Australian Journal of Basic and Applied Sciences, 5(11): 2272-2281, 2011 ISSN 1991-8178 Corresponding Author: Rand Alfaris, School of Mathematical Science, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia. E-mail: randalfaris@usm.my 2272 Two New Abelian Groups Based on JR-2CN And JR-3CN Rand Alfaris and Hailiza Kamarulhaili School of Mathematical Science, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia. Abstract. Two new Abelian groups are introduced in this paper. These Abelian groups are constructed based on the two families JR െ 2CN and JR െ 3CN. The elements of these Abelian groups are ordered pairs, each ordered pair represents an integer that can be represented as a summation of two signed cubic numbers which belongs to JR െ 2CN or three signed cubic numbers which belongs to JR െ 3CN. Two new binary operations have been defined to be associated with each family to form an Abelian group. Properties and propositions have been introduced. A one to one corresponding homomorphism function on JR െ 2CN and JR െ 3CN has been successfully defined. Key words: Abelian groups. Sum of two and three signed cubes. Isomorphism representation. INTRODUCTION The motivation behind constructing such groups was based on an unusual question: How to create new groups whose elements are ordered pair to employ them in public key Cryptography. The only known groups whose elements are ordered pairs are those which related to Elliptic curve cryptography. Elliptic curve cryptography groups are only used in Elliptic curve cryptography and cannot employ them in public key cryptography in general. In (Rand Alfaris and Hailiza Kamarulhaili, 2010), we introduced two new families JR െ 2CN and JR െ 3CN that generate two infinite subsets of integer numbers as a sum of two and three signed cubes respectively. Based on these two families, two new binary operations have been defined, followed by constructing two new Abelian groups. When constructing new groups, we start with defining the binary operation. Some subgroups, normal subgroups and fractional groups are introduced for each family. Most importantly, we found an isomorphism from JR െ 2CN to JR െ 3CN. This paper contains two main results. The first result is defining two new binary operations. These two new Abelian groups are based on special kind of numbers which they are the integer numbers that can be represented as a summation of two and three cubic numbers, The elements of the two groups are not numbers but they are ordered pairs which are considered rare in group theory. The second result is constructing an Isomorphism between the two Abelian groups. The Isomorphism indicates that there is a one to one correspondence between the two families. Isomorphism means transferring the properties and the features from one Abelian group to another. Finding an isomorphism between the two groups is suffice to study only one group and applied the results to the other. The work been done in this research, emerged the two areas, the number theory and group theory under one roof. As we mentioned earlier, the motivation behind constructing new Abelian groups, in addition to add two new Abelian groups to the list of the Abelian groups have been constructed so far, is to apply them in cryptography and to be specific, in public key cryptography and this will be our next step. The first application, as in (Andreas Enge, 2002) and (Eriault, 2003), is working on the Jacobian of hyperelliptic curve where the Jacobian is an Abelian group and as such it can serve as group for the discrete logarithm problem (DLP). The second application is using the Abelian groups in factorization methods in cryptography (Montgomery, 1987), such as elliptic curve, ൅1 and െ1 methods where each operates in an Abelian group. On the other hand, an application has been going on these new Abelian groups to construct finite Abelian groups based on JR െ 2CN and JR െ 3CN. This paper is organized as follows. The second section is on construction of the Abelian group that is related to the summation of two signed cubes as in (Rand Alfaris and Hailiza Kamarulhaili, 2010), denoted by ൫JR െ 2CN, ۩ JRଶCN ൯. The third section is on construction of an Abelian group that is related to the summation of three signed cubes, denoted by ൫JR െ 3CN, ۩ JRଷCN ൯, we mostly depend on (Stroppel, 2006) and (Zassenhaus, 1999). Section four is allocated to introducing the properties of each Abelian group, where we discussed some subgroups, normal subgroups and the fractional groups of each Abelian group. The last section discusses the Isomorphism between ൫JR െ 2CN, ۩ JRଶCN ൯ and ൫JR െ 3CN, ۩ JRଷCN ൯, the main reference was (Krylov, 2003).