Number Theory & Mathematical Cryptography

Diophantus of Alexandria - (Greek: Διόφαντος ὁ Ἀλεξανδρεύς , circa 200/214 – circa 284/298)  was a Greek mathematician of the Hellenistic era. Little is known of his life except that he lived in Alexandria, Egypt ...

He was known for his study of equations with variables which take on rational values and these Diophantine equations are named after him. Diophantus is sometimes known as the father of Algebra. He wrote a total of thirteen books on these equations. Diophantus also wrote a treatise on polygonal numbers.

In 1637, while reviewing his translated copy of Diophantus' Arithmetica (pub. ca.250) Pierre de Fermat wrote his famous Last Theorem in the page's margins. His copy with his margin-notes survives to this day.

Although little is known about his life, some biographical information can be computed from his epitaph. He lived in Alexandria and he died when he was 84 years old. Diophantus was probably a Hellenized Babylonian.

A 5^th and 6^th century math puzzle involving Diophantus' age: He was a boy for one-sixth of his life. After one-twelfth more, he acquired a beard. After another one-seventh, he married. In the fifth year after his marriage his son was born. The son lived half as many as his father. Diophantus died 4 years after his son. How old was Diophantus when he died?

What is the answer, with reasoning?   (It is in the source-file.)

Cryptography (or cryptology; derived from Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write") is the study of message secrecy. In modern times, it has become a branch of information theory, as the mathematical study of information...

... Steganography (i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus, concealed a message - a tattoo on a slave's shaved head - by regrown hair. More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information .

Ciphertexts produced by classical ciphers always reveal statistical information about the plaintext, which can often be used to break them. After the Arab discovery of frequency analysis (around the year 1000), nearly all such ciphers became more or less readily breakable by an informed attacker. ... Essentially all ciphers remained vulnerable to cryptanalysis using this technique until the invention of the polyalphabetic cipher by Leon Battista Alberti around the year 1467. Alberti's innovation was to use different ciphers (ie, substitution alphabets) for various parts of a message (often each successive plaintext letter). He also invented what was probably the first automatic cipher device, a wheel which implemented a partial realization of his invention. In the polyalphabetic Vigenère cipher, encryption uses a key word, which controls letter substitution depending on which letter of the key word is used.

In December 1932, the Polish Cipher Bureau first broke Germany's Enigma ciphers. Five weeks before the outbreak of World War II, on 25 July 1939, in Warsaw, the Polish Cipher Bureau gave Enigma-decryption techniques and equipment to French and British military intelligence. …[A]llied codebreakers were able to decrypt a vast number of messages that had been enciphered using the Enigma. The intelligence gleaned from this source was codenamed “Ultra” by the British.

[After World War II] Winston Churchill told Britain's King George VI: It was thanks to Ultra that we won the war.

Though the Enigma cipher had cryptographic weaknesses, in practice it was only in combination with other factors (procedural flaws, operator mistakes, occasional captured hardware and key tables, etc.) that those weaknesses allowed Allied cryptographers to be so successful.