold true. For example, this chapter is entirely devoted to the divisibility properties of integers, but it turns out that very similar definitions, methods and theorems are valid for certain other objects which can be added, subtracted and multiplied; some of these objects, such as polynomials, are very familiar, while others, such as Gaussian integers and quaternions, will be introduced in later chapters. These generalisations of the integers are also explored in algebra, under the heading of ring theory.KeywordsInteger SolutionDiophantine EquationGreat Common DivisorCommon MultipleUnique PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

Polynomial GCD (greatest common divisor) finding is an important problem in algebraic computation, especially in decoding error correcting codes. The authors show a new systolic array structure for the polynomial GCD problem using a systematic array synthesis technique. The VLSI implementation of the array structure is area-efficient and achieves maximum throughput with pipelining. The dependency graph (DG) of the Euclid GCD algorithm is drawn using iterated polynomial division. The resulting DG is data-dependent and variable-sized. The authors consider the worst-case implementation to make the DG data-dependent and fixed-size, where data-dependences are hidden inside by introducing four different working modes in each DG node. This novel approach requires just a few additional multiplexors and can be generalized for other data-dependent and variable-sized computation. The authors then map the DG to a one-dimensional systolic array using a linear mapping. The new array structure has m/sub 0/ + n/sub 0/ + 1 processing elements, where m/sub 0/ and n/sub 0/ are degrees of two polynomials. It can find a GCD of any two polynomials of total degree less than or equal to m/sub 0/ + n/sub 0/. The block pipeline period is one, which means that it can start a new GCD computation immediately in the next cycle. Unlike the array of Brent and Kung, a pre-processing step for extracting a common factor X/sup i/ is not necessary and the size of the processing element (PE) does not depend on m/sub 0/ and n/sub 0/. The authors extend this new array structure to the extended polynomial GCD algorithm, which is closely related to the decoding of BCH and Reed-Solomon codes. To verify the structure, they have used the VERILOG simulator, and implemented a 2 /spl mu/ CMOS test chip.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Let a, b and h be positive integers and S = {x 1, x 2, …, xh } be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ of {1, …, h} such that x σ(1)|…|x σ(h). We say that the set S consists of two coprime divisor chains if we can partition S as S = S 1 ∪ S 2, where S 1 and S 2 are divisor chains and each element of S 1 is coprime to each element of S 2. The matrix having the a-th power (xi , xj ) a of the GCD (GCD: greatest common divisor) of xi and xj as its i, j-entry is called a-th power GCD matrix defined on S, denoted by (Sa ). Similarly we can define the a-th power least common multiple (LCM) matrix [Sa ]. In this article, we show the following results: assume that S consists of two coprime divisor chains and 1∈S. We first show that if a|b, then the power GCD matrix (Sa ) divides the power GCD matrix (Sb ) in the ring Mh (Z) of h × h matrices over integers. But such factorization should not hold if . Consequently, we show that if a|b, the power LCM matrix [Sa ] divides the power LCM matrix [Sb ] in the ring Mh (Z). Finally we show that if a|b, the power GCD matrix (Sa ) divides the power LCM matrix [Sb ] in the ring Mh (Z). But such results fail to be true if . These results confirm partially Hong's conjectures.