The study of **cryptography** and cryptanalysis scientifically is known as **Cryptology. **It is math. For instance, number theory and algorithm that support** cryptography** and cryptanalysis. We will focus here on some of key math concepts at the back of the **cryptography**.

To secure the data for storage or transmission, it must be transformed in such a method that is for any unauthorized person would be hard to determine its true meaning. To do the said, certain math equation are applied. The difficulty level for solving the given equation is recognized as its intractability. The basis of **cryptography** is formed by these types of equations. Most of the important are:

**Discrete Logarithm problem**

The most excellent way to explain this problem is to show its inverse concept mechanism first. Suppose a prime number P (a number which is not dividable apart from 1 and itself, P). Over 300 digits this P is a large prime number. Let us now suppose that we have two other integers more, a and b. Now we need to find out the value of N, the value can be found by the following formula:

N = a^{b} mod P, when 0 <= N <= (P · 1)

This is recognized as discrete exponentiation and is very simple to calculate. Though, the reverse is true when we reverse it. If it is given P, a and N and is required to locate b so that the equation is legal, then we confront an incredible level of struggle. This problem figures the basis for several public key infrastructure algorithms, like Diffie-Hellman and EIGamal. For many years this puzzle has been studied and is the base of **cryptography**. It has survived many forms of attack.

**The integer Factorization problem**

This is a very simple idea. If one has two prime numbers P1 and P2, both are “large”. And then we multiply these two primes to generate the product, N. The difficulty occurs when, being given N, we strive and locate the original P1 and P2.

To a great degree simplification of this matter, the product N is the public key and the P1 and P2 both numbers are together the private key.

In all mathematical concepts this puzzle is one of the main basic. It has deeply been studied for the previous 20 years and the agreement appears to be that some of the mathematics laws are not proven or not discovered that forbids any single shortcuts. It is said, the simple fact that it has being studied and very much direct others to worry if one way or another breakthrough may be revealed.

**The Elliptic Curve Discrete Logarithm Problem**

A new cryptographic procedure based upon a logically well-known mathematical puzzle. For centuries the properties of elliptic curves have been familiar, but only recently their application to the field of cryptography has been taken on.

Imagine a giant piece of paper first, on that a series of vertical and horizontal lines are printed. With the vertical lines every line symbolizes an integer forming X class element. We get a pair of coordinates (x,y) with intersection of horizontal and vertical lines.

The elliptic curve is defined by the below highly simplified equation example:

y^{2} + y = x^{3} · x^{2} (to use in a real life application this is very small, but it will exemplify the general idea.

On the affirmative side, the puzzle comes to be quite difficult, having need of a shorter key length for equivalent security levels as measure up to the Integer Factorization Problem and the Discrete Logarithm Problem.

On the depressing side, critics challenged that this problem, since it has recently begun to put into practice in cryptography, has not had the strong study of many years that is necessary to give it a satisfactory level of trust as being secure.

The **cryptography software** is generally known as *Encryption Software.*