Fun computer math triva/facts


Tribe of Judah StarCraft Series Chapter Leader
Lately I've been studying a lot about cryptography and computer science, and I'd like to share some interesting things I've learned with you.

I'm going to provide several examples and demonstrations that are theoretical but are based on real-world applications; they should be accurate to an acceptable margin of error in most cases. If you notice an error just let me know!

How long would it take for a personal computer to crack[sup]1[/sup] a standard Wi-Fi password that's six characters long[sup]2[/sup]?

Less than 4 seconds.

What if the password was eight characters?

About 20 hours.

Approximately how long would it take for the world's fastest supercomputer[sup]3[/sup] to crack[sup]1[/sup] a standard AES encryption key[sup]4[/sup]?

Nearly six hundred billion centuries. (598,715,102,916,700 years)

If everyone on Earth had a supercomputer, how long would it take to crack[sup]1[/sup] an AES-encrypted message that was only 5 characters long? "Hello"

A little over half an hour.

What if the message were 6 characters long? "Hello!"

8206 years.

Fun facts:

Even the best desktop computers only allow for approximately 35-bit memory address allocation.[sup]5[/sup]

The supercomputers of 1997 are roughly equivalent to the very best gaming desktops in 2015 (not counting the processing power of graphics cards).

If IPv6 were measured as square inches, there would be enough addresses to fill the combined surface area of all the planets in Earth's solar system. Twice.[sup]6[/sup]

1. Using brute force
2. A WPA-2 encryption key is based on the ASCII character set, which allows up to 128 character choices per character space and is represented by 7 bit strings, or ~1 byte.
3. The Tianhe-2 is the world's fastest supercomputer as of 2015, and is rated at 33.86 petaflops. It can theoretically perform just under 36,044,800,000,000,000 operations per second.
4. This is based on the outdated 128-bit key length. The new standard, 256-bit, would take trillions of millenia. The key is the same as a password, and should not be confused with the encrypted content, which is far more difficult to crack.
5. A desktop computer has a maximum of 3.2768 * 10^7 RAM address space, or 32 GB. A fully-"unlocked" 64-bit computer would have 18 quintillion address spaces. Although a 64-bit computer processes much slower than a 32-bit computer it is much more efficient and accurate when dealing with very large data-sets.
6. Actually, it's just barely short. The surface area of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune combined, in inches, is ~187,428,995,527,809,000,000. (log(2^128)/log(surfaceAreaSolarSystem) = 1.90066

Password: ASCII = 128 character choices = 7 bits ≈ 1 byte ≈ 1 floating point operation
Six characters = 128^6 = 4,398,046,511,104
ASCII math where x is the character-length of a password.
Full expression is ((128^x)*(log(128^x)/log(2^32))/2/(3500000000*32*4*2)

Key = 128 bit / 2 = (2^128)/2 = 1.701*10^38 (log(2^128)/log(2^32)/2)/supercomputerFLOPs
Encrypted message = UTF-8 ≈ 24 bits = 2^24 = 16,777,216; (2^24)/2/(supercomputerFLOPs
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