## Diffie Hellman Key Exchange

### September 12, 2013

[ I apologize for not posting an exercise on Tuesday. I have had intermittent internet access at home since last week that has prevented me from working on the blog. And now I published it immediately instead of scheduling it for Friday morning. Oh my! ]

Diffie-Hellman key exchange is a method for two people, traditionally named Alice and Bob, to agree on a secret key known only to the two of them using an insecure communications channel; Wikipedia gives an explanation, complete with a cheesy diagram.

The key exchange begins when Alice and Bob agree on parameters *p*, which should be at least a few hundred decimal digits in a real-life application, and *g*, which is typically small, say 2 or 3 or 5; we will follow Wikipedia’s illustration and choose *p* = 23 and *g* = 5. Then Alice chooses a secret number *a*, computes *g ^{a}* (mod

*p*), and sends it to Bob; in a real application,

*a*will be of similar magnitude to

*p*, but for illustration we choose

*a*= 6, so Alice sends the number 5

^{6}(mod 23) = 8 to Bob. Bob follows the same procedure, choosing

*b*, computing

*g*(mod

^{b}*p*), and sending it to Alice; in a real application,

*b*will be of similar magnitude to

*p*, but for illustration we choose

*b*= 15, so Bob sends the number 5

^{15}(mod 23) = 19 to Alice. All of these exchanges can be conducted over an insecure channel, as long as Alice keeps

*a*secret and Bob keeps

*b*secret.

Once those exchanges have been made, Alice and Bob can both compute the same secret key *s*; Alice computes *s* = 19^{6} (mod 23) = 2, using Bob’s public number 19 and her secret number *a* = 6, and Bob computes *s* = 8^{15} (mod 23) = 2, using Alice’s public number 8 and his secret number *b* = 15. At this point, both *a* and *b* can be discarded. Using the shared secret number *s* = 2 as a key, Alice and Bob can encrypt and decrypt messages using any symmetric cipher, such as DES or AES, and send encrypted messages over insecure channels, confident that no one else knows their secret key. In practice, *p* is generally taken to be 2*q*+1, where *q* is a Sophie Germain prime of at least 512 bits, and *g* is generally taken to be a primitive root modulo *p*; these rules guarantee some desirable properties of the security of the secret key. The only way for someone with knowledge of *g*, *p*, *g ^{a}* (mod

*p*) and

*g*(mod

^{b}*p*) to determine the secret key

*s*is to solve for

*a*and

*b*, but that involves computing the

*discreet logarithm*and is intractable with current algorithms.

Your task is to write programs that perform Diffie-Hellman key exchange as described above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.