By Harald Niederreiter

This textbook equips graduate scholars and complicated undergraduates with the required theoretical instruments for using algebraic geometry to details conception, and it covers basic purposes in coding thought 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 prime to analyze within the box this day, but earlier no different textbook has featured whole proofs of it. Niederreiter and Xing disguise classical functions like algebraic-geometry codes and elliptic-curve cryptosystems in addition to fabric no longer handled by way of different books, together with function-field codes, electronic nets, code-based public-key cryptosystems, and frameproof codes. Combining a scientific improvement of conception with a vast number of real-world purposes, this is often the main complete but obtainable advent to the sector available.Introduces graduate scholars and complex undergraduates to the rules 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 comprises purposes to coding concept and cryptography Covers the most recent advances in algebraic-geometry codes positive factors purposes to cryptography no longer handled in different books

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

**Example text**

Otherwise, g has a positive degree and roots b1 , . . , bd ∈ k = A1 . Then V = {b1 , . . , bd }. Thus, the affine algebraic sets in A1 are A1 itself and the finite subsets of A1 . If V is defined over k, then I (V /k) is a principal ideal of k[X]. In the case where this ideal is nontrivial, we have I (V /k) = (g) with g ∈ k[X] of positive degree. For each monic irreducible factor of g in k[X], the roots of this factor form a k-closed point of V . In the proof of the following proposition, we use two fundamental results on polynomial rings, namely the Hilbert basis theorem and the Hilbert Nullstellensatz, which can be found, for example, in the books [23] and [29].

Proof. Let σ (P ) be an arbitrary point in P. Let V = Z(S) for some S ⊆ k[X]. 2, f (σ (P )) = σ (σ −1 (f ))(σ (P )) = σ (σ −1 (f )(P )) = σ (f (P )) = σ (0) = 0. This means that P is a subset of V . By the above result, it makes sense to speak of a k-closed point of an affine algebraic set defined over k. 4. (i) Let q be a power of 2. Let V ⊆ A2 (Fq ) be the affine algebraic set defined by the single equation y 2 + y − x = 0. 3(ii), for an element a ∈ Fq there exists an element b ∈ Fq such that (a, b) ∈ V if and only if a is in the kernel of the trace map TrFq /F2 .

Let {β1 = 1, β2 , . . , βn } be a basis of Fq n over Fq . Then for any τ ∈ Gal(Fq n /Fq ) and x = ni=1 βi xi ∈ Fn with all xi ∈ F , put n τ (x) = τ (βi )xi . i=1 Then τ is a Galois automorphism of Fn /F and all elements of Gal(Fn /F ) are obtained in this way. 2. Let Fn = F · Fq n be a constant field extension of F . Then for any place P of F and any place Q of Fn lying over P , the following holds: (i) (ii) (iii) (iv) e(Q|P ) = 1; deg(Q) = d/gcd(d, n), where d = deg(P ) is the degree of P ; f (Q|P ) = n/gcd(d, n); there are exactly gcd(d, n) places of Fn lying over P .