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If Alice and Bob want to agree on a secret key, each of them adds one liter
of their own secret color to their own pot of yellow paint. Alice might add
a peculiar shade of purple, while Bob might add crimson. Each sends their
own mixed pot to the other and we assume that Eve can see and even sample
these mixtures as they are sent between Alice and Bob. Finally, Alice takes
Bob's mixture and adds one liter of her own secret color, and Bob takes
Alice's mixture and adds one liter of his own secret color. Both pots should
now be the same color, because they both contain one liter of yellow, one
liter of purple and one liter of crimson. It is the exact color of the doubly
contaminated pots that is used as the key.
Does Eve know the secret key? No, she doesn't. She saw (and possibly
sampled) the two partial mixtures that passed by her: “yellow and purple” and
“yellow and crimson.” If Eve combines these mixtures - the only operation
she could do on her own - she will only end up with a mixture containing
“yellow and yellow and purple and crimson.” In order to find the secret key
she would need to remove or “unmix” one unit of yellow from this mixture.
Since she cannot unmix one unit of yellow she cannot generate the same
color as Alice and Bob and thus does not know the key. 13
So although Eve can intercept the pots of paint being exchanged, she can-
not work out Alice's and Bob's secret keys because mixing paint is a one-way
function.
The actual mathematical one-way function used in the Diffie and Hellman
key exchange proposal was based on modular arithmetic . Calculations in modular
arithmetic are done with a count that resets itself to zero every time a certain
number, known as the modulus , has been reached. Modular arithmetic is like
telling time using the numbers on a clock face. For example, 9 + 7 in normal
arithmetic equals 16. However, in modular arithmetic with a modulus of 12
(“mod 12” arithmetic, also called clock arithmetic ), the result of 9 + 7 is 4. If it is
9 o'clock in the morning then seven hours later it will be 4 o'clock in the after-
noon. Because the hour number starts over after it reaches 12, the modulus is
12. In normal arithmetic, the result of adding two numbers grows as the num-
bers being added are larger. With modular arithmetic, the numbers can grow
just to the value of the modulus. Although this key exchange system was a great
breakthrough in cryptography, it still required that Alice and Bob exchange
several messages to establish the shared secret key. The Diffie-Hellman key
exchange protocol was also fundamentally a two-party protocol rather than
a broadcast protocol that allowed Alice or Bob to communicate securely with
B.12.6. Whitfield Diffie and Martin Hellman are the inventors of the Diffie-Hellman key
exchange protocol. This is remarkable process by which Alice and Bob can agree on a
secret key using an open link that is vulnerable to access by an eavesdropper, Eve.
 
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