## Download e-book for kindle: Algebraic Geometry in Coding Theory and Cryptography by Harald Niederreiter

By Harald Niederreiter

ISBN-10: 0691102880

ISBN-13: 9780691102887

This textbook equips graduate scholars and complicated undergraduates with the mandatory theoretical instruments for making use of algebraic geometry to info conception, and it covers fundamental functions in coding conception and cryptography. Harald Niederreiter and Chaoping Xing give you the first certain dialogue of the interaction among nonsingular projective curves and algebraic functionality fields over finite fields. This interaction is prime to investigate within the box this present day, but before no different textbook has featured entire proofs of it. Niederreiter and Xing conceal classical purposes like algebraic-geometry codes and elliptic-curve cryptosystems in addition to fabric no longer taken care of by means of different books, together with function-field codes, electronic nets, code-based public-key cryptosystems, and frameproof codes. Combining a scientific improvement of concept with a large collection of real-world purposes, this can be the main complete but available creation to the sector available.Introduces graduate scholars and complicated undergraduates to the rules of algebraic geometry for purposes to details idea presents the 1st precise dialogue of the interaction among projective curves and algebraic functionality fields over finite fields comprises purposes to coding idea and cryptography Covers the newest advances in algebraic-geometry codes beneficial properties functions to cryptography now not handled in different books

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**Extra resources for Algebraic Geometry in Coding Theory and Cryptography**

**Sample text**

The dimension of a nonempty affine algebraic set V is equal to the Krull dimension of its coordinate ring k[V ]. Proof. 8, the irreducible closed subsets of V correspond to prime ideals of k[X] containing I (V ). These, in turn, correspond to prime ideals of k[V ]. Hence, dim(V ) is the length of the longest chain of prime ideals of k[V ], which is the Krull dimension of k[V ] by definition. 19. For every n ≥ 1, we have dim(An ) = dim(Pn ) = n. Proof. Since I (An ) = (0), the coordinate ring k[An ] is the polynomial ring k[X] = k[x1 , .

Consider the affine space A1 (Fq ). 4 that an Fq -closed point of degree m corresponds to the set of roots of an irreducible polynomial of degree m over Fq . 3. 6. For a positive integer m, let Iq (n, m) be the number of Fq -closed points of degree m in An (Fq ). Then Iq (n, m) = 1 m µ(d)q mn/d , d|m where the sum is over all positive integers d dividing m and µ is the Möbius function on N. Proof. 4 that d|m dIq (n, d) = q mn for all m ≥ 1. The Möbius inversion formula (see [72, p. 92]) gives the desired result.

A classical treatise on algebraic geometry is Hartshorne [51]. More recent books that discuss algebraic geometry from various angles are Bump [13], Cox, Little, and O’Shea [23], Kunz [65], and Smith et al. [115]. 1 Affine and Projective Spaces From now on in this chapter, we assume that k is a perfect field, that is, that every algebraic extension of k is separable. Note that, in particular, the finite field Fq and its algebraic closure Fq are perfect for any prime power q. For a given positive integer n, affine n-space over k, denoted by An , is the Cartesian n-space An := An (k) := {(a1 , .

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