Algebraic Curves in Cryptography by San Ling PDF

By San Ling

ISBN-10: 1420079468

ISBN-13: 9781420079463

The succeed in of algebraic curves in cryptography is going a long way past elliptic curve or public key cryptography but those different software parts haven't been systematically lined within the literature. Addressing this hole, Algebraic Curves in Cryptography explores the wealthy makes use of of algebraic curves in more than a few cryptographic purposes, similar to mystery sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate scholars in arithmetic and machine technology, this self-contained booklet is without doubt one of the first to target many subject matters in cryptography concerning algebraic curves. After offering the mandatory historical past on algebraic curves, the authors talk about error-correcting codes, together with algebraic geometry codes, and supply an advent to elliptic curves. each one bankruptcy within the rest of the e-book bargains with a specific subject in cryptography (other than elliptic curve cryptography). the themes coated contain mystery sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters start with introductory fabric prior to that includes the appliance of algebraic curves.

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From the definition, we know that, once this function is given, we have information on the number of Fqm -rational points on X , for all m ≥ 1. Before studying this zeta function, we first look at an example. 2 Let L be the projective line defined by aX + bY + cZ = 0 over Fq with (a, b, c) = (0, 0, 0). We may assume that a = 0. ” For each y = β ∈ Fqm , there is a unique solution Introduction to Algebraic Curves 17 for the equation ax + bβ + c = 0. This implies that we get q m “finite points” {[−(bβ +c)/a, β, 1] : β ∈ Fqm }.

I j  aij x y ∈ Fr2 [x, y]  f (x, y) =     0≤ri+(r+1)j≤m,i≥0, 0≤j≤r−1 We will investigate the parameters of this code later. For X a smooth curve over Fq and a divisor D = P ∈X mP P , the support of D, denoted by Supp(D), is defined to be the set {P : mP = 0}. Thus, for any f ∈ L(D) and any point Q ∈ Supp(D), Q is not a pole of f and we can evaluate f at the point Q. An algebraic geometry code, introduced by Goppa, is defined as follows. 2 Let X be a projective smooth curve over Fq and let D be a def divisor on X .

The desired result on the dimension now follows from the Riemann-Roch Theorem. 4 The dual code C(D; P)⊥ of an algebraic geometry code C(D; P) can be described in terms of the residues of certain differentials [151, 163]. In this book, we also refer to the dual code of an algebraic geometry code as a residual code. 43]). 5 (i) Let L be a projective line defined over Fq . Then it has genus g = 0. Thus, an algebraic geometry code C(D; P) from L has parameters [n, k ≥ m + 1, d ≥ n − m]. By the Singleton bound, we must have k = m + 1 and d = n − m.

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Algebraic Curves in Cryptography by San Ling

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