2740.sagews

# Diffie-Helmann Key Exchange Protocol:
# how to exchange a secret number over insecure channels

# Alice and Bob agree on a large prime p and a primitive root, r, modulo p.  This information is public.
# I am Alice.  You are Bob.

# We agree that p will be:

p = next_prime(19648400)
p

19648459

# And we ask sage to find a primitive root for us:
    
r = primitive_root(p)
r

2

# Let's exchange a secret number.  Your assignment is to send me the numbers B and C defined as follows.

# How to use sage:
    # click on each cell.
    # a blue "evaluate" appears beneath it.
    # click on "evaluate" to evaluate the cell.

# Alice chooses a secret random integer a  (1 < a < p-1) and sends A = r^a mod p to Bob.

# a = ???
# A = power_mod(r, a, p)

A = 17525235

# Bob chooses a secret random integer b (1 < b < p-1) and sends B = r^b mod p to Alice.

# b = pick a number, any number, fill in the blank below:
b = 
B  = power_mod(r, b, p)
B

# Give me B.

# Alice knows B.  She computes B^a mod p.
# C = power_mod(B, a, p)

# Bob knows A.  He computes A^b mod p.
C = power_mod(A, b, p)

# C is the secret number shared by both Alice and Bob.
# Give me C.