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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.

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    $\begingroup$ It is a graduate book. Ideal, varieties, and algorithm by Cox at al. has some nice part for Elliptic curves, Elliptic tales by Ash and Gross. Any good number theory book should be enough. It is hard to say that any elementary number theory is not used in Cryptography. $\endgroup$ – kelalaka Jul 1 at 8:34
  • $\begingroup$ Does it need to be a book? There are quite a few ECC primers on the internet that are worth looking at, e.g. the one from Certicom: An Elliptic Curve Cryptography (ECC) Primer. You can read those first and then look up more advanced mathematical books / papers. $\endgroup$ – Maarten Bodewes Jul 1 at 10:32
  • $\begingroup$ No it doesn't necessarily have to be a book. $\endgroup$ – Aiytan Jul 2 at 4:43
  • $\begingroup$ Not a book reference, but as you are an undergraduate student, probably, you've access to multiple resources that are otherwise under a paywall, these are for example: Springer(Link), Elsevier, IEEE and journals on information security and cryptography. There you can find multiple schemes based on multiple branches of mathematics like: group theory, (group) algebras over a field, linear algebra and modules, number theory and a large etc. Some provide the reader with a background in the theory, so my recommendation is to let yourself go. Arxiv and IACR's eprint are good sources to browse too. $\endgroup$ – kub0x Jul 2 at 12:48
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I think

Johannes Buchmann, Introduction To Cryptography, Springer, 2nd Ed, 2004

is very nice and pitched squarely at undergraduates. You can see a preview here

https://books.google.com.au/books/about/Introduction_to_Cryptography.html?id=BuQlBQAAQBAJ&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false

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.

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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
  • Lattices

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.

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  • $\begingroup$ Thanks for the pointer to Galbraith's book! $\endgroup$ – auspicious99 Jul 3 at 15:05

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