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• W101098056 abstract "1. INTRODUCTION Cryptology, the science of secret writing, is a great way to introduce students to different areas of mathematics such as number theory, linear algebra, probability and statistics. Cryptology consists of two branches: cryptography and cryptanalysis. Cryptography is the science of designing techniques for encrypting and decrypting a message. Cryptanalysis is the science of recovering plaintext (the original unencrypted form of a message) from ciphertext (the secret form of the message after encryption) without knowledge of the process used for encryption and decryption. Cryptography is separated into two main categories: private-key and public-key. For a discussion of these different categories see [1], [2], or [5]. Until 1978, when the paper Method for Obtaining Digital Signatures and Public Key Cryptosystems by Rivest, Shamir and Adelman was published in the journal Communications of the Association for Computing Machinery, all cryptography fell into the category of private-key. Since that time, private-key cryptography has largely given way to its public key counterpart [5]. However, some cryptosystems that employ both techniques have been developed. These cryptosy stems are typically called hybrids. In many hybrid cryptosystems the message is enciphered using a private-key algorithm, but the key for the private-key cryptosystem is transmitted using some type of public-key cryptography [2]. Hence it is still worthwhile to investigate private-key cryptosystems. One of the most famous, private-key cryptosystems is the Vigenere Cipher. Encryption and decryption using the Vigenere Cipher was originally described in terms of a table known as the Vigenere Square and a secret keyword. Like many other cryptosystems, it is quite easy to describe this cryptosystem mathematically using basic properties of modular arithmetic. A variety of techniques can be used to perform a cryptanalysis on a message that has been encrypted using the Vigenere Cipher. Most techniques require that a frequency analysis be done on either the entire message or on various parts of the message. After the frequency analysis is complete, there are ways to estimate the keyword length and use that information to determine the keyword. Once the keyword is determined a cryptanalyst can easily recover a message that was intended to be secret. More realistic applications of Vigenere Cipher encryption and decryption involve long messages, hence it is not practical to perform these processes by hand. Although it is common to use computers to do the work, the algorithms necessary to perform the computations required to encipher, decipher and cryptanalyze messages can easily be programmed into a TI-83 graphing calculator. The general concepts discussed above are available elsewhere, but it is our purpose here to show how they can be made available to students with lower level mathematics skills. We will also show how the TI-83 graphing calculator can perform the computations needed to implement the Vigenere Cipher, and a message encoded via the Vigenere Cipher will be cryptanalyzed using the TI-83. 3. THE VIGENERE CIPHER The Vigenere Cipher is named after the Frenchman Blaise de Vigenere and uses a table known as the Vigenere Square. An English form of the table is shown below. In the Vigenere Square, plaintext letters label the columns and keyword letters label the rows. The main entries of the table correspond to ciphertext letters. To use the Vigenere Cipher two correspondents must first agree upon a secret keyword or key phrase. We will briefly illustrate how one can encrypt and decrypt messages using the Vigenere Square. For a detailed explanation as well as some interesting historical background about the Vigenere Cipher, see [1], [2] or [5]. Suppose Michael wants to send Bill the message MATH IS COOL using the Vigenere Cipher with keyword DOG. Michael first writes down the plaintext and then writes the keyword repeatedly above the plaintext, character for character. …" @default.
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• W101098056 title "The Vigenere Cipher with the Ti-83" @default.
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