A new parallelization of Euclid’s greatest common divisor algorithm is proposed. It matches the best existing integer GCD algorithms since it can be achieved in parallel Oε(n/log n) time using only n processors on a Priority CRCW PRAM.

Blind image deconvolution refers to the process of determining both an exact image and the blurring function from its inexact image. This thesis presents a solution of the blind image deconvolution problem using polynomial computations. The proposed solution does not require prior knowledge of the blurring function or noise level. Blind image deconvolution is needed in many applications, such a...

As an effective solution to protect the privacy of the data, homomorphic encryption has become a hot research topic. Existing homomorphic schemes are not truly practical due to their high computational complexity and huge key size. In 2013, Coron et al. proposed a batch homomorphic encryption scheme, i.e. a scheme that supports encrypting and homomorphically evaluating several plaintext bits as...

Yiannis Moschovakis suggested the following example of a classical existence proof with a quantifier–free kernel which does not obviously contain an algorithm: the gcd of two natural numbers a1 and a2 is a linear combination of the two. Here we treat that example as a case study for program extraction from classical proofs. We apply H. Friedman’s A– translation [3] followed by a modified realiz...

This paper derives the mean squared length of vectors in a lattice relevant to the Approximate GCD problem and the related fully homomorphic encryption scheme.

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, ...,+∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characteri...

We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality ax mod b 6 x, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.

This paper considers the computation of greatest common divisor (GCD) dt1,t2(x,y) three bivariate Bernstein polynomials that are defined in a rectangular domain, where t1(t2) is degree when it written as polynomial x(y) whose coefficients y(x). The Sylvester resultant matrix and its subresultant matrices used for degrees GCD. It shown there four forms these they differ their computational prope...

Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the code length is smaller than the number of rational points on the curve, then this method can correct up to 12(d ∗ − 1)− s errors, where d∗ is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all ra...