Elliptic Curve Cryptology Used to Make Keys

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Elliptic Curve Cryptology

What and Why of ECC?

Elliptic curve cryptography (ECC) is a public key cryptography technique by making use of elliptic curve properties and their algebraic structure of over finite fields. It is one of the efficient ways of providing encryption of cryptographic keys.
Elliptic curves as algebraic/geometric entities have been studied extensively for the past 150 years, and from these studies has emerged a rich and deep theory. Elliptic curve systems as applied to cryptography were first proposed in 1985 independently by Neal Koblitz from the University of Washington, and Victor Miller, who was then at IBM, Yorktown Heights.[1]
These curves have allowed establishment of a new generation of asymmetric
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We take another point Q on the curve. We compute R = P + Q. It is a graphical operation consisting of tracing a line through P and Q. This line defines a unique intersection with the EC: the point –R. R is found by taking the symmetric point with respect to the x-axis (see figure 2).

Figure 2 – Example of addition
P = (x1,y1)
Q = (x2,y2)
R = P + Q = (x3,y3)
x3 = ө 2 – x1 – x2 y3 = ө (x1 + x3) – y1 with ө = (y2 – y1)/(x2 – x1).
If P = Q, R = P + Q is equivalent to adding a point to itself: doubling point P.

The operation needs a single point and consists of finding a point 2P. We draw a tangent line to the EC at point P. This line intersects the curve in a point –R. Reflecting this point across the x- axis gives R: R = 2P. See figure 3.
Formally: the only difference with an addition is the definition of ө. ө = (3x1 + a)/2y1

Figure 3 – Example of doubling a point P

Calculations over the real numbers are slow and inaccurate due to round-off error. Cryptographic applications require fast and precise arithmetic; thus elliptic curve groups over the finite fields of Fp and Fm2 are used in practice.[1]

The key generation and verification of elliptic curve analogue (ECDSA) of the U.S. government digital signature algorithm (DSA) is described below.

ECDSA Key Generation: E is an elliptic curve defined over Fq , and P is a point of prime order n in E (Fq ); these are system-wide

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