Hi All , so in a previous article i tried to explain how D-H works , and i hope I did a good job of it and hopefully there’s no questions , but how hard would it be for the person eavesdropping to reverse the “secret exponents” and guess the key?

Remember that image ? let’s recap:

Jerry and Simon agreed a **prime** and a **base** number , while they were chatting about this there’s a third person “Random” that is listening to everything.

Next step Simon and Jerry will calculate a number and tell it each other, for example jerry will tell Simon:

BASE to the SECRET EXPONENT modulo PRIME

which translates to:

3667 ** 6531 % 2833 = 2642

So Jerry will tell Simon my Computed number is **2642.**

So how hard is it for “Random” to obtain the “secret exponent” with all the values that she had right now? to clarify “Random” equation should be something like:

3667 **X% 2833 = 2642

**X** is the value we want to know , if “Random” knows **X** , then she will be able to know the secret exponent.

This is called discrete logarithms , so to examplify this i will use smaller numbers:

BASE=5 , PRIME=7 , RESULT=2 , SECRET = 105 ** 7 % 17 = 10

So far there’s no easy way to solve this backwards , it’s easy with small numbers , for example we could make a list of all numbers smaller than **prime:**

So that would work in a way … but as you can tell it is sort of brute forcing , the catch is when you use a large prime number **(+100 digits)** , something in the range of:

2074722246773485207821695222107608587480996474721117292752992589912196684750549658310084416732550077

you can see the complexity **increases** drastically , the other catch is once you build this table , and build it in **time**! , how do you search **efficiently** over it ? as you see not impossible but very very slow.

So long prime numbers would add a lot of complexity to the reversing this modulo.

One question that in a forum was how complex is to generate these numbers , and how fast is it ? specially **power** of something.

Well what i found is that if the a compute want’s to know:

3 ** 17 OR 3 power of 17

The computer won’t do this: (it’s too slow)

3x3x3x3x3x3x3x3x3....

Instead there’s an operation called **repeated squaring ,** which simplifies this greatly , the idea is the following:

This is way faster than multiplying **3** , **17** times :)

So to wrap up , Some arithmetic operations are very easy/doable one way but very complex the other way around , hence D-H popularity .