2212.sagews

Elliptic Diffie--Hellman Key Exchange

# A trusted party chooses and publishes a large prime p and an elliptic curve over F_q,
# where q=p^s, for some integer s.
q=next_prime(81)^2
F.<a>=GF(q,'a')
E=EllipticCurve(F,[0,0,0,F(4),F(5)])
E

# The trusted party also publishes a base point P.
P=E.points()[40]
P

# Bob and Alice each choose secret integers nB and nA, and compute nB*P and nA*P, respectively
# Bob first,
nB=23456
QB=nB*P
# then Alice,
nA=9876
QA=nA*P
# Then they sent each other their public values
QB;QA

# Now, each multiplies the point they received by their secret integer,
# and they should be the same.
nB*QA==nA*QB;nB*QA

# They discard the y value, and use the x value as a common key for some other encryption.

reset()

Elliptic ElGamal public key cryptosystem

# A trusted party chooses and publishes a large prime p and an elliptic curve over F_q,
# where q=p^s, for some integer s.
q=next_prime(100)
F.<a>=GF(q,'a')
E=EllipticCurve(F,[0,0,0,14,5])
E

# The trusted party also publishes a base point P.
P=E.points()[27]
P

# Alice chooses a private key, an integer, nA, and publishes QA=nA*P
nA=876432
QA=nA*P
QA

# Bob embeds a plain text into the x coordinate of a point M on E.  He chooses an integer
# k to be his ephemeral key and computes C1=kP and C2=M+kQA.
M=E.points()[31]
k=123456784
C1=k*P
C2=M+k*QA
# He then sends C1 and C2 to Alice.
M;C1;C2

# Alice computes C2-nA*C1=(M+k*QA)-nA*(k*P)=M+k*(nA*P)-nA*(k*P)=M
C2-nA*C1