This chapter provides a brief overview of selected topics in the exciting field of cryptology. In particular, we cover symmetric and asymmetric encryption cryptosystems. When talking about symmetric cryptography, we deal with the notion of a block cipher as one of the main cryptographic primitives, i.e. building blocks, for modern practical applications. Asymmetric cryptography is illustrated by the RSA and El Gamal encryption cryptosystems. Later, we present code-based cryptosystems due to McEliece and Niederreiter. In this chapter we will use the terms “cryptosystem” and “encryption schemes” as synonyms for simplicity. These schemes are good examples for a connection between the theory of error-correcting codes and cryptography. Whereas in the theory of error-correcting codes one deals with correcting errors in noisy channels, in the cryptographic applications a crypto designer intentionally adds noise to confuse an attacker and only a legitimate counterpart can remove the noise from a transmitted message.

We also take a look at other interesting topics such as authentication codes, secret sharing, and linear feedback shift registers. Here, we especially discuss connections to codes and related structures. Therewith we show how error-correcting codes can be used in other parts of cryptography as well.

Symmetric Encryption Schemes and Block Ciphers

Symmetric Encryption Schemes

This section is devoted to symmetric cryptosystems, symmetric encryption schemes to be more precise. The idea behind these is quite simple and thus has been known for ages. The problem to be addressed is to convey a secret message between two parties, called traditionally Alice and Bob, so that figuring out the secret message is not possible without knowledge of some additional information. This additional information is called a secret key and is supposed to be known only to the two communicating parties. The secrecy of the transmitted message rests entirely upon the knowledge of this secret key, and thus if an adversary or an eavesdropper, traditionally called Eve, is able to find out the key, then the whole secret communication is corrupted. Now let us take a look at the formal definition.

The idea of redundant information is a well-known phenomenon in reading a newspaper. Misspellings usually go unnoticed for a casual reader, while the meaning is still grasped. In Semitic languages such as Hebrew, and even older in the hieroglyphics in the tombs of the pharaohs of Egypt, only the consonants are written while the vowels are left out so that we do not know for sure how to pronounce these words nowadays. The letter “e” is the most frequent occurring symbol in the English language, and leaving out all these letters would still give in almost all cases an understandable text to the expense of greater attention of the reader.

The science of deleting redundant information in a clever way such that it can be stored in less memory or space and still can be expanded to the original message is called data compression or source coding. It is not the topic of this book. So we can compress data, but an error made in a compressed text would give a different message that is most of the time utterly meaningless.

The idea in error-correcting codes is the converse. One adds redundant information in such a way that it is possible to detect or even correct errors after transmission. In radio contacts between pilots and radar controls the letters in the alphabet are spoken phonetically as “Alpha, Bravo, Charlie, …” but “Adams, Boston, Chicago, …” is more commonly used for spelling in a telephone conversation. The addition of a parity check symbol enables one to detect an error, such as on the former punch cards that were fed into a computer, in the ISBN code for books, the European Article Numbering (EAN) and the Universal Product Code (UPC) for articles. Error-correcting codes are common in numerous modern applications where data are required to be stored, processed and transmitted in a reliable manner, such as in audiovisual media, quick response (QR) codes, fault-tolerant computer systems and deep space telecommunication, to name but a few.

In this chapter, we present an introduction to error-correcting codes focusing on the most commonly used codes, namely, block codes and linear codes, including fundamental concepts and procedures for code construction, encoding and decoding.

A lot of mathematical objects are closely related to each other. While studying certain aspects of a mathematical object, one tries to find a way to “view” the object in a way that is most suitable for a specific problem. Or in other words, one tries to find the best way to model the problem. Many related fields of mathematics have evolved from one another this way. In practice, it is very useful to be able to transform your problem into other terminology: it gives a lot more available knowledge that can be helpful to solve a problem.

In this chapter we give a broad overview of fields closely related to error-correcting codes. From various methods of determining the weight enumerator, as treated in Chapter 3, we naturally run into other ways to view an error-correcting code. We will introduce and link the following mathematical objects: