The theory of groups, rings and modules is developed to a great depth. Group theory results include Zassenhaus’s theorem and the Jordan-Hoelder theorem. The ring theory development includes ideals, quotient rings and the Chinese remainder theorem. The module development includes the Nakayama lemma, exact sequences and Tensor products.

For a positive composite integer n, we investigate the residues of ( mn k ) for positive integers m and k. First, we discuss divisibility of such coefficients. Then we study congruence identities relating products of binomial coefficients modulo n. Certainly the Chinese Remainder Theorem can be used in combination with prime power results to evaluate binomial coefficients modulo a composite. Ho...

Corresponding Author: Sayee Kumar, M., Muthayammal Engineering College, Rasipuram, Tamilnadu, India Email: [email protected] Abstract: In hasty growth of communication, security plays a central role in maintaining confidentiality of data in group communication. Keeping the data intended for the group in confidential manner is the most important security feature need to be sustained for t...

In this paper, a novel 3-moduli set { 2 − 1, 2, 2 − 1 } , which has larger dynamic range when compared to other existing 3-moduli sets is proposed. After providing a proof that this moduli set always results in legitimate RNS, we subsequently propose an associated reverse converter based on the New Chinese Remainder Theorem. The proposed reverse converter has a delay of (4n + 6)tFA with an area...

Broadcast authentication is a critical security service in wireless sensor networks (WSNs). However, due to resource constrained of sensor nodes, providing an authentication mechanism for broadcast message is difficult. This paper deals with the forwarding scheme for wireless sensor networks aimed at combining low computational complexity and high performance in terms of energy efficiency with ...

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the ...

We show that multiplication complexities of n-term Karatsuba-Like formulae of GF (2)[x] (7 < n < 19) presented in the above paper can be further improved using the Chinese Remainder Theorem and the construction multiplication modulo (x−∞).