Gröbner Bases, Coding, and Cryptography

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Gröbner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations.

This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will findthat all chapters fit nicely within a unified notation.

Gröbner Bases, Coding, and Cryptography: a Guide to the State-of-Art.- Gröbner Bases, Coding, and Cryptography: a Guide to the State-of-Art.- Invited Papers.- Gröbner Technology.- The FGLM Problem and Möller’s Algorithm on Zero-dimensional Ideals.- An Introduction to Linear and Cyclic Codes.- Decoding Cyclic Codes: the Cooper Philosophy.- A Tutorial on AG Code Construction from a Gröbner Basis Perspective.- Automorphisms and Encoding of AG and Order Domain Codes.- Algebraic Geometry Codes from Order Domains.- The BMS Algorithm.- The BMS Algorithm and Decoding of AG Codes.- A Tutorial on AG Code Decoding from a Gröbner Basis Perspective.- FGLM-Like Decoding: from Fitzpatrick’s Approach to Recent Developments.- An Introduction to Ring-Linear Coding Theory.- Gröbner Bases over Commutative Rings and Applications to Coding Theory.- Overview of Cryptanalysis Techniques in Multivariate Public Key Cryptography.- A Survey on Polly Cracker Systems.- Block Ciphers: Algebraic Cryptanalysis and Gröbner Bases.- Algebraic Attacks on Stream Ciphers with Gröbner Bases.- Notes.- Canonical Representation of Quasicyclic Codes Using Gröbner Bases Theory.- About the nth-Root Codes: a Gröbner Basis Approach to the Weight Computation.- Decoding Linear Error-Correcting Codes up to Half the Minimum Distance with Gröbner Bases.- Gröbner Bases for the Distance Distribution of Systematic Codes.- A Prize Problem in Coding Theory.- An Application of Möller’s Algorithm to Coding Theory.- Mattson Solomon Transform and Algebra Codes.- Decoding Folded Reed–Solomon Codes Using Hensel-Lifting.- A Note on the Generalisation of the Guruswami–Sudan List Decoding Algorithm to Reed–Muller Codes.- Viewing Multipoint Codes as Subcodes of One-Point Codes.- A Short Introduction to Cyclic Convolutional Codes.- On the Non-linearity of Boolean Functions.- Quasigroups as Boolean Functions, Their Equation Systems and Gröbner Bases.- A New Measure to Estimate Pseudo-Randomness of Boolean Functions and Relations withGröbner Bases.- Radical Computation for Small Characteristics.