Download PDF by San Ling: Algebraic Curves in Cryptography
By San Ling
The succeed in of algebraic curves in cryptography is going some distance 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 functions, comparable to mystery sharing, frameproof codes, and broadcast encryption.
Suitable for researchers and graduate scholars in arithmetic and laptop technological know-how, this self-contained ebook is among the first to target many subject matters in cryptography concerning algebraic curves. After providing 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. every one bankruptcy within the rest of the e-book bargains with a specific subject in cryptography (other than elliptic curve cryptography). the subjects lined 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 applying of algebraic curves.
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1(i). 1(i). It is smooth and hence its genus is (3 − 1)(3 − 2)/2 = 1. 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 .
As |wi | = r, we must have wi = −r for all 1 ≤ i ≤ 2g. Finally, we conclude that the zeta function of H is ZH (t) = (1 + rt)r(r−1) . (1 − t)(1 − r2 t) In particular, N2 = (r2 )2 + 1 − 2g(−r)2 = r3 + 1 = N1 . This implies that all the Fr4 -rational points are also Fr2 -rational. 20 Algebraic Curves in Cryptography For an algebraic curve X , we denote by N (X ) the number of Fq -rational points on X . By the Hasse-Weil bound, we know that the number of Fq rational points on a curve is upper bounded in terms of q and its genus.
15]) The Hamming code Ham(m, q) is a [(q m −1)/(q−1), (q m − 1)/(q − 1) − m, 3]-code. 16 The dual of the q-ary Hamming code Ham(m, q) is called a q-ary simplex code. It is sometimes denoted by S(m, q). The next bound, the Singleton bound, is an interesting one that is related to the well-known Reed-Solomon codes. 1]) For any integer q > 1, any positive integer n and any integer d such that 1 ≤ d ≤ n, we have Aq (n, d) ≤ q n−d+1 . In particular, when q is a prime power, the parameters [n, k, d] of any linear code over Fq satisfy k + d ≤ n + 1.
Algebraic Curves in Cryptography by San Ling