## Big numbers in crypto

One big problem I always have is I have no sense of how big are numbers used in cryptography. Yes I always now that it can reach $2^{571}$. But HOW BIG IS IT??? And I also get lost on comparing that 2 power of something is equal to 10 power of something.

Last night I had a very unexpected intellectually challenging conversation with a professor of chemistry. Surprisingly he’s very good at general cryptography and math. And probably everything 😀 He also play chess 🙂

As Rosing wrote in his book that elliptic curve crypto systems are in the range of astronomical, it means that numbers used in crypto is large enough compared to the large numbers in the universe.

Here’s is the example given by the prof. He get a piece of paper and rip it into two pieces, piled it and rip it into two pieces and again for about 4 times. After doing that he said that how thick would you think the paper would be if I rip the paper for 50 times with the assumption that the paper is 1mm thick?

First I thought that it’s maybe just as thick as The Codebreakers book but later I think that it’s $2^{50}$. But how big it is? Let’s grab a calculator.

$2^{50}$ = 1125899906842624. So the pile of papers is 1.1 x $10^{15}$ mm.

Still have no idea how big it is.

Now let’s compare it to some astronomical numbers. From here I get the info that the distance from earth to sun is 92,935,700 miles or 149565139.43800 km or 149565139438000 mm or 1.5 x $10^{12}$ mm.

So then the pile of paper wouldn’t fit the space between earth and the sun! WOW!!!

Now let’s pick another example. For a $2^{512}$ bit length key, $2^{512}$ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096, which is equal to 1.3 x $10^{154}$. Which is MUCH bigger than the number of atoms in all the earth, which is 8.87 x $10^{49}$.

Well, now I have a better sense of how big those numbers are. Hoping you will too 🙂