Algebraic Curves in Cryptography by San Ling

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 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 more than a few cryptographic purposes, 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 e-book is among the first to target many subject matters in cryptography regarding algebraic curves. After providing the required heritage 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 bargains 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 start with introductory fabric prior to that includes the applying of algebraic curves.

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4 Let X be a smooth projective curve of genus g with L2g 2g function L(t) = i=0 ai ti . Write L(t) into a product i=1 (1 − wi t) and let Nm be the number of Fqm -rational points on X . Then (i) a2g−i = q g−i ai for all 0 ≤ i ≤ 2g, in particular, a2g = q g ; 18 Algebraic Curves in Cryptography 2g (ii) Nm = q m + 1 − i=1 wim for all m ≥ 1, in particular, N1 = q + 1 − 2g i=1 wi = q + 1 + a1 ; √ (iii) (Hasse-Weil bound) |Nm − q m − 1| ≤ 2g q m for all m ≥ 1, in par√ ticular, N1 ≤ q + 1 + 2g q. Proof.

On the other hand, by the fact that y is algebraic over the field Fq (x) of degree r, we know that the set {xi y j : 0 ≤ ri + (r + 1)j ≤ m, i ≥ 0 and 0 ≤ j ≤ r − 1} is linearly independent over Fr2 and it is contained in L(m∞). By counting the number of elements in this set, we conclude that it is a basis of L(m∞). Therefore, the code is   (f (α1 , β1 ), . . , f (αn , βn )) :       . i j  aij x y ∈ Fr2 [x, y]  f (x, y) =     0≤ri+(r+1)j≤m,i≥0, 0≤j≤r−1 We will investigate the parameters of this code later.

Hence, the equation y r + y − αr+1 = 0 becomes Tr(y − β) = 0. , there are r solutions for this equation. This implies that, for any given α ∈ Fr2 , we get r points on H. Together with the “point at infinity” [0, 1, 0], we have 1 + r · r2 = 1 + r3 Fr2 -rational points. 4(ii) that 2g 3 2 N1 = 1 + r = r + 1 − wi , i=1 where {wi }2g i=1 stand for the reciprocal roots of the L-function of H. 2g From the above identity, we have − i=1 wi = r2 (r − 1) = 2gr. As |wi | = r, we must have wi = −r for all 1 ≤ i ≤ 2g.

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