9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups of the Integers . . . . . . . . . . . . . . . . . . . . 1 9.2 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . 2 9.3 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . 3 9.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9.5 The Fundamental Theorem of Arithmetic . . . . . . . . ....

When the image data are regularly changed, existing image encryption algorithm is easy to be decrypted. In order to improve the security of encryption algorithm, this paper proposes an image encryption algorithm based on chaotic mapping and Chinese remainder theorem. The encryption algorithm is divided into pixel scrambling and image diffusion. Firstly, the Chebyshev mapping is used to generate...

A recent work of Harn and Fuyou presents the first multilevel (disjunctive) threshold secret sharing scheme based on the Chinese Remainder Theorem. In this work, we first show that the proposed method is not secure and also fails to work with a certain natural setting of the threshold values on compartments. We then propose a secure scheme that works for all threshold settings. In this scheme, ...

In this paper, we propose a new efficient reverse converter for the 4-moduli set { 2, 2 + 1, 2 − 1, 2 − 1 } based on a modified Chinese Remainder Theorem and Mixed Radix Conversion. Additionally, the resulting architecture is further reduced to obtain a reverse converter that utilizes only carry save adders, a multiplexer and carry propagate adders. The proposed converter has an area cost of (1...

Using a prime number p of memory banks on a vector processor allows a conflict-free access for any slice of p consecutive elements of a vector stored with a stride not multiple of p. To reject the use of a prime number of memory banks, it is generally advanced that address computation for such a memory system would require systematic Euclidean division by the number p. The Chinese Remainder The...

In this paper, a novel method for image encryption based on a Generalized Chinese Remainder Theorem (GCRT) is presented. The proposed method is based on the work developed by Jagannathan et al. Some modifications are proposed in order to increase the method's encryption quality and its robustness against attacks. Specifically, the inclusion of a vector to reduce the segment pixel space and a Ge...

We give some new q-supercongruences on truncated forms of squares basic hypergeometric series. Most them are modulo the cube a cyclotomic polynomial, and two fourth power polynomial. The main ingredients our proofs creative microscoping method, lemma El Bachraoui, Chinese remainder theorem for coprime polynomials. also propose several related conjectures further study.

With the help of creative microscoping method recently introduced by Guo and Zudilin Chinese remainder theorem for coprime polynomials, we establish a q -supercongruence with two parameters modulo [ n ] ? ( ) 3 . Here = 1 ? / is -th cyclotomic polynomial in In particular, confirm recent conjecture give complete -analogue Long's supercongruence. The latter also generalization obtained Schlosser.

In this paper, we propose two novel memoryless reverse converters for the moduli set {2 − 1, 2, 2 − 1}. The first proposed converter does not entirely cover the dynamic range while the second proposed converter covers the entire dynamic range. First, we simplify the Chinese Remainder Theorem in order to obtain a reverse converter that utilizes mod-(2 − 1) operation. Second, we further reduce th...

This work presents a mathematical framework for the development of efficient algorithms for cyclic convolution computations. The framework is based on the Chinese Reminder Theorem (CRT) and the Winograd’s Minimal Multiplicative Complexity Theorem, obtaining a set of formulations that simplify cyclic convolution (CC) computations. In particularly, this work focuses on the arithmetic complexity o...