In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is ...

An access control mechanism is a technology to protect the confidential files stored in a database by restricting the access rights of different approved users of these files. In this paper, we propose a novel access control mechanism using the single-key-lock system and the generalized Aryabhata remainder theorem (GART), in which each user is associated with a key and each digital file with a ...

Based on unique decoding of the polynomial residue code with non-pairwise coprime moduli, a polynomial with degree less than that of the least common multiple (lcm) of all the moduli can be accurately reconstructed when the number of residue errors is less than half the minimum distance of the code. However, once the number of residue errors is beyond half the minimum distance of the code, the ...

In view of the problems that the plaintext space is too small in the existing schemes. In this paper, a new improved scheme is presented by improving the DGHV scheme. The plaintext space of the improved scheme is extended from finite prime field F2 in the original scheme to finite prime field Fp. Combine and apply the method of encryption in the batch encryption scheme was proposed in 2013, and...

The aim of this paper is to find more Karatsuba-like formulae for a fixed set of moduli polynomials in GF (2)[x]. To this end, a theoretical framework is established. We first generalize the division algorithm, and then present a generalized definition of the remainder of integer division. Finally, a previously generalized Chinese remainder theorem is used to achieve our initial goal. As a by-p...

We propose different approaches of PDF files based steganography, essentially based on the Chinese Remainder Theorem. Here, after a cover PDF document has been released from unnecessary characters of ASCII code A0, a secret message is hidden in it using one of the proposed approaches, making it invisible to common PDF readers, and the file is then transmitted through a non-secure communication ...

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. Thi...

Given n relatively prime integers p1 < · · · < pn and an integer k < n, the Chinese Remainder Code, CRTp1,...,pn;k, has as its message space M = {0, . . . , ∏k i=1 pi − 1}, and encodes a message m ∈ M as the vector 〈m1, . . . ,mn〉, where mi = m(mod pi). The soft-decision decoding problem for the Chinese remainder code is given as input a vector of residues ~r = 〈r1, . . . , rn〉, a vector of wei...

We show that the fractional Sobolev inequality for the embedding H̊ s 2 (R ) →֒ L 2N N−s (R ), s ∈ (0, N) can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak L N N−s -norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fract...

In this note we consider the following Sobolev inequality