From Math2033

Cryptology is a Greek word whose root means "hidden". It is best described as the science of secret communications.

Contents 1 Overview 2 Classic cryptography 2.1 Ciphers 2.2 In-class presentation 3 Modern cryptography 4 Try it yourself! 5 References

Overview

Cryptology includes the the study of Cryptography, producing or enciphering codes, and Cryptanalysis the study and practice of deciphering. Typically, each is carried out in the presence of third party adversaries. These third party adversaries can be described as enemies, opponents, or anyone planning to intervene, prevent or steal such communication efforts.

Before modern times, cryptography was considered the same as encryption; simply encoding an illegible message and sharing techniques for another party to decode said message. Since World War I, the methods of secret communication have become increasingly complex.

As you may have guessed, it takes a lot of brains to be a cryptanalyst. A person with the gift for "breaking" or "cracking" codes is in high demand for many organizations; particularly in military operations, homeland security, cyber-defense and the like. The interesting thing about simply creating a key and encrypting a message, however, is that just about anyone can do it! As long as you can create a method by which to change your original message into a ciphertext and relay that key to the message recipient, you have created your own system for secret communication!

Classic cryptography

In classic cryptography, ciphers are the algorithms for performing encryption and decryption. Some of the oldest known ciphers include substitution ciphers and transposition ciphers. Substitution ciphers use messages in plaintext that are replaced with units of ciphertext in a particular order. Transposition ciphers also use messages of plaintext, however, the units (characters, letters, etc.) are shifted according a regular system. Order is not consistent with transposition ciphers.

Ciphers

Ciphers are sometimes referred to as codes today. However, in classic cryptography, codes and ciphers were not similar. Codes are typically in sync with a codebook that link numbers and characters to words and phrases. Words and phrases converted to code are more specifically related to meaning.

Ciphers operate on a lower level, one involving individual letters or small groups of letters. When using a cipher, original information is known as plaintext. The encrypted form of the information one would not understand is called ciphertext. Each message contains the same information, however, not in means legible by a human or computer without the proper techniques of decryption.

A great example of a substitution cipher is the Caesar shift cipher.

It is named after the Roman general Julius Caesar who used it as a tool to protect messages of military significance. A study conducted by a student of Stony Brook University found that Caesar used the cipher with an alphabetical shift of three when encoding messages. ^[1]

Check out this example of how the cipher works:

 Plain:    ABCDEFGHIJKLMNOPQRSTUVWXYZ
 Cipher:   DEFGHIJKLMNOPQRSTUVWXYZABC

Here's an example of encryption:

 Ciphertext: WKH TXLFN EURZQ IRA MXPSV RYHU WKH ODCB GRJ
 Plaintext:  the quick brown fox jumps over the lazy dog

Similar to the Caesar cipher is the Vigenére cipher. A Vigénere cipher is a tool for encrypting alphabetic text using a series of different Caesar ciphers based on the letters of a keyword. The Vigenère cipher is named for Blaise de Vigenère, although Giovan Battista Bellaso had invented the cipher earlier. Vigenère just made stronger innovations. For more information, see this cool page on Vigenère ciphers.

Here's an example from another student's project page of how a Vignénere cipher works:

The phrase, "To be or not to be that is the question," with the key word, "drama," would look like this:

T

O

B

E

O

R

N

O

T

T

O

B

E

T

H

A

T

I

S

T

H

E

Q

U

E

S

T

I

O

N

D

R

A

M

A

D

R

A

M

A

D

R

A

M

A

D

R

A

M

A

D

R

A

M

A

D

R

A

M

A

The ciphered text would then read:
w f b q o u e o f t r s e f h d k i e t k v q g e v k i a n ^[2]

A first step in discovering the possible keys to a Vigenère cipher is to examine the repetitions in the text encryption. Short words can be encrypted differently multiple times. However, the code can still be discovered if the message is long enough. Research says that the letter “e” is the most often used letter in the English alphabet. One could guess that this is true with all ciphers, however letters like "e" can be used in multiple keys for the same message. With this type of cipher, multiple encryptions could exist.

One mathematician explains an effective way to understand the process: “We know that the key is of length N, then we know that every Nth character is encrypted the same way. We can split the whole encrypted message into N messages by lumping the characters that are encrypted with the same shift. Then we essentially have N different messages that are normal Caesar shifts. Each is shifted by one value - a single character of the key. We can apply frequency analysis to each individual message to guess at those single characters. We combine our best guesses at each character to get the best keyword guess.” ^[3]

Assumptions aren’t always stable. In many cases with Vigenère ciphers, even best guesses do not match the key. Letters that form actual words are typically considered the strongest guesses.

In-class presentation

For an in-class presentation, I brought each classmate their very own miniature models of both the Caesar shift cipher and the Vigenère cipher. In my presentation, I included the instructions of how to use each cipher.

I also provided a worksheet with two exercises involving encryption and decryption with each handout cipher. You can see the exercises below in the Try it yourself! section.

Modern cryptography

Modern cryptography is heavily based on mathematical theory and computer science. Information security, such as data integrity and data authentication, is a large concept of such secret communication.

Today, cryptography involves math, computer science and engineering. ATM cards, credit cards, online usernames and passwords including bank accounts and social media, and even electronic commerce are all synonymous with modern cryptography. As you can see, cryptography has more of a presence in our lives than we may notice.

Modern cryptographic algorithms related to security are uniquely designed to make them extremely difficult to break by any adversary. Theoretically, it is possible to break such a system, but not without an appropriate cryptanalysis system and much understanding.

In today's age, the ciphers mentioned above are not cryptographically secure. However, one could argue that they were secure before the extensive availability of computers. There are several ciphers that are believed to be adequately secure but not without manipulation techniques that require computerized software.

A popular algorithmic cipher used today is the Data Encryption Standard (DES), which was developed in the early 1970s for the encryption of electronic data. The United States government has since adopted the DES, although it is now being proven to be somewhat insecure. Many weaknesses have been uncovered, particularly the theory that the key size is now too small.

Try it yourself!

Below is a worksheet similar to the one provided during class. Have some quick and simple fun learning how to encrypt and decode these very simple quotes from two very famous mathematicians. Feel free to use this template to create your own methods of encryption and decryption. Check it out!

Caesar shift cipher:

Using an alphabetical shift of 8, encode this plaintext quote from a famous mathematician:

“The essence of mathematics is its freedom.” – Geoge Cantor

Your answer in ciphertext:

“__ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __.” – Geoge Cantor

Vigénere cipher:

Keyword – MATH

MA THMAT HM ATHMATHMATHM, ATH MA THMA TH MATHMAT.” – Alfred Whitehead

Ciphertext quote:

IE MOUND UN ZLZEKHXIMPQS, UBF WX SUVX PZ PEMHULL.” – Alfred Whitehead

Your decoded answer: “__ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __, __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __.” – Alfred Whitehead

References

1. http://www.math.sunysb.edu/~scott/papers/MSTP/crypto/3Caesar_Cipher.html

2. http://math2033.uark.edu/wiki/index.php/Vigen%C3%A8re_cipher

1. ↑ Julius Caesar is reported to have used such a cipher with a shift of 3 for his military communications.
2. ↑ Unlike Caesar Cipher that uses a single left shift for all letters, a Vigenère cipher uses multiple shifts based on a key word.
3. ↑ http://www.mygeocachingprofile.com/codebreaker.vigenerecipher.aspx