Class fields generated by coordinates of elliptic curves

Lambda Coordinates for Binary Elliptic Curves

In this work we present the λ-coordinates, a new system for representing points in binary elliptic curves. We also provide efficient elliptic curve operations based on the new representation and timing results of our software implementation over the field F2254 . As a result, we improve speed records for protected/unprotected single/multi-core software implementations of random-point elliptic c...

control of the optical properties of nanoparticles by laser fields

در این پایان نامه، درهمتنیدگی بین یک سیستم نقطه کوانتومی دوگانه(مولکول نقطه کوانتومی) و میدان مورد مطالعه قرار گرفته است. از آنتروپی ون نیومن به عنوان ابزاری برای بررسی درهمتنیدگی بین اتم و میدان استفاده شده و تاثیر پارامترهای مختلف، نظیر تونل زنی(که توسط تغییر ولتاژ ایجاد می شود)، شدت میدان و نسبت دو گسیل خودبخودی بر رفتار درجه درهمتنیدگی سیستم بررسی شده اشت.با تغییر هر یک از این پارامترها، در...

Elliptic Curves over Finite Fields

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

Frobenius fields for elliptic curves

Let E/Q be an elliptic curve over the field of rational numbers, with EndQ̄(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fp. Let ΠE(K;x) be the number of primes p ≤ x such that the field extension Q(πp(...

Elliptic Curves over Finite Fields