I am an undergraduate student looking for resources (books or lectures) explaining mathematics involved in cryptography, such as number theory, elliptic curves etc. I found the book 'A Course in Number Theory and Cryptography by Neal Koblitz' hard to follow.
Johannes Buchmann, Introduction To Cryptography, Springer, 2nd Ed, 2004
is very nice and pitched squarely at undergraduates. You can see a preview here
Its contents are:
Integers Congruences and Residue Class Rings Encryption Probability and Perfect Secrecy DES AES Prime Number Generation Public Key Encryption Factoring Discrete Logarithms Cryptographic Hash Functions Digital Signatures Other Systems Identification Secret Sharing Public Key Infrastructures
It is broad, as opposed to very deep, but really well written.
Nigel Smart has written Cryptography Made Simple. If you have institutional access, the ebook can be downloaded for free from SpringerLink. To quote the book regarding prerequisites:
The background I assume is what one could expect of a third or fourth year undergraduate in computer science. One can assume that such students have already met the basics of discrete mathematics (modular arithmetic) and a little probability. In addition, they will have at some point done (but probably forgotten) elementary calculus. Not that one needs calculus for cryptography, but the ability to happily deal with equations and symbols is certainly helpful. Apart from that I introduce everything needed from scratch. For those students who wish to dig into the mathematics a little more, or who need some further reading, I have provided an appendix which covers most of the basic algebra and notation needed to cope with modern cryptosystems.
While I have not fully read the book, it has suitable references to (most) of the popular "advanced mathematics" underlying cryptography schemes. It in particular has sections on:
- (Finite field) DLOG/Factoring based schemes
- Elliptic Curves
Steven Galbraith has written The Mathematics of Public-Key Cryptography. This is written at a more advanced level (and mentions Smart's book itself as an intended pre-requisite), but is available free at the author's website. As it is at a more advanced level I would not recommend it for your situation, but there are certain topics covered in it (Isogenies for example) which are often not covered in other books, so depending on your particular interests there are sections of it which may still be appropriate.