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 components 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 quite a number cryptographic functions, corresponding to mystery sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate scholars in arithmetic and laptop technology, this self-contained publication is among the first to target many themes in cryptography related to algebraic curves. After offering the mandatory history on algebraic curves, the authors speak 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 offers with a particular subject in cryptography (other than elliptic curve cryptography). the subjects coated contain mystery sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters start with introductory fabric ahead of that includes the appliance of algebraic curves.

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**Additional resources for Algebraic Curves in Cryptography**

**Example text**

If G is equivalent to D with D = G + div(z), then it is easy to verify that L(D) = zL(G). Hence, dim L(D) = dim L(G). In this book, the dimension dim L(G) is also denoted by ℓ(G). 6 Let X be a smooth projective curve over Fq and let D be a divisor of X over Fq . Then Introduction to Algebraic Curves 15 (i) L(D) is a subspace of L(G) if D ≤ G; (ii) L(0) = Fq ; (iii) L(D) = {0} if deg(D) < 0. Proof. (i) is clear. (ii) It is clear that Fq is contained in L(0). Now let a be a nonzero element of L(0), then div(a) ≥ 0 by definition.

3]) For all integers r ≥ 0, a sphere of radius r in An contains exactly Vqn (r) vectors, where A is an alphabet of size q > 1. We are now ready to state the sphere-covering bound. 4]) For an integer q > 1 and integers n, d such that 1 ≤ d ≤ n, we have qn d−1 n i=0 i (q − 1)i = qn ≤ Aq (n, d). , for linear codes) known since the 1950s. 5), which concerns infinite sequences of codes whose lengths tend to infinity. For a long time, the asymptotic GilbertVarshamov bound was the best lower bound known to be attainable by an infinite family of linear codes, so it became a kind of benchmark for judging the “goodness” of an infinite sequence of linear codes.

2 Let x and y be words of length n over an alphabet A. The (Hamming) distance from x to y, denoted by d(x, y), is defined to be the number of places at which x and y differ. 1) where xi and yi are regarded as words of length 1, and d(xi , yi ) = 1 if xi = yi 0 if xi = yi . 3 (i) Let A = {0, 1} and let x = 000000, y = 011101, z = 111111. Then d(x, y) = 4, d(y, z) = 2, d(z, x) = 6. (ii) Let A = {0, 1, 2} and let x = 1200, y = 1012, z = 0011. Then d(x, y) = 3, d(y, z) = 2, d(z, x) = 4. 4 Let x, y, z be words of length n over A.

### Algebraic Curves in Cryptography by San Ling

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