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 details conception, and it covers fundamental functions in coding conception and cryptography. Harald Niederreiter and Chaoping Xing give you the first unique 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 beforehand no different textbook has featured whole proofs of it. Niederreiter and Xing hide classical functions like algebraic-geometry codes and elliptic-curve cryptosystems in addition to fabric no longer handled 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 huge choice of real-world purposes, this is often the main finished but obtainable creation to the sector available.Introduces graduate scholars and complex undergraduates to the principles of algebraic geometry for functions to info concept offers the 1st special dialogue of the interaction among projective curves and algebraic functionality fields over finite fields comprises functions to coding conception and cryptography Covers the newest advances in algebraic-geometry codes beneficial properties purposes to cryptography no longer handled in different books

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

**Sample text**

1≤i≤n Then b1 = 0 and n bi b1−1 z i = 0. z1 + i=2 By the condition on νP (b1 ), we have bi b1−1 ∈ OP for 2 ≤ i ≤ n. Passing to the residue classes modulo MQ , we get n ρ(bi b1−1 + MP )(z i + MQ ) = 0 + MQ , (z 1 + MQ ) + i=2 a contradiction to the choice of z 1 , . . , z n . 13, the following definition is meaningful. 14. The degree deg(P ) of a place P of F /k is defined to be the degree of the residue class field of P over k. A place of F /k of degree 1 is also called a rational place of F /k.

I) Let Fq n = Fq (β) and let f be the minimal polynomial of β over Fq . We will prove that f is irreducible over F . Suppose that f had a factorization f = gh over F with g and h monic and deg(g) ≥ 1, deg(h) ≥ 1. It is obvious that all roots of g and h are elements of Fq n . Hence, from the fact that the coefficients of a monic polynomial are polynomial expressions of its roots, it follows that g and h are polynomials over Fq n . Thus, all coefficients 28 CHAPTER 1 of g and h are algebraic over Fq .

Pr } ⊂ An is a finite set. Let Pj = (a1j , . . , anj ) for 1 ≤ j ≤ r and put gi = rj =1 (xi − aij ) for 1 ≤ i ≤ n. Then gi ∈ I (V ). Thus, in the coordinate ring k[V ] the monomial xir is a linear combination of 1, xi , . . , xir−1 . This means that the set { ni=1 xiei : 0 ≤ ei ≤ r − 1} generates the whole vector space k[V ]. This implies that k[V ] has a finite k-dimension. The following theorem provides an algebraic characterization of the dimension of an affine algebraic set. 2 for background on the Krull dimension of rings.

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