4442.sagews

# ECCN: 5D002 under US License Exception TSU (740.13(e))
# Curve P-256 (published by NIST):

# Modulus
p = 115792089210356248762697446949407573530086143415290314195533631308867097853951

# Order
r = 115792089210356248762697446949407573529996955224135760342422259061068512044369

# Coefficients
a = -3
b = 41058363725152142129326129780047268409114441015993725554835256314039467401291

# Base Point
Gx = 48439561293906451759052585252797914202762949526041747995844080717082404635286
Gy = 36134250956749795798585127919587881956611106672985015071877198253568414405109

#Generate Curve
E = EllipticCurve(GF(p),[a,b])  # y^2 = x^3 + ax + b
g = E.point((Gx,Gy))
(E,g)

(Elliptic Curve defined by y^2 = x^3 + 115792089210356248762697446949407573530086143415290314195533631308867097853948*x + 41058363725152142129326129780047268409114441015993725554835256314039467401291 over Finite Field of size 115792089210356248762697446949407573530086143415290314195533631308867097853951, (48439561293906451759052585252797914202762949526041747995844080717082404635286 : 36134250956749795798585127919587881956611106672985015071877198253568414405109 : 1))

# Diffie Hellman Key Exchange Example

# Alice generates a secret number and multiplies times the base point
rA = randrange(2**256);RA = rA*g

# Bob generates a secret number and multiplies times the base point
rB = randrange(2**256); RB = rB*g

# Alice and bob exchange their new points RA and RB

# Bob multiplies the point from Alice times his secret number
kA = rA*RB

# Alice multiplies the point from Alice times his secret number
kB = rB*RA

# Both Alice and Bob have the shared secret kA = kB
(kA,kB)

((92858997737304457521252065746379931338696374205359539900134625392925349057012 : 25826257329444039229011399285538065758919635841044136678692840454595768569900 : 1), (92858997737304457521252065746379931338696374205359539900134625392925349057012 : 25826257329444039229011399285538065758919635841044136678692840454595768569900 : 1))

# El Gamal Example

# Encode message
message=E.random_element();

# Generate ephemeral key
ek = randrange(2**256)

# e1
e1 = ek*g

# e2
e2 = message + ek*RA

message

(33716410284378234666415809925922089422719337098388252201854934211257541142654 : 64870628577544083902154579481388089116632823396012869459173603062127465342661 : 1)

# Decrypt El Gamal

# Alice decrypts the message using her secret key (rA)
decoded=e2-rA*e1

decoded

(33716410284378234666415809925922089422719337098388252201854934211257541142654 : 64870628577544083902154579481388089116632823396012869459173603062127465342661 : 1)