This book makes a very accessible introduction to a very important contemporary application of number theory, abstract algebra, and probability. It contains numerous computational examples throughout, giving learners the opportunity to apply, practice, and check their understanding of key concepts. KEY TOPICS Coverage starts from scratch in treating probability, entropy, compression, Shannon¿s theorems, cyclic redundancy checks, and error-correction. For enthusiasts of abstract algebra and number theory. This book is intended to be accessible to undergraduate students with two years of typical mathematics experience, most likely meaning calculus with a little linear algebra and differential equations. Thus, specifically, there is no assumption of a background in abstract algebra or number theory, nor of probability, nor of linear algebra. All these things are introduced and developed to a degree sufficient to address the issues at hand. We will address the fundamental problem of transmitting information effectively and accurately. The specific mode of transmission does not really play a role in our discussion. On the other hand, we should mention that the importance of the issues of efficiency and accuracy has increased largely due to the advent of the internet and, even more so, due to the rapid development of wireless communications. For this reason it makes sense to think of networked computers or wireless devices as archetypical fundamental practical examples. The underlying concepts of information and information content of data make sense independently of computers, and are relevant in looking at the operation of natural languages such as English, and of other modes of operation by which people acquire and process data. The issue of efficiency is the obvious one: transmitting information costs time, money, and bandwidth. It is important to use as little as possible of each of these resources. Data compression is one way to pursue this efficiency. Some well known examples of compression schemes are commonly used for graphics: GIFs, JPEGs, and more recently PNGs. These clever file format schemes are enormously more efficient in terms of filesize than straightforward bitmap descriptions of graphics files. There are also general-purpose compression schemes, such as gzip, bzip2, ZIP, etc. The issue of accuracy is addressed by detection and correction of errors that occur during transmission or storage of data. The single most important practical example is the TCP/IP protocol, widely used on the internet: one basic aspect of this is that if any of the packets composing a message is discovered to be mangled or lost, the packet is simply retransmitted. The detection of lost packets is based on numbering the collection making up a given message. The detection of mangled packets is by use of 16-bit checksums in the headers of IP and TCP packets. We will not worry about the technical details of TCP/IP here, but only note that email and many other types of internet traffic depend upon this protocol, which makes essential use of rudimentary error-detection devices. And it is a fact of life that dust settles on CD-ROMs, static permeates network lines, etc. That is, there is noise in all communication systems. Human natural languages have evolved to include sufficient redundancy so that usually much less than 100% of a message need be received to be properly understood. Such redundancy must be designed into CD-ROM and other data storage protocols to achieve similar robustness. There are other uses for detection of changes in data: if the data in question is the operating system of your computer, a change not initiated by you is probably a sign of something bad, either failure in hardware or software, or intrusion by hostile agents (whether software or wetware). Therefore, an important component of systems security is implementation of a suitable procedure to detect alterations in critical files. In pre-internet times, various schemes were used to reduce the bulk of communication without losing the content: this influenced the design of the telegraphic alphabet, traffic lights, shorthand, etc. With the advent of the telephone and radio, these matters became even more significant. Communication with exploratory spacecraft having very limited resources available in deep space is a dramatic example of how the need for efficient and accurate transmission of information has increased in our recent history. In this course we will begin with the model of communication and information made explicit by Claude Shannon in the 1940's, after some preliminary forays by Hartley and others in the preceding decades. Many things are omitted due to lack of space and time. In spite of their tremendous importance, we do not mention convolutional codes at all. This is partly because there is less known about them mathematically. Concatenated codes are mentioned only brie