By Harald Niederreiter

ISBN-10: 0691102880

ISBN-13: 9780691102887

This textbook equips graduate scholars and complex undergraduates with the mandatory theoretical instruments for utilising algebraic geometry to info concept, and it covers basic functions in coding idea and cryptography. Harald Niederreiter and Chaoping Xing give you the first precise dialogue of the interaction among nonsingular projective curves and algebraic functionality fields over finite fields. This interaction is key to investigate within the box at the present time, but beforehand no different textbook has featured entire proofs of it. Niederreiter and Xing disguise classical functions like algebraic-geometry codes and elliptic-curve cryptosystems in addition to fabric no longer handled via different books, together with function-field codes, electronic nets, code-based public-key cryptosystems, and frameproof codes. Combining a scientific improvement of thought with a huge collection of real-world functions, this is often the main accomplished but obtainable advent to the sector available.Introduces graduate scholars and complicated undergraduates to the principles of algebraic geometry for purposes to details concept offers the 1st particular dialogue of the interaction among projective curves and algebraic functionality fields over finite fields contains functions to coding conception and cryptography Covers the most recent advances in algebraic-geometry codes beneficial properties purposes to cryptography now not handled in different books

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**Example text**

2). (i) Let W ⊆ An be an affine variety. Then the closure W in Pn is a projective variety and W = W ∩ An . (ii) Let V ⊆ Pn be a projective variety. Then V ∩ An is either empty or an affine variety with V ∩ An = V . (iii) If an affine (respectively projective) variety V is defined over k, then V (respectively V ∩ An ) is also defined over k. 48 CHAPTER 2 Proof. 9(ii), the closure of an affine variety is irreducible in Pn . Hence W is a projective variety. 9(iii), if V ∩ An = ∅, then it is an irreducible closed subset of An , and so an affine variety.

11, the degree of a finite place p(x) of F is equal to the degree of the polynomial p(x) and the degree of the infinite place of F is equal to 1. If k = Fq , then the rational function field F has thus exactly q + 1 rational places. Next we prove the approximation theorem for valuations of algebraic function fields of one variable. The following two preparatory results are needed for the proof. 16. If P1 and P2 are two distinct places of F /k, then there exists an element z ∈ F such that νP1 (z) > 0 and νP2 (z) ≤ 0.

For a projective algebraic set V , we let I (V ) be the ideal of k[X] generated by the set {f ∈ k[X] : f is homogeneous and f (P ) = 0 for all P ∈ V }. 10. We prove a lemma on homogeneous ideals, which will be used in the next section. We observe that, by collecting the terms of the same degree, any polynomial in k[X] can be written as a sum of finitely many homogeneous polynomials. 14. (i) An ideal I of k[X] is homogeneous if and only if the following condition is satisfied: for every polynomial f = d f [d] ∈ I with f [d] being homogeneous of degree d, we also have f [d] ∈ I for all d.

### Algebraic Geometry in Coding Theory and Cryptography by Harald Niederreiter

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