# Cryptography and Coding Theory

What is the difference between Cryptography and Coding Theory? What books are good books on these topics? Especially from the algebraic viewpoint.

Cryptography: Here is the essence of cryptography (both sides of the story). I have some information, and it's encoded in a text file. You really want to know what it says, but I can't let you do that. So what happens is I'm going to try to translate it into my own secret language that I can share with my pals, but not with you. Now, you want to do whatever you can to obtain and read my message. Once you have the message you have some different things. You can try to work backwards through my encoding scheme (if you know it), or you could try every possible decoding scheme until you find one that makes sense. What you're doing here is called cryptanalysis.

Coding Theory: I have some information, in the form of a binary string (a sequence of 0's and 1's) that I want to store for a long time on my hard drive. It's, like, a really long string. But I'm pretty sure my hard drive is a safe place to store it, so we can take away some of the redundancy. This compresses the file, and makes it easier to store.

Now, I want to take that file, and send it over the internet, but the connection isn't too good - it's a noisy channel. How can I make sure that you receive the correct file? I can do this by adding redundancy - finding an efficient way to send my message with the following property: If some percentage of the bits get "flipped" (some 0's turn into 1's and vice versa) when I send the message, you'll still be able to recognize what I said.

• Thank the answer is very clear – zacarias Dec 7 '12 at 23:22

There are many other excellent (both theoretical and applied in both these areas and there are combined books too), but these should get you started.

Cryptography Books

A Course in Number Theory and Cryptography, Neal Koblitz

Algebraic Aspects of Cryptography, Neal Koblitz

The mathematics of ciphers: number theory and RSA cryptography, Coutinho S C, Severino Collier Coutinho

An Introduction to Cryptography, Richard A. Mollin

Introduction to Cryptography with Coding Theory (2nd Edition) by Wade Trappe and Lawrence C. Washington (Jul 25, 2005)

Coding Theory

Introduction to Coding Theory (Graduate Texts in Mathematics), J.H. van Lint (Author)

Information and Coding Theory (Springer Undergraduate Mathematics Series), Gareth A. Jones (Author), J.Mary Jones (Author)

Coding Theory and Cryptography: The Essentials, Second Edition (Chapman & Hall/CRC Pure and Applied Mathematics) by D.C. Hankerson, Gary Hoffman, D.A. Leonard and Charles C. Lindner (Aug 4, 2000)

I can add more applied books to each list and there are many, so just ask.

Regards -A

• Thanks this is very helpful for me – zacarias Dec 7 '12 at 23:21

Cryptography is concerned with secure, Coding Theory with efficient, communication.

• I would add that coding theory also deals with reliable communication. – robjohn Jan 11 '13 at 11:49

Cryptography is a specific art of secret communication where the shape of the signal to be sent is not of main importance. The communication channel is also assumed to be secured, e.g, by wired cables or the existence of secret communication is public.

Another art of secret communication where the existence of the secret communication is not shared by the third parties is called steganograpy.

The former one uses mainly number theory whereas the latter one uses mainly reverse-coding-theory. I call it as reverse coding theory because the theory from channel coding is not directly applicable and should be a bit changed.

The art of science which targets at exposing the existence of secret messages is called steganalysis and uses mainly the theory from detection/classification theory whereas the the art of science which targets at decrypting the secret messages is called cryptanalysis and uses mainly the theory from probability and number theory.

The coding theory is divided into two subsections. One of them is called source coding and the other is called channel coding.It is mostly attributed to channel coding because it is wider and more comprehensive in my opinion.

Source coding deals with coding a given source with least possible number of bits. It is also known as lossless compression. The limits of this coding scheme is given By Shanon. No source coding sheme can encode a given source below the entropy of that source with an error free decoding option. The entropy of a source $$H(X)=\sum_i p(x_i) \log \frac{1}{p(x_i)}$$

is also known as the average information and for a given cardinality, random sources maximize it.

Channel coding on the other hand is opposite to the idea of soure coding. While source coding removes the redunancy to reach least possible bits to represent the source, channel coding aims at introducing redundancy in a clever way to provide error free communication over a noisy channel. The main aim here is to provide some channel codes which can achieve the chanel capacity which is defined between the transmit $X$ and received signal $Y$ as

$$C=\max_{p_x}I(X,Y)$$

where $I$ is called mutual information. There exist no channel codes which can transfer more number of bits from a noisy channel having a rate $R$ which is greater than $C$. As it can be seen there is an analogy between source and channel coding as well as cryptography and steganograpy although the theories that are used are changing from one to the other.

• Thank the answer is very clear – zacarias Dec 7 '12 at 23:24
• What knowledge is required for Coding Theory? – Evinda Feb 7 '16 at 21:32

As to undergraduate textbooks, Trappe/Washington is a good one that covers both coding theory and cryptography (with more emphasis on cryptography).

To appreciate the difference, look at Claude Shannon's seminal papers on Cryptography and on the foundations of Coding Theory.

this book is useful too : Designs and their Codes, J. D. Key

• How does this answer the question? – Ram Jan 11 '13 at 11:18
• @Ram: It addresses the request for books, I assume. – robjohn Jan 11 '13 at 11:36
• @robjohn when I wrote my comment there is no link, it seems he added (link) later. – Ram Jan 11 '13 at 11:41
• @Ram: I looked it up on amazon.com and added the link – robjohn Jan 11 '13 at 11:42
• The book appears as if it might be useful. Would the downvoter care to comment? – robjohn Jan 11 '13 at 11:45