Fast Construction of Irreducible Polynomials of degree n over any Finite Field

In summary, the construction of irreducible polynomials over finite fields is crucial for applications in coding theory, cryptography, and finite field arithmetic. Several efficient algorithms exist for constructing these polynomials of degree n, which can be applied in various fields of mathematics and computer science. These polynomials are used to construct error-correcting codes, generate secure keys, and perform finite field arithmetic operations. They also have practical uses in digital signal processing, error-correcting circuits, and pseudo-random number generation.
  • #1
burritoloco
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Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then.

http://www.shoup.net/papers/fastirred.pdf

After that, I'm not really sure. Any ideas? Thank you.
 
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  • #2
Just found this paper by Jean-Marc Couveignes and Reynald Lercier (2009):

http://www.math.univ-toulouse.fr/~couveig/publi/couveignes-lercier.pdf
 
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1. What is the purpose of constructing irreducible polynomials over finite fields?

The construction of irreducible polynomials over finite fields is important in many areas of mathematics and computer science, particularly in applications involving coding theory, cryptography, and finite field arithmetic. These polynomials serve as building blocks for constructing algebraic structures and performing computations in finite fields.

2. How can we efficiently construct irreducible polynomials of degree n over any finite field?

There are several algorithms that have been developed for fast construction of irreducible polynomials over finite fields. These include the Cantor-Zassenhaus algorithm, the Berlekamp algorithm, and the Niederreiter algorithm. These algorithms use different techniques and have different time complexities, but all are efficient in producing irreducible polynomials of degree n.

3. What is the significance of constructing irreducible polynomials of degree n over finite fields?

The significance of constructing irreducible polynomials over finite fields lies in their applications in various fields of mathematics and computer science. In coding theory, these polynomials are used to construct error-correcting codes, while in cryptography, they are used to generate keys for secure communication. They also have applications in algebraic geometry, number theory, and theoretical computer science.

4. Can irreducible polynomials be constructed for any degree and any finite field?

Yes, it is possible to construct irreducible polynomials for any degree and any finite field. However, the construction process may vary depending on the degree and finite field in question. For example, the Berlekamp algorithm is efficient for constructing irreducible polynomials of small degree over binary fields, while the Cantor-Zassenhaus algorithm is more suitable for larger degrees and prime fields.

5. Are there any practical uses for irreducible polynomials over finite fields?

Yes, irreducible polynomials over finite fields have a wide range of practical uses in various fields of mathematics and computer science. As mentioned earlier, they are used in coding theory and cryptography, but they also have applications in areas such as digital signal processing, error-correcting circuits, and pseudo-random number generation. They are also essential for implementing finite field arithmetic operations in computer algebra systems.

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