By San Ling

The succeed in of algebraic curves in cryptography is going a long way past elliptic curve or public key cryptography but those different program 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 a number cryptographic functions, resembling mystery sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate scholars in arithmetic and laptop technology, this self-contained publication is likely one of the first to target many subject matters in cryptography concerning algebraic curves. After offering the required historical past on algebraic curves, the authors talk about error-correcting codes, together with algebraic geometry codes, and supply an advent to elliptic curves. every one bankruptcy within the rest of the publication offers with a particular subject in cryptography (other than elliptic curve cryptography). the subjects coated comprise mystery sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters commence with introductory fabric earlier than that includes the appliance of algebraic curves.

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28 Algebraic Curves in Cryptography (ii) Consider the binary linear code C2 = {0000, 0111, 0011, 0100}. It has parameters [4, 2, 1]. The extended code C2 = {00000, 01111, 00110, 01001} is a binary [5, 2, 2]-linear code. This example shows that the minimum distance d(C) can achieve both d(C) and d(C) + 1. The Golay codes were discovered by Golay in the late 1940s. 23 Let G be the 12 × 24 matrix G = (I12 |A), where I12 is the 12 × 12 identity matrix and A is the 12 × 12 matrix 0 1 11 1 11 1 1 11 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 .

1); or (iii) C is equivalent to the dual of the code generated by 1. Otherwise, C is said to be nontrivial. An interesting family of examples of MDS codes is given by the (generalized) Reed-Solomon codes. 23 Let n ≤ q. Let α = (α1 , α2 , . . , αn ), where αi (1 ≤ i ≤ n) are distinct elements of Fq . Let v = (v1 , v2 , . . , vn ), where vi ∈ F∗q for all 1 ≤ i ≤ n. For k ≤ n, the generalized Reed-Solomon code GRSk (α, v) is defined to be {(v1 f (α1 ), v2 f (α2 ), . . , vn f (αn )) : f (x) ∈ Fq [x] and deg(f (x)) < k}.

6]) Let n, k, and d be integers satisfying 2 ≤ d ≤ n and 1 ≤ k ≤ n. 3) then there exists an [n, k]-linear code over Fq with minimum distance at least d. 7]) For a prime power q > 1 and integers n, d such that 2 ≤ d ≤ n, we have n−1 Bq (n, d) ≥ q n−⌈logq (Vq (d−2)+1)⌉ ≥ q n−1 , Vqn−1 (d − 2) where, for a real number x, ⌈x⌉ is the smallest integer greater than or equal to x. The first upper bound for Aq (n, d) that we discuss is the Hamming bound, also known as the sphere-packing bound. 1]) For an integer q > 1 and integers n, d such that 1 ≤ d ≤ n, we have Aq (n, d) ≤ qn ⌊(d−1)/2⌋ n i=0 i (q − 1)i = qn , − 1)/2⌋) Vqn (⌊(d where, for a real number x, ⌊x⌋ is the largest integer less than or equal to x.