Fundamental Math; Elliptic Curve

ECClines-3.svg

 

Math is the underpinning of all technology. Whilst math itself is simply a man made creation, those rules and systems are the foundations of security. It is sad, then that by the time student’s reach college in the UK, many have a deep fear of mathematics.

Unlike linguistics, math has few ambiguities which is why it is a great foundation for technology.

Today we will be learning about the elliptic curve.

The math behind an elliptic curve is very complex and can take some time to digest, so we will be working from a high level and getting more complex as we progress.

The elliptic curve is a set of points that satisfy an equation based on two variables with the points marking out a set of degree.

The example equation we often see is – y2 = x3 + ax + b

Which creates the representation seen in the image below.

Curve

 

We can think of this diagram as a drawn version of the points mapped out to give a visual representation.

A deeper view would show us the individual points of the equation marked out.

Some interesting qualities about elliptic curves are that they offer horizontal symmetry and any non-vertical line will intersect the curve in at least three places.

These idiosyncrasies are what go into making the elliptic curve one of the most secure forms of cryptography today.

Elliptic curves are important because Asymmetric schemes like RSA and Elgamal require exponentiation in integer rings  which are not only computationally demanding but many experts predict that these forms of cryptography could be broken within years.

Elliptic curves can be defined over finite fields, not just real numbers, For purposes of cryptography we are interested module a prime b – or rather, elliptic curves over prime fields.

 

 Curve22

http://cs.ucsb.edu/~koc/cren/docs/w03/09-ecc.pdf

To make a graphical representation we first generate points on an elliptic curve through point addition P+Q=R – (Xp, Yp) + (Xq, Yq) = (XrYr).

Then to get a geometric graphical representation we draw a straight line through P and Q (if P = Q we draw a tangent line) mirror third intersection point of drawn line with elliptic curve along x-axis.

Elliptic curve can be used for key exchange and offers performance advantages over public key encryption methods such as RSA.

Elliptic curve is based on the discrete logarithm model.

The discrete logarithm is

 

 

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