Chapter

## Abstract

This chapter will discuss the Linear Sieve and Quadratic Sieve, algorithms for factoring the product of two distinct prime integers, or any other composite number. The main purpose of the algorithm is to break the famous cryptosystem RSA. The algorithms use matrices over $$\mathbb{G}\mathbb{F}$$(2), but they will be sparse matrices rather than dense matrices.

This chapter is here for several reasons. First, we have written primarily of dense matrices over $$\mathbb{G}\mathbb{F}$$(2), and the exposition would be incomplete without discussing sparse matrices. The sparse matrix techniques described in Appendix D can be used anywhere that sparsity occurs in cryptanalysis, but many were designed for the Quadratic Sieve. Second, we have written of how to break block ciphers and stream ciphers, so it would be a pity not to discuss how to break public-key systems as well. Third, the Quadratic Sieve algorithm, when taken with all its variants and modifications, stands as one of the most sophisticated algorithms in all of computer science, and fourth, it uses some elegant number theory.

This is only the tip of a very large iceberg. There are many variations, improvements, and enhancements which are omitted here. Many of those are crucial in factoring larger numbers. Furthermore, we exclude many other important factoring algorithms, because they are unrelated to the Quadratic Sieve. While the NFS (Number Field Sieve) has eclipsed the Quadratic Sieve as an algorithm, understanding the NFS is much easier after studying the QS. We hope this section will inspire the reader to read further on this vital topic.

## Keywords

Null Space Stream Cipher Sparse Matrice Chinese Remainder Theorem Euclidean Algorithm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.