How do one-way hash functions work? (Edited)

Question

I read the Wikipedia article about md5 hashes but I still can\'t understand how a hash can\'t be \"reconstituted\" back to the original text.

Could someone explain t

Answer 1

Shooting for a simple analogy here instead of a complex explanation.

To start with, let's break the subject down into two parts, one-way operations and hashing. What is a one-way operation and why would you want one?

One way operations are called that because they are not reversible. Most typical operations like addition and multiplication can be reversed while modulo division can not be reversed. Why is that important? Because you want to provide a output value which 1) is difficult to duplicate without the original inputs and 2) provides no way to figure out the inputs from the output.

Reversible

Addition:

4 + 3 = 7  

This can be reversed by taking the sum and subtracting one of the addends

7 - 3 = 4  

Multiplication:

4 * 5 = 20  

This can be reversed by taking the product and dividing by one of the factors

20 / 4 = 5

Not Reversible

Modulo division:

22 % 7 = 1  

This can not be reversed because there is no operation that you can do to the quotient and the dividend to reconstitute the divisor (or vice versa).

Can you find an operation to fill in where the '?' is?

1  ?  7 = 22  
1  ?  22 = 7

With that being said, one-way hash functions have the same mathematical quality as modulo division.

Why is this important?

Lets say I gave you a key to a locker in a bus terminal that has one thousand lockers and asked you to deliver it to my banker. Being the smart guy you are, not to mention suspicious, you would immediately look on the key to see what locker number is written on the key. Knowing this, I've done a few devious things; first I found two numbers that when divided using modulo division gives me a number in the range between 1 and 1000, second I erased the original number and written on it the divisor from the pair of numbers, second I chose a bus terminal that has a guard protecting the lockers from miscreants by only letting people try one locker a day with their key, third the banker already knows the dividend so when he gets the key he can do the math and figure out the remainder and know which locker to open.

If I choose the operands wisely I can get near to a one-to-one relationship between the quotient and the dividend which forces you to try each locker because the answer spreads the results of the possible inputs over the range of desired numbers, the lockers available in the terminal. Basically, it means you can't acquire any knowledge about the remainder even if you know one of the operands.

So, now I can 'trust' you to deliver the key to its rightful owner without worrying that you can easily guess to which locker it belongs. Sure, you could brute force search all the lockers but that would take almost 3 years, plenty of time for my banker to use the key and empty the locker.

See the other answers for more specifics on the different hash functions.