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 program parts 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, equivalent to mystery sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate scholars in arithmetic and desktop technological know-how, this self-contained booklet is among the first to target many themes in cryptography related to algebraic curves. After offering the required history on algebraic curves, the authors speak about error-correcting codes, together with algebraic geometry codes, and supply an creation to elliptic curves. every one bankruptcy within the rest of the publication 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 commence with introductory fabric ahead of that includes the applying of algebraic curves.

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Example text

Clearly, this dual code is GRS1 (α, v′ ). 4) where v = (v1 , . . , vn ). Now, for arbitrary k, we claim that GRSk (α, v)⊥ = GRSn−k (α, v′ ). A typical codeword in GRSk (α, v) is (v1 f (α1 ), . . , vn f (αn )), where f (x) ∈ Fq [x] with degree ≤ k − 1, while a typical codeword in GRSn−k (α, v′ ) has the form (v1′ g(α1 ), . . , vn′ g(αn )), with g(x) ∈ Fq [x] of degree ≤ n − k − 1. Since deg(f (x)g(x)) ≤ n − 2 < n − 1, we have (v1 f (α1 ), . . , vn f (αn )) · (v1′ g(α1 ), . . 4). Therefore, GRSn−k (α, v′ ) ⊆ GRSk (α, v)⊥ .

6). Let D be the divisor i=0 Pi . Then D is an Fq -rational divisor and L(D) = f (y) : f (y) ∈ Fq [y], deg(f (y)) ≤ d . p(y) It is easy to see that it is a vector space of dimension d + 1 over Fq since {y i /p(y)}di=0 is a basis. 7. 5 Rational Points and Zeta Functions Let X be a projective curve over Fq defined by a finite set S of polynomials. We are interested in the number of Fq -rational points on X . In fact, there are 16 Algebraic Curves in Cryptography only finitely many Fq -rational points on X .

6 Let P, Q be two nonsingular points of an affine (or projective) curve X over Fq (P, Q may not be Fq -rational). Then these two points correspond to the same discrete valuation of Fq (X ) if and only if they are conjugate to each other. 15] for the detailed proof of this theorem. Let P be a closed point. For a point P ∈ P and an element t of Fq (X ) with νP (t) = 1, we have νQ (t) = 1 for all points Q conjugate to P since νP and νQ are identical. In this case, t is called a local parameter at P.

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