Keys and guessing.

Step 1 of 6

A locked message

Alice writes Bob a message, locks it with a key, and sends it down the tube. Eve copies every capsule that passes, but without the key she sees only scrambled bytes. Bob, who has the same key, opens the message and reads it. The key is a binary number, a row of bits, punched on a card both sides hold.

But where did Bob get the key from?

Step 2 of 6

Counting keys

Eve can try guessing: try every possible key until the message makes sense. Her machine, on the board at the back, tries a trillion keys a second, $10^{12}$⁠, and shows how many keys there are and how long it takes to try them all.

An $n$⁠-bit key has $2^n$⁠ possible values, because each bit has two possibilities, and every extra bit doubles the number. Key size is now fixed at $8$⁠ bits: a toy key.

Step 3 of 6

One more byte

The DES encryption standard, adopted in the United States in 1977, used a $56$⁠-bit key. Eve’s machine goes through all its keys in about twenty hours. Eight bits make one byte.

Step 4 of 6

Eight more bits

A $40$⁠-bit key, like those the United States once allowed for export, falls to Eve’s machine in about a second. Work out how long a key one byte longer would take.

Step 5 of 6

Out of reach

Because every bit doubles the time, a few more bits take a key from “broken in a second” to “broken after the codebreaker is long dead”. Today’s keys are $128$⁠ or $256$⁠ bits long. The glass cube beside Eve shows what share of all the keys she could try in eighty years.

Step 6 of 6

Twice as long

A long enough key shuts the door on guessing. But the problem from the start of the Lesson remains: to use a key, both sides must have the same key, and it must not travel down the tube. The next Lesson solves that.