# QC-LDPC Code-Based CryptographyQuasi-Cyclic Codes

QC-LDPC Code-Based Cryptography: Quasi-Cyclic Codes [In this chapter, we recall the main definitions concerning quasi-cyclic codes, which will be used in the remainder of the book. We introduce the class of circulant matrices, and the special class of circulant permutation matrices, together with their isomorphism with polynomials over finite fields. We characterize the generator and parity-check matrices of quasi-cyclic codes, by defining their “blocks circulant” and “circulants block” forms, and show how they translate into an encoding circuit. We define a special class of quasi-cyclic codes having the parity-check matrix in the form of a single row of circulant blocks, which will be of interest in the following chapters. Finally, we describe how to achieve efficient encoding algorithms based on fast polynomial multiplication and vector-circulant matrix products.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# QC-LDPC Code-Based CryptographyQuasi-Cyclic Codes

18 pages      /lp/springer-journals/qc-ldpc-code-based-cryptography-quasi-cyclic-codes-xJpJPVU6hM
Publisher
Springer International Publishing
ISBN
978-3-319-02555-1
Pages
23 –40
DOI
10.1007/978-3-319-02556-8_3
Publisher site
See Chapter on Publisher Site

### Abstract

[In this chapter, we recall the main definitions concerning quasi-cyclic codes, which will be used in the remainder of the book. We introduce the class of circulant matrices, and the special class of circulant permutation matrices, together with their isomorphism with polynomials over finite fields. We characterize the generator and parity-check matrices of quasi-cyclic codes, by defining their “blocks circulant” and “circulants block” forms, and show how they translate into an encoding circuit. We define a special class of quasi-cyclic codes having the parity-check matrix in the form of a single row of circulant blocks, which will be of interest in the following chapters. Finally, we describe how to achieve efficient encoding algorithms based on fast polynomial multiplication and vector-circulant matrix products.]

Published: May 3, 2014

Keywords: Quasi-cyclic codes; Circulant matrices; Generator matrix; Parity-check matrix; Polynomial representation; Fast vector-by-circulant-matrix product