By Harald Niederreiter

This textbook equips graduate scholars and complicated undergraduates with the mandatory theoretical instruments for utilizing algebraic geometry to info concept, and it covers fundamental purposes in coding concept and cryptography. Harald Niederreiter and Chaoping Xing give you the first special dialogue of the interaction among nonsingular projective curves and algebraic functionality fields over finite fields. This interaction is key to investigate within the box this present day, but beforehand 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 now not taken care of via 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 vast number of real-world functions, this is often the main finished but obtainable creation to the sphere available.

- Introduces graduate scholars and complicated undergraduates to the principles of algebraic geometry for functions to info idea
- Provides the 1st distinct dialogue of the interaction among projective curves and algebraic functionality fields over finite fields
- Includes functions to coding concept and cryptography
- Covers the most recent advances in algebraic-geometry codes
- Features functions to cryptography no longer handled in different books

**Read or Download Algebraic Geometry in Coding Theory and Cryptography PDF**

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

**Example text**

Bn z n ) are all distinct, and so the strict triangle inequality yields n ν = min ν(bi z i ) < ∞, bi z i 1≤i≤n i=1 which is again a contradiction. In the above proof, we have shown, in particular, that the restriction of a valuation of F /k to k(x) yields a valuation of k(x). Obviously, for equivalent valuations of F the restrictions are again equivalent. Thus, a place Q of F corresponds by restriction to a unique place P of k(x). We say that Q lies over P or that P lies under Q. Therefore, every place of F lies either over a place of k(x) corresponding to a monic irreducible polynomial in k[x] or over the infinite place of k(x).

7, |{σ (P ) : σ ∈ Gal(Fq /Fq )}| = |{σ (P ) : σ ∈ Gal(Fq m /Fq )}| ≤ |Gal(Fq m /Fq )| = m. This completes the proof. We define the degree of an Fq -closed point P to be the cardinality of P, denoted by deg(P). For a point P = [a0 , a1 , . . , an ] ∈ Pn with ai = 0, the definition field of P over k, denoted by k(P ), is defined by k(a0 /ai , a1 /ai , . . , an /ai ). 10. (i) The definition field of a point is well defined. First of all, if (b0 , b1 , . . , bn ) = (λa0 , λa1 , . . , λan ) with λ = 0, we have bj /bi = aj /ai for all j = 0, 1, .

Define the map θi : Ui → An , [a0 , a1 , . . , ai ai ai ai . 2) It is clear that θi is well defined and bijective. 1). Via θi , the space An can be viewed as an open subset of Pn . We claim that the map θi is a homeomorphism from Ui (with its induced Zariski topology from Pn ) onto An (with the Zariski topology). It suffices to check this for i = 0. Indeed, let V = Zh (S) ⊆ U0 be an algebraic set of Pn . Then it is easy to verify that θ0 (V ) = Z({f (1, y1 , . . , yn ) : f (x0 , x1 , . . , xn ) ∈ S homogeneous}).