Download e-book for iPad: Algebraic Curves in Cryptography by San Ling

By San Ling

ISBN-10: 1420079468

ISBN-13: 9781420079463

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

Suitable for researchers and graduate scholars in arithmetic and desktop technological know-how, this self-contained booklet is without doubt one of the first to target many subject matters in cryptography concerning algebraic curves. After offering the required history 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 ebook offers with a particular 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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Let v = (v1 , . . , vn ) ∈ C be a codeword of weight e > 0. Suppose the nonzero coordinates are in the positions i1 , . . , ie , so that vj = 0 if j ∈ {i1 , . . , ie }. Let ci (1 ≤ i ≤ n) denote the ith column of H. By the definition of a parity-check matrix, it is easy to see that C contains a nonzero word v = (v1 , . . , vn ) of weight e (whose nonzero coordinates are vi1 , . . , vie ) if and only if 0 = vH T = vi1 cTi1 + · · · + vie cTie , which is true if and only if there are e columns of H (namely, ci1 , .

There are (q n+1 − 1)/(q − 1) points in Pn . ✷ Let X be a smooth projective plane curve defined by f (X, Y, Z) = 0, where f (X, Y, Z) is a polynomial of degree d. For each x = α ∈ Fq , there are at most d solutions for the equation f (α, y, 1), thus there are at most dq “finite points” on this curve. Together with those “points at infinity,” there are at most dq + q + 1 Fq -rational points (note that all the “points at infinity” of a projective plane are [β, 1, 0], with β ∈ Fq , and [1, 0, 0]).

Thus, for any k ≥ 1 and Fq rational point P , we have dimFq (L(kP )) = k + 1 − 1 = k. 1(ii). 7(iii) is smooth. Hence, its genus is (r + 1 − 1)(r + 1 − 2)/2 = r(r − 1)/2. 1(ii). 10(v). 3]). , X is an elliptic curve, then X is smooth and hence its genus is (3 − 1)(3 − 2)/2 = 1. Next, we generalize the Riemann-Roch spaces defined above. Let X be a smooth curve over Fq . A divisor is a formal sum P ∈X nP P , with nP ∈ Z for all P ∈ X , and nP = 0 for all but finitely many points P ∈ X . For a divisor D = P ∈X nP P , we also denote the coefficient nP by νP (D).

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Algebraic Curves in Cryptography by San Ling

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