Finite fields

The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

Introduction to finite fields and their applications

We combine well-known techniques from the areas of error-correcting codes and cryptography to achieve a new type of cryptographic primitive that we refer to as a fuzzy commitment scheme. Like a conventional cryptographic commitment scheme, our fuzzy commitment scheme is both concealing and binding: it is infeasible for an attacker to learn the committed value, and also for the committer to decommit a value in more than one way. In a conventional scheme, a commitment must be opened using a unique witness, which acts, essentially, as a decryption key. By contrast, our scheme is fuzzy in the sense that it accepts a witness that is close to the original encrypting witness in a suitable metric, but not necessarily identical.This characteristic of our fuzzy commitment scheme makes it useful for applications such as biometric authentication systems, in which data is subject to random noise. Because the scheme is tolerant of error, it is capable of protecting biometric data just as conventional cryptographic techniques, like hash functions, are used to protect alphanumeric passwords. This addresses a major outstanding problem in the theory of biometric authentication. We prove the security characteristics of our fuzzy commitment scheme relative to the properties of an underlying cryptographic hash function.

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A fuzzy commitment scheme

Sphere Packings, Lattices and Groups

The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment. In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

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Quantum error correction via codes over GF(4)

Preface 1. Basic concepts of linear codes 2. Bounds on size of codes 3. Finite fields 4. Cyclic codes 5. BCH and Reed-Soloman codes 6. Duadic codes 7. Weight distributions 8. Designs 9. Self-dual codes 10. Some favourite self-dual codes 11. Covering radius and cosets 12. Codes over Z4 13. Codes from algebraic geometry 14. Convolutional codes 15. Soft decision and iterative decoding Bibliography Index.

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Fundamentals of Error-Correcting Codes

Preface. List of Contributors. An introduction to algebraic codes (V.S. Pless, W.C. Huffman, R.A. Brualdi). Coding constructions (V.S. Pless). Self-dual codes (E.M. Rains, N.J.A. Sloane). Bounds on the size of linear codes (A.E. Brouwer). An updated table of the best binary codes known (S. Litsyn). Universal bounds for codes and designs (V.I. Levenshtein). Complexity issues in coding theory (A. Barg). Covering radius (R.A. Brualdi, S. Litsyn, V.S. Pless). Quadratic residue codes and divisibility (H.N. Ward). Algebraic geometry codes (T. Hoholdt, J.H. van Lint, R. Pellikaan). Open problems on cyclic codes (P. Charpin). The algebraic theory of concolutional codes (R.J. McEliece). Author index. Subject index.

http://www.gbv.de/dms/ilmenau/toc/269730176.PDF

Handbook Of Coding Theory

Introductory Concepts. Useful Background. A Double--Error--Correcting BCH Code and a Finite Field of 16 Elements. Finite Fields. Cyclic Codes. Group of a Code and Quadratic Residue (QR) Codes. Bose--Chaudhuri--Hocquenghem (BCH) Codes. Weight Distributions. Designs and Games. Some Codes Are Unique. Appendix. References. Index.

Vera Pless

Papers

Fundamentals of Error-Correcting Codes

Handbook Of Coding Theory

Introduction to the Theory of Error-Correcting Codes

A classification of self-orthogonal codes over GF(2)

On the uniqueness of the Golay codes