The prime
number Internet codes are based on an invention
made several hundred years by Gauss: the clock
calculator.
Internet cryptography uses clock calculators
to scramble your credit card.
To add times on a clock is the same as working
out what time it will be 4 hours on from 9. The
answer is 1 o’clock.
9+4 = 1 (modulo 12)
We write modulo 12 because we can consider clocks
with different numbers of hours than just 12.
For example on a clock with 10 hours
9+4 = 3 (modulo 10)
How do we do multiplication on this calculator?
What does a calculation like 4 x 9 mean? It means
taking four 9's and adding them together. Multiplication
consists of doing addition a certain number of
times. So where does the hand on the clock end
up after adding together four 9's? At each turn
we fall back three hours until we get eventually
to 12 o'clock which it's better to call 0 o'clock.
Multiplication is possible since we can do addition.
We get the strange answer: 4x9=0(modulo 12).
Let's see how to raise a number to some power,
for example 94. What does this mean: multiply
9 together 4 times. But we just learnt how to
do multiplication so we should be able to perform
this calculation. To make this calculation, because
the numbers are getting quite big, it is easier
here to take the remainder after division by 12.
Lets start with 9x9 which is 81. What is the remainder
on division by 12, i.e. what is 81 o'clock? Actually
it turns out to be 9 again! How ever many times
we multiply 9 together we always end up with 9
again. So 9x9=9x9x9=9x9x9x9=9 (modulo 12).
The answers are got by calculating the answer
on a normal calculator and then taking the remainder
after division by the number of hours on the clock.
But the power of the clock calculator is that
you don’t have to calculate things on the
conventional calculator first.
See if you can work out what 799 is
on a 12 hour clock calculator? Hint: work out
7 x 7 first, then multiply the answer by 7 again.
Do you see the pattern?
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