Algebraic Curves in Cryptography by San Ling

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 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, similar to mystery sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate scholars in arithmetic and machine technological know-how, this self-contained booklet is without doubt one of the first to target many issues in cryptography concerning 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 advent to elliptic curves. every one bankruptcy within the rest of the booklet bargains with a specific subject in cryptography (other than elliptic curve cryptography). the subjects lined 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 applying of algebraic curves.

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Example text

Ihara [78] first showed, by using modular curves, that A(q) ≥ q − 1 for any square power q. , for any square power q, we have √ A(q) = q − 1. 6) On the other hand, no single value of A(q) is known if q is a nonsquare. However, some lower bounds have been obtained so far. For instance, by using modular curves and explicit function fields, Zink [182] and Bassa-GarciaStichtenoth [7] showed that A(q 3 ) ≥ 2(q 2 − 1) . 7) Serre [141, Part II, Theorem 24] made use of class field theory to show that there is an absolute positive constant c such that A(q) ≥ c · log(q) for all prime powers q, where, here and throughout this book, log is taken to mean log2 .

We now interpret this space using discrete valuations. 3. Then Pk is the set {f ∈ Fq (L) = Fq (y) : ν∞ (f ) ≥ −k, νQ (f ) ≥ 0 for all other “finite” points Q}. 1(i). 4(ii), there is a discrete valuation 12 Algebraic Curves in Cryptography νO corresponding to O. For k ≥ 0, consider the set def Vk = {f ∈ F2 (E) = F2 (x, y) : νO (f ) ≥ −k, νQ (f ) ≥ 0 for all other “finite” points Q}. Then it is easy to verify that Vk is an F2 -linear space. In fact, it has a basis xi y j : i ≥ 0, 0 ≤ j ≤ 1, 2i + 3j ≤ k .

This means that, once we know the numbers of Fqm -rational points for m = 1, . . , g, we can totally determine the zeta function of X and hence the number of Fqm -rational points for all m ≥ 1. 5 (i) Consider the plane curve E over F3 defined by the affine equation y 2 = x3 − x + 1. It is easy to check that all points Introduction to Algebraic Curves 19 (including the “point at infinity” [0, 1, 0]) on E are nonsingular. Hence, the genus of this curve is 1. 4, to find the zeta function of this curve, it is sufficient to determine the number of F3 rational points.