Consider the elliptic curve group based on the equation y^2 ≡ x^3 + ax + b mod p where a=1190, b=213, and p=2011. We will use these values as the parameters for a session of Elliptic Curve Diffie-Hellman Key Exchange. We will use P=(2,51) as a subgroup generator. You may want to use mathematical software to help with the computations, such as the Sage Cell Server (SCS). On the SCS you can construct this group as:
Consider the elliptic curve group based on the equation
where a=1190, b=213, and p=2011.
We will use these values as the parameters for a session of Elliptic Curve Diffie-Hellman Key Exchange. We will use P=(2,51) as a subgroup generator.
You may want to use mathematical software to help with the computations, such as the Sage Cell Server (SCS).
On the SCS you can construct this group as:
G=EllipticCurve(GF(2011),[1190,213])
Here is a working example.
(Note that the output on SCS is in the form of homogeneous coordinates. If you do not care about the details simply ignore the 3rd coordinate of output.)
Alice selects the private key 34 and Bob selects the private key 22.
What is A, the public key of Alice?
What is B, the public key of Bob?
After exchanging public keys, Alice and Bob both derive the same secret elliptic curve point ???. The shared secret will be the x-coordinate of ???. What is it?
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