Public-key cryptography uses an encryption algorithm that is made by a mathematical formula. Elliptic Curve is one of the formulas. Why is the math problem used for this cryptography system?

Mathematical Formulas

Those math problems in an encryption area have a feature like this.

“Finding a result with inputs is easy, but finding the input(s) is not easy, almost impossible.”

For example, let’s assume that you are trying to solve a problem:

Problem: $y = x^2 - 66x + 1073$

If $x = 29$ , what is $y$ value?

$y = (29 \times 29) - (66 \times 29) + 1073 \\ y = 841 - 1914 + 1073 \\ y = 0$

And next, what is the other $x$ value for $y = 0$ ?

$y = x^2 - 66x + 1073 \\ y = x ^2 - (29 + 37)x + (29 \times 37) \\ y = (x - 29)(x-37) \therefore x = 29 \; or \; x = 37$

Easy? Then, if you didn’t know that when $x = 29$ , $y = 0$ at first, could you solve this problem nimbly? How about if $y = 7842$ ?

In these problems, finding $x$ value(s) with input $y$ is always taking longer than finding $y$ value with input $x$ whether you are a genius or not. And, the computer is also! But, the example is so easy to the computer!

If so, how about a math problem has no inverting solution to find $x$ value? The final solution is assigning all number for $x$ until finding presented $y$ value. For instance, someone wants to find out my secret value $x$ , and I made the secrete value from very large value like 9,223,372,036,854,775,807 = possible maximum value for $2^8$ in binary system = 8 bytes in the computer using a mathmatical formula, $F(x)$ . Additionally, the computer spends 0.1 second to solve the $F(x)$ for every single value $x$ . How long does it take to find out my secret value? At the very worst, it takes almost 29,247,120,867 years. So, it’s practically impossible!

In the world, there are a lot of unsolved problems. Elliptic curve is the one of that kind.

Month: April 2017

Elliptic Curve Cryptography

Mathematical Formulas