By Ivan Soprunov

Show description

Read Online or Download Algebraic Curves and Codes [Lecture notes] PDF

Similar cryptography books

Read e-book online Military Cryptanalysis PDF

This publication offers a good origin for fixing cipher platforms. The textual content describes the elemental ideas of cipher resolution plus use of the unilateral frequency distribution within the resolution technique is roofed in a few aspect. a variety of unilateral and multilateral structures are conscientiously mentioned.

Kazumaro Aoki, Jens Franke, Thorsten Kleinjung, Arjen K.'s Advances in Cryptology – ASIACRYPT 2007: 13th International PDF

ASIACRYPT 2007 used to be held in Kuching, Sarawak, Malaysia, in the course of December 2–6, 2007. This used to be the thirteenth ASIACRYPT convention, and used to be subsidized through the foreign organization for Cryptologic learn (IACR), in cooperation with the knowledge protection study (iSECURES) Lab of Swinburne collage of know-how (Sarawak Campus) and the Sarawak improvement Institute (SDI), and used to be ?

Integer Algorithms in Cryptology and Information Assurance by Boris S Verkhovsky PDF

Integer Algorithms in Cryptology and knowledge coverage is a set of the author's personal leading edge methods in algorithms and protocols for mystery and trustworthy conversation. It concentrates at the "what" and "how" in the back of enforcing the proposed cryptographic algorithms instead of on formal proofs of "why" those algorithms paintings.

Additional info for Algebraic Curves and Codes [Lecture notes]

Sample text

FIELDS AND POLYNOMIAL RINGS 31 Proof. Consider f, g, h as elements of F(x)[y]. By Gauss’s lemma f is irreducible in F(x)[y]. Now F(x)[y] is the ring of univariate polynomials over a field, so by the usual Euclid’s lemma f |gh implies f |g or f |h in F(x)[y]. 2) g(x, y) = f (x, y)q(x, y) for some q ∈ F(x)[y]. Let’s clear the denominators: there exists c(x) ∈ F[x] such that c(x)q(x, y) ∈ F[x, y]. As before, factor out the gcd of its coefficients to make it primitive: c(x)q(x, y) = d(x)q1 (x, y) for a primitive q1 ∈ F[x][y].

4) (Euclid’s lemma) If p ∈ F[x] is irreducible and p|f g then either p|f or p|g. Moreover when R = Z we have (5) (Gauss’ lemma) If p ∈ Z[x] factors in Q[x] then it factors in Z[x]. How do we define F[x, y], the ring of polynomials in two variables? One way would be to say that F[x, y] consists of finite linear combinations of monomials xi y j with coefficients in F and i ≥ 0, j ≥ 0: � f (x, y) = ai,j xi y j , aij ∈ F, all but finitely many aij are zero. i,j≥0 Another way is to set R = F[x], which is a commutative ring with 1, and define F[x, y] = R[y] = F[x][y].

Recall that in Uz we have u = xz , v = yz , so the two equations become yz − 2 xz = 0 and yz − 2 xz − 2 = 0. If we clear the denominators we obtain y − 2x = 0 and y − 2x − 2z = 0. Notice that now they make sense for z = 0 as well. Thus they define two lines in the projective space: ¯ 1 = {(x : y : z) ∈ P2 | y − 2x = 0}, L ¯ 2 = {(x : y : z) ∈ P2 | y − 2x − 2z = 0}, L which coincide with L1 and L2 on Uz . However each of them has one extra point ¯ 1 and L ¯ 2 intersect in P2 at one point (1 : 2 : 0).

Download PDF sample

Algebraic Curves and Codes [Lecture notes] by Ivan Soprunov

by Jason

Download e-book for iPad: Algebraic Curves and Codes [Lecture notes] by Ivan Soprunov
Rated 4.89 of 5 – based on 24 votes