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Voiceover: Congratulations. We've reached a branching
point in our lesson. Now a few different ideas
have been introduced, so it's important to orient
ourselves here before moving forward in various directions. Now, what motivated this lesson? Well, we learned about RSA encryption, and RSA depended on two things. One, that prime
factorization is difficult. So if I multiply two big
primes together, P1 and P2 and I give you N, I can trust
or feel secure in knowing that you'll take a long
time to find those primes, and maybe more than your lifetime. Also two, RSA requires
that those large primes I generated was easy, meaning I could just quickly generate a large prime. So let's return to the first problem. Difficulty of prime factorization. Well what is it about prime
factorization or just prime numbers themselves which
make problems hard? And to find out we begin
with a core problem. Given X, is X prime or composite,
which is a primality test? Now we ended up building some
solutions to this problem. And in doing so, we realize that this problem was computationally expensive. That is, there is no instant
solution to this problem. As our input number
grew, the amount of time or the amount of steps involved for an algorithm to find the solution also grew. And, how much it grew,
we actually understand better now because we can predict this search space using the
prime number theorem. Though, the real issue can
be thought of as a graph, where on the vertical axis
we have the number of steps our algorithm takes, so you
can just think of it as time. And on the X-axis is the input size and as input size increased, so did time. And the problem we had is that shape of this curve is unacceptable. Because in our case, once
N hit even 20 digits, we can no longer solve the
problem in the worst case. And if we threw in input
hundreds of digits long at our algorithm we can all
agree it would not work. In our case it would crash. So it's practically impossible
to check if large input is prime using our current strategies. Now let's return to the
first point, factorization. Realize based on our current understanding in this lesson, factorization has to be harder than a primality test. That is there are more steps
involved in finding the prime factorization of some number, versus just telling me if a number is prime. Since, remember, factorization
requires us to find all the prime factors for some input N, whereas a primality test only
requires we find one divisor And a nice reminder of this
is that some users have actually turned these
primality tests into prime factorization algorithms, simply by repeating after it
finds its first divisor. So the primality test is
just kind of a sub-routine inside the main factorization algorithm. So if we return to this
graph, a factorization algorithm would be something like this. As input grows, our
factorization algorithm would be above this primality test line. And realize that in any
situation we always have a computational limit, that is the number of primitive steps we can
calculate which is based on our computing power in any given situation and the amount of time we have. And you could think of
this as a horizontal line, or a threshold on this graph. Anything above this line is out of reach, not feasible to solve. And in this lesson we were
limited by the rover's on-board computer which was fairly slow,
which is why we couldn't run primality tests on numbers
with even 20 digits. However, even if we had,
say, 1,000 computers running for a year, this would
simply just push this horizontal line up to
some other threshold. And this would allow us
to run tests on larger numbers, but as you can see,
we would always hit some limit where the input is large enough that we can no longer solve the problems again. Now, this leads to many
interesting question paths. However, I'll identify the
two I'm going to explore next. First of all, so far I have not been very precise about these growth curves. Though, it would be
helpful if, imagine for any algorithm you give me, no
matter what it's trying to solve, and no matter what
machine it's even running on, we could draw some
corresponding growth curve on this graph, simply by looking at the description of the algorithm. If we could do this, we
could identify categories of certain growth
shapes, which tell us how difficult a problem would be to solve. And in this way, we could
understand an algorithm before we even run it, very
important to think about. And you could hand me your
algorithm written on a piece of paper and I could look
at it and I'll say, "I know "this algorithm is not solvable
with the input size needed." And this leads us to a
lesson on time complexity, an incredibly important
conceptual tool we will need. And if you've ever heard this
runs in polynomial time or exponential time, these are terms which come out of time complexity. Next, many of you have
been wondering, "well, how "are we generating these
random primes in practice," the second point I made about RSA. Well let's just walk through it quickly. To generate a large random
prime number hundreds of digits long, we need to
follow these instructions. Generate a random number,
test if it's prime, if it's prime, we're done. If not, try again until we hit a prime. Now in this three-step
procedure, what's most important is the second
step, test if it's prime. If we can't do that step, this won't work. So how is this working today? Well, it's based on a subtle
alteration to our problem definition, and more importantly,
the use of randomness. And this leads us to another
lesson on random algorithms. And now finally, if there are any other lingering question paths you have, please post them below and we can plan lessons. For example, there are
some deeper mathematics we have yet to explore in
order to speed up our existing trial division primality test. And what else are you thinking of? Please share below.