We present a new algorithm for computing a greatest common divisor of any two non-zero Gaussian integers. It is Euclidean in the sense that it computes a sequence of quotients and remainders. It is approximative in the sense that it approximates the exact quotient of any two successive remainders, Ai and Ai+1, and then uses a nearest Gaussian integer, Qi, to that approximate quotient to compute...

In this paper we propose a technique to blindly synthesize the generator polynomial of BCH codes. The proposed technique involves finding Greatest Common Divisor (GCD) among different codewords and block lengths. Based on this combinatorial GCD calculation, correlation values are found. For a valid block length, the iterative GCD calculation results either into generator polynomial or some of i...

The task of determining the greatest common divisors (GCD) for several polynomials which arises in image compression, computer algebra and speech encoding can be formulated as a low rank approximation problem with Sylvester matrix. This paper demonstrates a method based on structured total least norm (STLN) algorithm for matrices with Sylvester structure. We demonstrate the algorithm to compute...

1 RSA-cryptosystem, and the underlying theory The aim of this section is to recall the RSA cryptographic method and the necessary theory showing its correctness, computational complexity, etc. 1.1 Creating an RSA system RSA (Rivest, Shamir, Adleman) is a particular method which can be used for the public-key cryptography. A concrete RSA system is created as follows: 1. Choose randomly two diffe...

A novel bit level block cipher based symmetric key cryptographic technique using G.C.D is proposed in this research paper. Entire plain text file is read one character at a time and according to the binary representation of ASCII value of the characters, entire plain text file is divided into n number of 16 bit blocks. Then an agreed-upon symmetric key file is formed by dividing each 16 bit blo...

We introduce concepts of “recursive polynomial remainder sequence (PRS)” and “recursive subresultant,” and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate “recursively” with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresult...

We show asymptotic upper and lower bounds for the greatest common divisor of N and σ(N). We also show that there are infinitely many integers N with fairly large g.c.d. of N and σ(N).

• Greatest common divisor (GCD) algorithms. We begin with Euclid’s algorithm, and the extended Euclidean algorithm [2, 12]. We will then discuss variations and improvements such as Lehmer’s algorithm [14], the binary algorithms [12], generalized binary algorithms [20], and FFT-based methods. We will also discuss how to adapt GCD algorithms to compute modular inverses and to compute the Jacobi a...

Combination of algebraic and numerical techniques for improving the computations in algebra and geometry is a popular research topic of growing interest. We survey some recent progress that we made in this area, in particular , regarding polynomial roottnding, the solution of a polynomial system of equations, the computation of an approximate greatest common divisor of two polynomials as well a...

The atdvent of practical parallel processors has caused a reexamination of many existing algorithms with'the hope of discovering a parallel implementation. One of the oldest and best know algorithms is Euclid's algorithm for computing the greatest common divisor (GCD). In this paper we present a parallel algorithm to compute the GCD of two integers. Although there have been results in the paral...