Number Theory – Recommended Books on Mathematics Behind Cryptography

cryptographynumber theorystatistics

I am currently reading the Book Understanding Cryptography from Cristof Paar. I am enjoying the book but i don't like to scratch the surface when it comes to cryptography. I would like do dig a little deeper on the mathematics behind it. I read on:

https://www.schneier.com/crypto-gram/archives/1999/1015.html

that cryptography demands an understanding of multiple fields of mathematics, such as:

-Number theory

-Complexity theory

-Algorithms

-Statistics

-Abstract algebra

I have a masters on Telecommunications and Informatics, so i have an understanding of mathematics, calculus, statistics, algebra and complex calculus. I am no pro, but i think i am good. Could you advise me on books i should buy? I would like books that explain theorems and demonstrate those theorems. Do you think i am doing the right thing? Should i read only applied cryptography books, or should go deeper on the subject? I would like to go beyond the basic understanding. I would like to develop algorithms and break other algorithms. If you know books that talk about all these subjects, they are more than welcome.

Thank you

Kind regards

Best Answer

I would recommend a series of books, specifications, libraries and CAS programs.

Books

  • A Course in Number Theory and Cryptography, Neal Koblitz (very dense, but an amazing book)
  • An Introduction to Mathematical Cryptography, Jeffrey Hoffstein, Jill Pipher, J.H. Silverman (very readable and excellent book, which is more up-to-date)
  • An Introduction to Cryptography, Second Edition (Discrete Mathematics and Its Applications), Richard A. Mollin (easier on the math)
  • Handbook of Applied Cryptography, Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone (note, there is also a variant for Elliptic Curve Cryptography and both are just an excellent way to describe and detail cryptographic algorithms)
  • As a side note, these should be supplemented with books on Number Theory, Abstract Algebra, Algebraic Geometry and the like, but there are many excellent posts for such books on MSE

Specifications

Software Libraries

Computer Algebra Systems

  • Mathematica, Maple, ... (not free, professional packages with large integer math)
  • SAGE, Pari/Gp, ..., Python (free)
  • Here is a list of CAS programs, which is very helpful for experimenting with number theory topics and quickly coding things up (enormously helpful)

There are many others, but these are the ones I go back to often.

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