San Ling's Algebraic Curves in Cryptography PDF

By San Ling

ISBN-10: 1420079468

ISBN-13: 9781420079463

The achieve 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 a number cryptographic functions, similar to mystery sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate scholars in arithmetic and desktop technology, this self-contained e-book is without doubt one of the first to target many subject matters in cryptography regarding algebraic curves. After delivering the mandatory historical past on algebraic curves, the authors speak about error-correcting codes, together with algebraic geometry codes, and supply an advent to elliptic curves. each one bankruptcy within the rest of the booklet bargains with a particular subject in cryptography (other than elliptic curve cryptography). the themes lined comprise mystery sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters commence with introductory fabric sooner than that includes the appliance of algebraic curves.

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Example text

Vie ) if and only if 0 = vH T = vi1 cTi1 + · · · + vie cTie , which is true if and only if there are e columns of H (namely, ci1 , . . , cie ) that are linearly dependent. Here, T denotes the transpose of a matrix or a vector. To say that the distance of C is ≥ d is equivalent to saying that C does not contain any nonzero word of weight ≤ d − 1, which is in turn equivalent to saying that any ≤ d − 1 columns of H are linearly independent. This proves (i). Similarly, to say that the distance of C is ≤ d is equivalent to saying that C contains a nonzero word of weight ≤ d, which is in turn equivalent to saying that H has ≤ d columns (and hence d columns) that are linearly dependent.

Fn−1 ∈ Fq [X0 , X1 , . . , Xn ]. Then X is a curve over Fq . Since these polynomials belong to Fq [X0 , X1 , . . , Xn ], it is natural to view these polynomials over Fqm [X0 , X1 , . . , Xn ], for all m ≥ 1. Thus, X is also a curve defined over Fqm . Hence, we can speak of Fqm -rational points of X . It is clear that all Fq -rational points are Fqm -rational points, for any m ≥ 1. Let Nm denote the number of Fqm -rational points on X , then Nm ≥ N1 for all m ≥ 1. 4) where EXP(x) stands for the function ex .

C ⊥ = {x ∈ Fnq : x · c = 0 for all c ∈ C}, where · denotes the usual inner product. The dual minimum distance of C, denoted by d⊥ (C) or d(C ⊥ ), is defined to be the minimum distance of C ⊥ . If C is a subspace of C ⊥ , then C is called a self-orthogonal code. Furthermore, if C = C ⊥ , then C is said to be self-dual. 14 The binary code {0000, 1111} is self-orthogonal. Its dual code is {0000, 1111, 1100, 0110, 0011, 1001, 0101, 1010}. The binary code {0000, 1100, 0011, 1111} is self-dual. Note that, in the Euclidean space Rn (where R denotes the real field), the intersection of a subspace with its orthogonal complement is always 0.

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Algebraic Curves in Cryptography by San Ling

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