By San Ling
The succeed in of algebraic curves in cryptography is going a ways past elliptic curve or public key cryptography but those different software components 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 various cryptographic purposes, resembling mystery sharing, frameproof codes, and broadcast encryption.
Suitable for researchers and graduate scholars in arithmetic and desktop technology, this self-contained ebook is without doubt one of the first to target many themes in cryptography regarding algebraic curves. After delivering the mandatory heritage on algebraic curves, the authors talk about error-correcting codes, together with algebraic geometry codes, and supply an creation to elliptic curves. every one bankruptcy within the rest of the ebook offers with a particular subject in cryptography (other than elliptic curve cryptography). the themes coated comprise mystery sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters start with introductory fabric sooner than that includes the appliance of algebraic curves.
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Example text
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). Note that the points in a divisor may not be Fq -rational. A divisor P ∈X nP P is said to be Fq -rational if nP = nQ for any two conjugate points P, Q. In this book, except for Chapter 3 where divisors may not necessarily be Fq -rational, we always mean an Fq -rational divisor whenever a divisor is mentioned.
Then, writing x ∈ Fnq as x = (x1 , x2 , . . , xn ), the Hamming weight of x can also be equivalently defined as wt(x) = wt(x1 ) + wt(x2 ) + · · · + wt(xn ). 10 If x, y ∈ Fnq , then d(x, y) = wt(x − y). Proof. For x, y ∈ Fq , d(x, y) = 0 if and only if x = y, which is true if and only if x − y = 0 or, equivalently, wt(x − y) = 0. 2). 11 Let C be a code (not necessarily linear). The minimum (Hamming) weight of C, denoted wt(C), is the smallest of the weights of the nonzero codewords of C. 12 Let C be a linear code over Fq .
A code, of length n, size M , and distance d, is referred to as an (n, M, d)code. The numbers n, M, d are called the parameters of the code. , with code alphabet {0, 1, 2}). Then d(C) = 4 since d(00000, 10112) = 4, d(00000, 22222) = 5, d(10112, 22222) = 4. Hence, C is a ternary (5,3,4)-code. (ii) Let C = {000000, 111111, 222222} be a ternary code. Then d(C) = 6 since d(000000, 111111) = 6, d(000000, 222222) = 6, d(111111, 222222) = 6. Hence, C is a ternary (6,3,6)-code. 7 For a q-ary (n, M, d)-code, the minimum distance d in fact measures its error detection and correction capabilities (see [96]).