So for those of you having trouble understanding the video of the DH Key exchange I posted, here is how 3^15 mod 17 = 6. First, calculate 3^15, which is our "generator" times the private number of in our applied case, the enrollee... = 14,348,907. Now, we take 14,348,907 and divide it by our prime modulus (known by both the enrollee and registrar, (17) and get 844,053.3529. Now, we remove the remainder from that number, .3529, so we are left with only 844,053 and multiply it by our prime modulus... 844,053 * 17=14,348901. Now, take our initial value (3^15) which is 14,348,907 and subtract the value we just calculated, 14,348,901... 14,348,907 - 14,348,901 = 6!
Therefore:
3^15 mod 17 = 6
Now that the enrollee has generated a public number from its private number, it sends that to the registrar. Now it is the registrars turn to come up with its own private number (13) and make a public number.
3^13 mod 17 = 12
3^13 = 1,594,323
Now, divide by our prime modulus (17)... 1,594,323 / 17 = 93,783.70588. Remove the decimal and get 93,783. Now multiply that number by 17... 93,783 * 17 = 1,594,311. Now, take our initial value and subtract the value we just created to get the registrar's public number... 1,594,323 - 1,594,311 = 12! The registrar then sends this publicly to the enrollee.
Enrolee public number: 6
Enrolee private number: 15
Registrar public number: 12
Registrar private number: 13
Now, the registrar and the enrollee both know each others public numbers, 6 and 12. So what they do now is they take their public number from the other, and raise it to the power of their own private number.
So for the enrollee to calculate the shared secret, it would do 12^15 mod 17 = 10
And for the registrar to calculate the shared secret, it would do 6^13 mod 17 = 10
Now they both know the shared secret, which I THINK may also be the seed that the PRNG uses (as stated before) but I'm not sure yet, or maybe this shared secret in conjunction with something else is the seed.
