A report on Computer science, Mathematics and Cryptography

Charles Babbage, sometimes referred to as the "father of computing".
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
Lorenz cipher machine, used in World War II to encrypt communications of the German High Command
Ada Lovelace published the first algorithm intended for processing on a computer.
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
Alphabet shift ciphers are believed to have been used by Julius Caesar over 2,000 years ago. This is an example with k = 3. In other words, the letters in the alphabet are shifted three in one direction to encrypt and three in the other direction to decrypt.
The quadratic formula expresses concisely the solutions of all quadratic equations
Reconstructed ancient Greek scytale, an early cipher device
Rubik's cube: the study of its possible moves is a concrete application of group theory
First page of a book by Al-Kindi which discusses encryption of messages
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
16th-century book-shaped French cipher machine, with arms of Henri II of France
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
Enciphered letter from Gabriel de Luetz d'Aramon, French Ambassador to the Ottoman Empire, after 1546, with partial decipherment
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Symmetric-key cryptography, where a single key is used for encryption and decryption
A page from al-Khwārizmī's Algebra
One round (out of 8.5) of the IDEA cipher, used in most versions of PGP and OpenPGP compatible software for time-efficient encryption of messages
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Public-key cryptography, where different keys are used for encryption and decryption.
Leonhard Euler created and popularized much of the mathematical notation used today.
Whitfield Diffie and Martin Hellman, authors of the first published paper on public-key cryptography.
Carl Friedrich Gauss, known as the prince of mathematicians
In this example the message is only signed and not encrypted. 1) Alice signs a message with her private key. 2) Bob can verify that Alice sent the message and that the message has not been modified.
The front side of the Fields Medal
Variants of the Enigma machine, used by Germany's military and civil authorities from the late 1920s through World War II, implemented a complex electro-mechanical polyalphabetic cipher. Breaking and reading of the Enigma cipher at Poland's Cipher Bureau, for 7 years before the war, and subsequent decryption at Bletchley Park, was important to Allied victory.
Poznań monument (center) to Polish cryptanalysts whose breaking of Germany's Enigma machine ciphers, beginning in 1932, altered the course of World War II
NSA headquarters in Fort Meade, Maryland
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, electrical engineering, communication science, and physics.

- Cryptography

The fields of cryptography and computer security involve studying the means for secure communication and for preventing security vulnerabilities.

- Computer science

Mathematics is essential in many fields, including natural sciences, engineering, medicine, finance, computer science and social sciences.

- Mathematics

Computer science research also often intersects other disciplines, such as cognitive science, linguistics, mathematics, physics, biology, Earth science, statistics, philosophy, and logic.

- Computer science

Coding theory, including error correcting codes and a part of cryptography

- Mathematics
Charles Babbage, sometimes referred to as the "father of computing".

1 related topic with Alpha

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Flowchart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" or "true" (more accurately, the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).

Algorithm

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Flowchart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" or "true" (more accurately, the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
Ada Lovelace's diagram from "note G", the first published computer algorithm
Logical NAND algorithm implemented electronically in 7400 chip
Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test THEN GOTO step xxx, shown as diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf. Tausworthe 1977:100, 114).
The example-diagram of Euclid's algorithm from T.L. Heath (1908), with more detail added. Euclid does not go beyond a third measuring and gives no numerical examples. Nicomachus gives the example of 49 and 21: "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7." Heath comments that "The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'."(Heath 1908:300).
A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650.
"Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2–4. Depending on the two numbers "Inelegant" may compute the g.c.d. in fewer steps than "Elegant".
Alan Turing's statue at Bletchley Park

In mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation.

Arabic mathematicians such as al-Kindi in the 9th century used cryptographic algorithms for code-breaking, based on frequency analysis.