Number Theory and Cryptography

The Graduate Texts in Mathematics brand is a good brand. They're generally pretty easy reads for textbooks. I looked at the Table of Contents on Amazon, and it looks pretty self-contained. Though since it's geared at graduate students and advanced undergraduates, I can understand why you might feel the coverage is insufficient.

I think the table of contents for chapters 1-2 provide a good study guide, if you don't find the book's treatment sufficient. Number Theory is the study of the integers. It sounds intimidating, but you've really been doing it at some level since grade school. Remember long division where you have remainders? That's what modular arithmetic encapsulates. A lot of basic number theory deals with divisibility. In terms of a good extra reference, I think a Kenneth Rosen's Elementary Number Theory text is a good book for this. It will also address other topics like quadratic residues and the discrete log problem. I'm pretty sure there is coverage of RSA and a couple other cryptosystems in there.

In terms of the Abstract Algebra, the big hitter is with finite fields. To understand what a field is, you first have to understand what a group is. A group is a set of elements with an associative operation. There is a unique identity in the group and every element in the group is invertible. So the integers over addition form a group. Here, the identity element is 00. A group is called Abelian if the operator is commutative. That is, for all a,b∈Ga,b∈G, a+b=b+aa+b=b+a. So the integers over addition are commutative.

Now a field is a set of elements has two operations. We will call them addition and multiplication. The field is an Abelian group over addition. We refer to 00 as the additive identity. The field is also an Abelian group over multiplication, but we ignore 00 in the considerations. The identity on multiplication is 11. Some good examples of fields are the real numbers, rational numbers, complex numbers, and ZpZp (for pp a prime). If you look at the reals, note that r∗1r=1r∗1r=1, right? So 1r1r is the multiplicative inverse of rr. Notice that we cannot consider 00 for multiplicative inverses as we cannot divide by 00. The reals are invertible on addition just as the integers are.

In terms of books, Durbin's Modern Algebra text is a very introductory textbook, geared more towards sophomores and inexperienced juniors in mathematics. You may find it helpful and not so intimidating.

Hope this helps give you some direction and insights into what you're studying. Best of luck!