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Algebraic complexities and algebraic curves over finite fields.

D V Chudnovsky1, G V Chudnovsky

  • 1Department of Mathematics, Columbia University, New York, NY 10027.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 1987
PubMed
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This study determines the minimal multiplicative complexity for polynomial multiplication over finite fields. New linear bounds are established, connecting coding theory and algebraic curves.

Area of Science:

  • Algebraic Geometry
  • Coding Theory
  • Computational Algebra

Background:

  • The minimal multiplicative complexity for polynomial multiplication over infinite fields is established.
  • Understanding complexity over finite fields is crucial for applications in computer science and cryptography.

Purpose of the Study:

  • To establish lower and upper bounds for the minimal multiplicative complexity of polynomial multiplication over finite fields.
  • To explore the relationship between polynomial multiplication complexity and concepts from coding theory and algebraic curves.

Main Methods:

  • Utilizing connections between polynomial multiplication and linear coding theory.
  • Applying principles of algebraic curves over finite fields to complexity analysis.

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Main Results:

  • Proving lower and upper bounds on the minimal multiplicative complexity.
  • Demonstrating that these bounds are linear in the number of inputs.

Conclusions:

  • The minimal multiplicative complexity over finite fields can be bounded linearly.
  • The study highlights the utility of coding theory and algebraic geometry in understanding computational complexity.