Elliptic curves

Objectives of Chapter 3: To define the terms and the concepts of symmetric key ciphers; to emphasize the two categories of traditional ciphers: substitution and transposition ciphers; to describe the categories of cryptanalysis used to break the symmetric ciphers.

12p levuphongqn 18-08-2015 98 5   Download

Objectives of Chapter 1: To define three security goals; to define security attacks that threaten security goals; to define security services and how they are related to the three security goals; to define security mechanisms to provide security services; to introduce two techniques, cryptography and steganography, to implement security mechanisms

4p levuphongqn 18-08-2015 80 3   Download

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K− /K) by inversion). We prove (under mild hypotheses on p) that if the Zp -rank of the pro-p Selmer group Sp...

35p noel_noel 17-01-2013 66 8   Download

Elliptic units, which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincar´e upper half-plane, provide us with a rich source of arithmetic questions and insights. They allow the analytic construction of abelian extensions of imaginary quadratic fields, encode special values of zeta functions through the Kronecker limit formula, and are a prototype for Stark’s conjectural construction of units in abelian extensions of number fields.

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Let E be an elliptic curve over Q, let p be an ordinary prime for E, and let K be an imaginary quadratic field. Write K∞/K for the anticyclotomic Zp-extension of K and set G∞ = Gal(K∞/K). Following a construction of Section 2 of [BD1] which is recalled in Section 1, one attaches to the data (E,K, p) an anticyclotomic p-adic L-function Lp(E,K) which belongs to the Iwasawa algebra Λ := Zp[[G∞]]. This element, whose construction was inspired by a formula proved in [Gr1], is known, thanks to work of Zhang ([Zh, §1.

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At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi’s conjecture relates modified Selmer groups, which he defined, with modified padic L-functions defined by the first author.

19p tuanloccuoi 04-01-2013 51 5   Download

ON THE ALGEBRAIC DIFFERENCE EQUATIONS un+2 un = ψ(un+1 ) IN R+ , RELATED TO A FAMILY ∗ OF ELLIPTIC QUARTICS IN THE PLANE G. BASTIEN AND M. ROGALSKI Received 20 October 2004 and in revised form 27 January 2005 We continue the study of algebraic difference equations of the type un+2 un = ψ(un+1 ), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in “on some algebraic difference equations un+2 un = ψ(un+1 ) in R+ , related to families of...

35p sting12 10-03-2012 33 5   Download

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Supporting Symmetric 128-bit AES in Networked Embedded Systems: An Elliptic Curve Key Establishment Protocol-on-Chip

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Việc đầu tiên bạn mở hình ảnh dưới đây: Bấm Shift + Ctrl + I để nghịch đảo vùng chọn Go to Filter - Blur - Radial Blur, thiết lập như thế này: Bạn sẽ nhận được: Nhấn Ctrl + M để mở Curves, thiết lập như thế này Sử dụng Elliptical Marquee Tool tạo một lựa chọn với Feather = 10px như thế này:

13p iiduongii2 05-04-2011 89 11   Download