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“Fibonacci Expansions and ‘F-adic’ integers,” The Fibonacci Quarterly, v. 34, 1996

“On the Sums of Digits of Fibonacci Numbers,” The Fibonacci Quarterly, v. 35, 1996..

A Modification of Shanks’ Baby-Step Giant-Step Algorithm,” Mathematics of Computation, v. 69, 2000

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History of Cryptography

Cryptography is the science of making and breaking secret codes. Cryptography has a long history, as cryptosystems have become more and more difficult to break and methods for breaking them have improved more and more.

Caesar's Cipher
Caesar's cipher is one of the simplest types of cryptosystems. With this cipher, plaintext is encrypted by replacing each letter with a new letter a fixed number of places further in the alphabet, wrapping around if necessary. Julius Caesar used this cipher with three place substitution. Thus, CAESAR would be encrypted as FDHVDU. Unfortunately, Caesar ciphers are very easy to break, since there are only 25 possible such ciphers.

Substitution Ciphers
Caesar's cipher is an example of a substitution cipher, in which each letter is substituted with a different letter. There are a total of 25!, or 15,511,210,043,330,985,984,000,000 substitution ciphers. Thus, an exhaustive search of all of them is impractical. Nevertheless, substitution ciphers are easy to break by using frequency analysis of the letters in the English language, or in whatever language the text is in. Frequency analysis was discovered by the Arabs around 1000 AD.

Vignere Cipher
The Vignere cipher was invented by Giovan Batista Belaso in 1553 and later mistakenly attributed to Blase de Vignere in the 19th century. This cipher uses a secret keyword to encrypt the plaintext. The numerical value of each letter of the keyword is added to the value of each letter of the plaintext to obtain the ciphertext. Frequency analysis for the Vignere cipher is much harder than for the substitution cipher.

One-Time Pad
The one-time pad is the most secure possible cryptosystem. It was invented in 1917. It works similarly to the Vignere cipher, except that a one-time pad at least as long as the text is used in place of the keyword. The main disadvantage of this method is that as the name implies, the one-time pad may only be used once, and it better not fall into the wrong hands!

During World War II, the Germans used a machine called Enigma to encrypt secret messages. The Enigma resembled a typewriter and employed a 4-letter secret code set by the user. The Enigma cryptosystem proved nearly impossible to break, but the allied forces did manage to analyze it and break it and all improved variants thereafter. By the end of the war, they could break the German's codes set by Enigma within a day or two.

Symmetric-Key Cryptosystems
Symmetric-key cryptography, in which the sender and receiver share the same key, were used from World War II until 1976. Symmetric-key cryptosystems include block ciphers such as the Data Encryption Standard (DES) and the Advanced Encryption Standard (AES), as well as stream ciphers such as RC4.

Public-Key Cryptosystems
Public-key cryptosystems are the type of cryptosystems currently in use. The first public-key cryptosystem was the Diffie-Hellman cryptosystem, invented by Whitfield Diffie and Martin Hellman in 1976. In 1978, Ronald Rivest, Adi Shamir, and Len Adleman invented the RSA cryptosystem, which has become the standard cryptosystem used to encrypt private data such as e-mail messages and credit card information. The security of RSA relies on the difficulty of factoring large numbers. There are other types of public-key cryptosystems, such as the El Gamal cryptosystem and various cryptosystems involving elliptic curves.