Mathematical Codes

Mathematical Codes

Mathematical codes are used by millions everyday for a variety reasons, but all intending

to keep something private. The coding theory has actual applications in consumer

electronics and with other areas of mathematics. Encryption, which involves enciphering

and encoding, is used to protect data against organized crime, government and

multinational institutions. A use of arithmetic, prime numbers, and prime factorization

is used within coding theory.

The study of enciphering and encoding, and deciphering and decoding is called cryptography

(Gardner 17). Encryption is encoding or enciphering a message so that the contents are

hidden from outsiders (Frösen 10). Strong encryption is not a technical standard, it

means that current known methods within feasible time without the data being outdated

cannot break the encryption. Strong encryption is used to protect data against organized

crime, government and multinational institutions. Strong encryption brings possible

applications into daily life. Electric money, secure communications, passwords, and

others are among many. Applications that require privacy, trust and access control should

all use strong encryption methods when possible. It is suggested that people’s legal,

medical, personal data about themselves should stay confidential to the instances that

have a permit to collect the databases.

Mathematical Codes

Mathematical codes are used by millions everyday for a variety reasons, but all intending

to keep something private. The coding theory has actual applications in consumer

electronics and with other areas of mathematics. Encryption, which involves enciphering

and encoding, is used to protect data against organized crime, government and

multinational institutions. A use of arithmetic, prime numbers, and prime factorization

is used within coding theory.

The study of enciphering and encoding, and deciphering and decoding is called cryptography

(Gardner 17). Encryption is encoding or enciphering a message so that the contents are

hidden from outsiders (Frösen 10). Strong encryption is not a technical standard, it

means that current known methods within feasible time without the data being outdated

cannot break the encryption. Strong encryption is used to protect data against organized

crime, government and multinational institutions. Strong encryption brings possible

applications into daily life. Electric money, secure communications, passwords, and

others are among many. Applications that require privacy, trust and access control should

all use strong encryption methods when possible. It is suggested that people’s legal,

medical, personal data about themselves should stay confidential to the instances that

have a permit to collect the databases.

Encryption is not a new concept. Militaries and diplomatic forces have been using it for

thousand of years, trying to keep information from the enemy. Given, it was more

simplistic back then, but it was still used during War. For example, the Americans have

used Morse code for years.

There is a distinct difference between ciphers and codes. Substituting one word for

another word or sentence is using a code (Gardner 18). Mixing up or substituting existing

letters for one word or sentence is using a cipher (18). The majority of encryptions use

ciphers versus codes. The algorithm is the method used to encipher the original message,

known as the plaintext (20). A key is used with the algorithm to allow the plaintext to

be both enciphered and deciphered (20).

Ciphers are broken into two main categories: substitution ciphers and transportation

ciphers. Substitution ciphers replace letters in the plaintext with other letters or

symbols, keeping the order in which the symbols fall the same (25). By definition,

substitution ciphers could be, in most cases, called codes. Transposition ciphers keep

all of the original letters intact, but mix up the order (25). The resulting text is

referred to as the ciphertext.

Some cryptographic methods rely on the secrecy of algorithms used in the cipher, security

by obscurity (Frösen 2). All modern algorithms use a key to control the encryption and

decryption. The message can only be decrypted if the key matches the on it was encrypted

with. The key used for decryption can be different from the key used in encryption, and

this divides the algorithms in symmetric and asymmetric classes (2).

Symmetric cryptosystems use the same key, the secret key, to encrypt and decrypt a

message. Since it uses the same key for both encryption and decryption, the key should be

changed often and be sufficiently random (2). Symmetric algorithms use different length

keys, which usually means higher security. Symmetric algorithms can be divided into two

categories: stream ciphers, which take and encrypt one bit of the original data at a time,

and block ciphers, which take a number of bits and encrypt them as a single block (2).

The majority of ciphers belong to the block cipher class. Symmetric algorithms are

generally faster and use a much shorter key than asymmetric ones.

DES, Data Encryption Standard, is the notorious symmetric cryptosystem. It has been

certified by NIST, National Institution of Standards and Technology, for use as an

official US Government encryption standard for less-than-top-secret secret material (2).

DES was first certified for government use in 1977 (2). DES is a strong cipher, which

encrypts a block of 64 bits at a time. DES encryption consists of many rounds of

different transformations and permutations, which are linear and easy to reverse (3).

Performing a permutation involves arranging elements in different arrangements, where

order does matter. The critical encryption is done using S-boxes. S-boxes, or

substitution boxes, are sets of highly non-linear functions, implemented in DES as a set

with. The key used for decryption can be different from the key used in encryption, and

this divides the algorithms in symmetric and asymmetric classes (2).

Symmetric cryptosystems use the same key, the secret key, to encrypt and decrypt a

message. Since it uses the same key for both encryption and decryption, the key should be

changed often and be sufficiently random (2). Symmetric algorithms use different length

keys, which usually means higher security. Symmetric algorithms can be divided into two

categories: stream ciphers, which take and encrypt one bit of the original data at a time,

and block ciphers, which take a number of bits and encrypt them as a single block (2).

The majority of ciphers belong to the block cipher class. Symmetric algorithms are

generally faster and use a much shorter key than asymmetric ones.

DES, Data Encryption Standard, is the notorious symmetric cryptosystem. It has been

certified by NIST, National Institution of Standards and Technology, for use as an

official US Government encryption standard for less-than-top-secret secret material (2).

DES was first certified for government use in 1977 (2). DES is a strong cipher, which

encrypts a block of 64 bits at a time. DES encryption consists of many rounds of

different transformations and permutations, which are linear and easy to reverse (3).

Performing a permutation involves arranging elements in different arrangements, where

order does matter. The critical encryption is done using S-boxes. S-boxes, or

substitution boxes, are sets of highly non-linear functions, implemented in DES as a set

of lookup tables (3). After the S-boxes, the results are still permutated (3).

There are two known ways to decode DES. The first way consists of a search of the

keyspace, which consists of 2^56 possible keys (3). If one could test one million keys

every second, it would take about two thousand years to go through the keyspace. With

special hardware, a chip could be designed that does a billion tests per second, reducing

the time to two years (3). The more recent method of decoding DES is differential

cryptanalysis. This method reduces the number of keys that must be tested, but it

requires 2^47 chosen plaintexts encrypted with the key that is trying to be recovered.

Since it is unlikely that anyone would agree to encrypt 2^47 chosen plaintexts with their

secret DES key, this attack is impracticable in practice (3).

When used properly, DES is secure against all but the most powerful organizations. Proper

use means avoiding weak keys. Weak keys are a result of the key being split to sixteen

pieces, one for each round of encryption (4). Using simple DES for top-secret data is not

a good idea with today's technology; however, it is sufficient of everyday use (4).

Asymmetric cryptosystems, also known as public key cryptosystems, use one key, the public

key, to encrypt a message and a different key, the private key, to decrypt it.

An efficient and reliable solution is a public key cryptosystem is RSA. Ron Rivest, Adi

Shamir, and Leonard Adleman invented the RSA algorithm in 1978. RSA is the most widely

There are two known ways to decode DES. The first way consists of a search of the

keyspace, which consists of 2^56 possible keys (3). If one could test one million keys

every second, it would take about two thousand years to go through the keyspace. With

special hardware, a chip could be designed that does a billion tests per second, reducing

the time to two years (3). The more recent method of decoding DES is differential

cryptanalysis. This method reduces the number of keys that must be tested, but it

requires 2^47 chosen plaintexts encrypted with the key that is trying to be recovered.

Since it is unlikely that anyone would agree to encrypt 2^47 chosen plaintexts with their

secret DES key, this attack is impracticable in practice (3).

When used properly, DES is secure against all but the most powerful organizations. Proper

use means avoiding weak keys. Weak keys are a result of the key being split to sixteen

pieces, one for each round of encryption (4). Using simple DES for top-secret data is not

a good idea with today's technology; however, it is sufficient of everyday use (4).

Asymmetric cryptosystems, also known as public key cryptosystems, use one key, the public

key, to encrypt a message and a different key, the private key, to decrypt it.

An efficient and reliable solution is a public key cryptosystem is RSA. Ron Rivest, Adi

Shamir, and Leonard Adleman invented the RSA algorithm in 1978. RSA is the most widely