Letters being replaced by numbers is what encryption systems are

based on. An example of an encryption system is a one-time pad. This is when a

random key and plaintext are combined together. Using a one-time pad means that

from the message each character is missed with one character from a key stream.

If a key is random, this means that the result will also be random. One time pads are based

upon codes and keys which can only be used once, this means that they provide a

sense of security since they are essentially crack-proof. However, one-time

pads are unrealistic in some ways for general use since there is a large number

of codes and keys that are needed.

Cryptography was introduced thousands of years ago, and is a

secret way or writing a code so that only the intended user and read the

message. Classic cryptography was the method of encryption that used pen and

paper. With advances in technology, new methods are being used to encrypt

messages, for example the Enigma rotor machine provides an efficient way of

encryption, and this was introduced in the early 20th century during

the world war. Classic cryptography is becoming more and more unsuitable due to

the complex schemes generated through computing.

In World War II, for example, the

Allied Forces gained important strategic and tactical advantages from being

able to intercept and read the secret messages of Nazi Germany that had been

encoded with a cipher machine called Enigma (a machine invented by a German and

used by Britain’s codebreakers as a way of deciphering German signal traffic

during WWII). In addition, the United States gained a decided advantage over

Japanese forces through the development of operation MAGIC (Allied cryptanalysis project during World War II).

The RSA algorithm first came about in

1977. Modern computers encrypt and decrypt messages by using the RSA algorithm.

The difficulty of the composite numbers is essential for the security of the

system. A composite number is a integer which is not prime. Towards the end of

the twentieth century the RSA algorithm became the most commonly used

encryption and authentication algorithm. It was used in the development

of spreadsheets, emails, and word processing programs.

Towards the end of the twentieth century there

was a growth in computing technologies and a decline in pen and paper record

keeping, inventions such as hard drives and solid state drives have meant that

data is stored on large servers and computers, this has led to a major increase

in the importance of cryptography. More and more data began to have permanent

storage only in computer memory. Although the technological revolution and rise

of the internet presented unique security challenges, there were also challenges

to keeping information safe, which was stored electronically. As electronic communication and data storage

increased, cryptographic systems are becoming relied on and demanded a lot more.

Uses of cryptography have increased to individuals and companies from military

users, are more people want to keep their privacy.

Number theory has been applied to scientific

research organisation that are looking for advancements in cryptography. This

research will allow more privacy and much more security of data for all users.

The information has been kept safe, along with details on the sender of the

message. This has been through applications of number theory. A key is an algorithm

where only verified users can view the full information, users which aren’t verified

only receive limited information.

There are numerous ways of accomplishing

encryption, for example, you could choose certain prime numbers with the

product of those prime numbers. This means data is divided into specific

sections and limited lengths, so the information gathered is incomplete.

Decryption involves a trace back process that requires mathematics operations

which are unique. Decryption can be achieved through the inverse mathematical operations

used whilst encrypting.

Only the intended reader can fully read the messages, and the

unintended users will be provided with incomplete information, this originates

from the applications of number theory. Physical security is also provided,

which allows users which are authorised to deleted or update the data which is

sent. The time it takes to factor large numbers has been reduced through

computing power.

Generally, the larger the key size used in PGP-based RSA

public-key cryptology systems, the longer the time it will take computers to

factor the composite numbers which were used in the keys. A PGP-based RSA

public key cryptology system is a system which achieves two functions:

authentication, this is when the public key verifies a holder of the paired

private key which was sent to the message, and encryption, where only the

private key holder can decrypt the message which was encrypted by the public

key. The RSA algorithm is reliable because composite numbers which consist of

prime number, and the infinite number of primes there are make the RSA

algorithm very hard to crack.

Generally larger keys provide better security. Applications

of number theory and elliptical curves to cryptological algorithms make the use

of smaller keys which are easier to use, without the loss of any security.