A report on Computer science, Mathematics and Algorithm

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)

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 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.

Ada Lovelace's diagram from "note G", the first published computer algorithm

The quadratic formula expresses concisely the solutions of all quadratic equations

Logical NAND algorithm implemented electronically in 7400 chip

Rubik's cube: the study of its possible moves is a concrete application of group theory

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 Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

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).

Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.

A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650.

The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.

"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".

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.

Leonhard Euler created and popularized much of the mathematical notation used today.

Carl Friedrich Gauss, known as the prince of mathematicians

Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

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.

- Algorithm

Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory and automation) to practical disciplines (including the design and implementation of hardware and software).

- 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

Algorithms - especially their implementation and computational complexity - play a major role in discrete mathematics.

- Mathematics

2 related topics with Alpha

Lorenz cipher machine, used in World War II to encrypt communications of the German High Command

Cryptography

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Practice and study of techniques for secure communication in the presence of adversarial behavior.

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.

Reconstructed ancient Greek scytale, an early cipher device

First page of a book by Al-Kindi which discusses encryption of messages

16th-century book-shaped French cipher machine, with arms of Henri II of France

Enciphered letter from Gabriel de Luetz d'Aramon, French Ambassador to the Ottoman Empire, after 1546, with partial decipherment

Symmetric-key cryptography, where a single key is used for encryption and decryption

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

Public-key cryptography, where different keys are used for encryption and decryption.

Whitfield Diffie and Martin Hellman, authors of the first published paper on public-key cryptography.

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.

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

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

Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in actual practice by any adversary.

Computability theory

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Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

Nowadays these are often considered as a single hypothesis, the Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function.

Many problems in mathematics have been shown to be undecidable after these initial examples were established.