Book contents
- Frontmatter
- Contents
- Introduction
- 1 Cyclohexane, cryptography, codes, and computer algebra
- I Euclid
- 2 Fundamental algorithms
- 3 The Euclidean Algorithm
- 4 Applications of the Euclidean Algorithm
- 5 Modular algorithms and interpolation
- 6 The resultant and gcd computation
- 7 Application: Decoding BCH codes
- II Newton
- III Gauß
- IV Fermat
- V Hilbert
- Appendix
- Sources of illustrations
- Sources of quotations
- List of algorithms
- List of figures and tables
- References
- List of notation
- Index
- The Holy Qur'ān (732)
5 - Modular algorithms and interpolation
from I - Euclid
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Introduction
- 1 Cyclohexane, cryptography, codes, and computer algebra
- I Euclid
- 2 Fundamental algorithms
- 3 The Euclidean Algorithm
- 4 Applications of the Euclidean Algorithm
- 5 Modular algorithms and interpolation
- 6 The resultant and gcd computation
- 7 Application: Decoding BCH codes
- II Newton
- III Gauß
- IV Fermat
- V Hilbert
- Appendix
- Sources of illustrations
- Sources of quotations
- List of algorithms
- List of figures and tables
- References
- List of notation
- Index
- The Holy Qur'ān (732)
Summary
An important general concept in computer algebra is the idea of using various types of representation for the objects at hand. As an example, we can represent a polynomial either by a list of its coefficients or by its values at sufficiently many points. In fact, this is just computer algebra lingo for the ubiquitous quest for efficient data structures for computational problems.
One successful instantiation of the general concept are modular algorithms, where instead of solving an integer problem (more generally, an algebraic computation problem over a Euclidean domain R) directly one solves it modulo one or several integers m. The general principle is illustrated in Figure 5.1. There are three variants: big prime (Figure 5.1 with m = p for a prime p), small primes (Figure 5.2 with m = p1…pr for pairwise distinct primes p1,…,pr), and prime power modular algorithms (Figure 5.3 with m = pl for a prime p). The first one is conceptually the simplest, and the basic issues are most visible in that variant. However, the other two variants are computationally superior.
In each case, two technical problems have to be addressed:
○ a bound on the solution in R,
○ how to find the required moduli.
- Type
- Chapter
- Information
- Modern Computer Algebra , pp. 97 - 140Publisher: Cambridge University PressPrint publication year: 2013