Elliptic Curve Encryption

Choose one of the two following elliptic curves:

Curve over GF(p)

Given a prime p, the elliptic curves modulo p that we use are of the form: y² = x³ + ax + b.

The curve and key information used for this page is:

Curve over GF(2^m)

Given a field GF(2^m) and a irreducable polynomial, the curve is of the form: y² + xy = x³ + ax² + b.

The curve and key information used for this page is:

A common use of elliptic curve cryptography is to generate a shared secret key for use in secret key systems. The Diffie-Hellman protocol can be applied to elliptic curves as follows:

Alice generates a secret key r_A
Alice calculates P_A = r_A*e₁ and sends it to Bob.
Bob generates a secret key r_B
Bob calculates P_B = r_B*e₁ and sends it to Alice.
Alice receives P_B and calculates P = r_A*P_B
Bob receives P_A and calculates P = r_B*P_A
Alice and Bob can use P to generate a shared secret key suitable for AES or some other system. Some protocols use the x coordinate of P as the shared key, or a hash can be applied to P to generate the key.
Note that if Alice already has published an elliptic curve public key, then Bob and Alice can agree to use Q in place of P_A.

A's random value:
B's random value:

Details:

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