HackPack CTF 2021: Euler's Identity

Euler’s Identity

Category: misc

50 points

One night, a man named Euler gave you three strings in your dream: MD5:d8d540ae49aadd151b96feb4e0ff124f, SHA1:abb07ca45c9e7719e66e766b958d943f561b8de6, SHA2(SHA1(MD5)):c037c03ee627047a85df540c42d59c6b6028841704a7c706feff584a997fd2a3.

Hint 1: People ask:‘How does great Euler find the relationship among pi, imaginary number, and Euler’s number.’

Hint 2: On the following night, the man appeared in your dream again. He told you concatenating secrets together foreshadowed the future. For example, given MD5(This), SHA1(is), and SHA2(SHA1(MD5(future))), the key is flag{Thisisfuture}.

Solution

Decoded value for d8d540ae49aadd151b96feb4e0ff124f MD5 hash is e^.

echo -n 'e^' | md5sum
d8d540ae49aadd151b96feb4e0ff124f  -

For abb07ca45c9e7719e66e766b958d943f561b8de6 SHA1 hash the value is ipi=.

echo -n 'ipi=' | sha1sum 
abb07ca45c9e7719e66e766b958d943f561b8de6  -

Both hashes can be simply bruteforced. However third hash isn’t that simple, but let’s take a look into what we have so far. After a while I’ve realized it’s a mathematical formula. I was never good in math, so I needed to search for Euler's identity in wikipedia and found the answer there.

e^ipi+1=0

So, if we move the 1 to the other side of equation, we got…

e^ipi=-1

And bingo! It was the flag :-)

Flag

flag{e^ipi=-1}