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WHAT AN ELLIPTICAL CURVE DOES WITH SECRETS?

              ADVT: SMART DOORBELL IN AMAZON

    With proper keys, we can lock and unlock a safe or secret.  Consider the following two strings of the alphabet.
      ABCDEF.............Z AB
            ABCDEF.................ZYZ
    In the second string, each letter is shifted to third place.  A corresponds to C, B to D; C to E, and so on.  Suppose, you want to send the message 'BAD' you send DCF and also send 'Three' as the key.  The receiver can decode the message by shifting the alphabets to third place in the backward direction. Hence key plays key-role in cryptography.
     To further strengthen our secrets, we can have a double key.  One is called the public key and the other is the private key.  Using the public key, anyone can encrypt the message and sent it through public channels.  No one can decrypt the secret message except one who has the private key.

ECC:
    The elliptic curve cryptography(ECC) is a very good one and it is employed by 'Bitcoin' and many others.  We are going to see how to get private and public keys from that curve.
    The equation of an elliptic curve is y^2 =x^2 +ax+b.  That is, x and y coordinates of the points in the curve satisfy this equation.
                         
Also, y = + or - squreroot of x^3+ax+b
So, there are two values of y as +x or -x.  Also, all the elliptical curves are symmetrical about the x-axis.
Point addition:
    We know, adding two numbers gives the third number.  Similarly, you can add two points on an elliptic curve and get a third point on the curve.
1. Select two points that you want to add on the curve.
2. Find the straight line that goes through those two points.
3. Find the third point where that line intersects the curve.
4. Reflect that third point across the x-axis.  The point you get is the result of the addition of two points.


               

Hence P+Q =R
Adding a point to itself:
    In ECC, we specify a base point in the curve and add that point to itself.
1. Mark the point p on the curve.
2. Draw a tangent to the point P.
3. Find the next point that this line intersects and reflects it across the x-axis.

                                                            
That is P+P =2P
4, We can add point P to point 2p and get 3p.
                                     

5. We can continue to add p to itself to compute 4p and 5p and so on.

    Let us say, we compute n.p where n is a random large number.  The result will be point s on the curve.
    If you know s, p, and the curve,  can you find out n? That is, can you find out how many times (n) I added p to itself to get the point s?  No one can find out 'n'.  Even if one has supercomputer and infinity time, one can not find out n - the number of times.
    Now, we can choose n for the private key and s for the public key.  The public key is the x and y coordinate of point s on the curve.  The n and s satisfy the definition.
   "It is computationally infeasible to derive the private key corresponding to a given public key'.

     Now that we have the required keys, we can encrypt messages 
using the elliptic curve and maintain secrecy.
          
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