A primality test is an algorithm to decide whether an input number is prime. Some primality tests are deterministic. They always correctly decide if a number is prime or composite.The fastest known deterministic primality test was invented in 2004. There are three computer scientists, such as Agrawal, Kayal, and Saxena, invented the AKS primality test that operated in O˜ (log(n)6 ) time, where O˜ (f(n)) is represented as O(f(n).log(f(n))k) for some integer k [1]. Although a significant breakthrough, this speed is rather slow when compared to information security requirement.The advantage of prime numbers are that they are utilized in cryptography. ... Read More

Euler's theorem is a generalization of Fermat's little theorem handling with powers of integers modulo positive integers. It increase in applications of elementary number theory, such as the theoretical supporting structure for the RSA cryptosystem.This theorem states that for every a and n that are relatively prime −$$\mathrm{a^{\phi \left ( n \right )}\, \equiv\, 1\left ( mod \, n \right ) }$$where $\mathrm{\phi}$(n) is Euler's totient function, which counts the number of positive integers less than n that are relatively prime to n.Consider the set of such integers −R = {x1, x2, … x$\mathrm{\phi}$(n)}, i.e., each element xi of R ... Read More

S-Box Substitution is a procedure that accepts the 48-bit input from the XOR operation containing the compressed key and expanded RPT and creates a 32-bit output utilizing the substitution technique.The substitution is implemented by the eight substitution boxes (also known as the S-boxes). Each 8-S-boxes has a 6-bit input and a 4-bit output. The 48-bit input block is divided into 8 sub-blocks (each including 6 bits), and each sub-blocks is provided to an S-box.The substitution in each box follows a pre-decided rule depends on a 4-row by 16- column table. The sequence of bits one and six of the input ... Read More

The public key cryptography is treated as the most secure cryptography to make digital signatures and to implement encryption process. The management of digital signature will be treated as the most secured service in future for on-line communications. Therefore, it can perform secure online communications the public key cryptography plays an essential role in cryptography.There are various application of Public key cryptography which are as follows −Digital signatures − It is a message produced by user’s private key used as authenticity of a user. The digital signature generated by the private key of a user and hash algorithm. First the ... Read More

Public-key cryptography is also known as asymmetric cryptography. It is a form of cryptography in which a user has a set of cryptographic keys including public key and a private key. The functions of Public key Cryptography are as follows −Encryption − The encryption application supports the confidentiality and integrity security services for the information. The public key supports the security services including authentication and non-repudiation.Encryption can be used to transform the plaintext message into unreadable format with the support of a key, again the message transform back to original message by using the decryption procedure.In this process, each user ... Read More

A public-key cryptosystem is that the messages is encrypted with one key and can only be decrypted with a second key, etc. A strong public-key system is one in which control of both the algorithm and the one key provides no beneficial information about the other key and thus no indication as to how to decrypt the message.In public key cryptography, it is not applicable to calculate private key using the available public key in the network. Because of this reason public key can be completely available in the network.If a user encrypts a message with the public key of ... Read More

Let G be a finite cyclic set with n elements. It consider that the group is written multiplicatively. Let b be a generator of G and thus each element g of G can be written in the form g = bk for some integer k.Moreover, any two such integers defining g will be congruent modulo n. It can represent a function logb: G → Zn(where Zn indicates the ring of integers modulo n) by creating to g the congruence class of k modulo n. This function is a group isomorphism known as the discrete algorithm to base b.In mathematics, particularly ... Read More

Random numbers are the numbers which play an essential role for various network security applications. There are some approaches that are used to generating random numbers including pseudorandom number generator and linear congruent generator also cryptographically generated random numbers etc.The main objective of the Chinese Reminder Theorem is to generating Random numbers. In essence, CRT says it is applicable to reconstruct integers in a certain range from their residues modulo a set of pair wise relatively prime modulo.CRT is one of the main theorems of mathematics. This can be utilized in the area of cryptography. It is a perfect set ... Read More

Fermat's little theorem is a fundamental theorem in elementary number theory, which provides compute powers of integers modulo prime numbers. It is a specific case of Euler's theorem, and is essential in applications of elementary number theory, such as primality testing and public-key cryptography. This is referred to as Fermat’s little theorem.Fermat’s theorem is also called a Fermat’s little theorem defines that is P is prime and ‘a’ is a positive integer not divisible by P then −aP−1 ≡ 1 mod PSecond condition says that if P is a prime and a is an integer then aP ≡ 1 mod ... Read More

AES encryption defines the phase of hiding electronic information using an approved 128-bit, 192-bit, or 256-bit symmetric encryption algorithm from the Advanced Encryption Standard (AES), also called a FIPS 197.In the 1990s, the US Government required to standardize a cryptographic algorithm which was to be used globally by them. It is known as the Advance Encryption Standard (AES).Several proposols were submitted and after multiple debate, an algorithm known as Rijndael was accepted. Rijandael was invented by Joan Daemen and Vincent Rijmen. The need for appearing up with a new algorithm was actually because of recognized weakness in DES.The 56-bit keys ... Read More