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https://github.com/jezachen/mathematical-base-for-information-safety

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https://github.com/jezachen/mathematical-base-for-information-safety

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# Mathematical Base for Information Safety

The programming homework of Mathematical Base for Information Safety



## the Sieve of Eratosthenes 2017/9/7



In mathematics, the sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to any given limit.

It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.

The earliest known reference to the sieve (Ancient Greek: κόσκινον Ἐρατοσθένους, kóskinon Eratosthénous) is in Nicomachus of Gerasa's Introduction to Arithmetic, which describes it and attributes it to Eratosthenes of Cyrene, a Greek mathematician.

One of a number of prime number sieves, it is one of the most efficient ways to find all of the smaller primes. It may be used to find primes in arithmetic progressions.

![Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from prime's square).](https://upload.wikimedia.org/wikipedia/commons/b/b9/Sieve_of_Eratosthenes_animation.gif)



## Euclidean algorithm 2017/9/12

In mathematics, the Euclidean algorithm[a], or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's Elements (c. 300 BC). It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.



The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e.g., 21 = 5 × 105 + (−2) × 252. The fact that the GCD can always be expressed in this way is known as Bézout's identity.



The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Additional methods for improving the algorithm's efficiency were developed in the 20th century.



The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.



![Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus' example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).](https://upload.wikimedia.org/wikipedia/commons/3/37/Euclid%27s_algorithm_Book_VII_Proposition_2_3.png)



## Bézout's identity 2017/9/14

Bézout's identity (also called Bézout's lemma) is a theorem in elementary number theory: let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x and y such that

![PICTURE](https://wikimedia.org/api/rest_v1/media/math/render/svg/a28ae26d7a10b7cfe504c7984b59be51c130e1f5)



In addition,

the greatest common divisor d is the smallest positive integer that can be written as ax + by

every integer of the form ax + by is a multiple of the greatest common divisor d.

The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm. If both a and b are nonzero, the extended Euclidean algorithm produces one of the two pairs such that ![PICTURE](https://wikimedia.org/api/rest_v1/media/math/render/svg/1443ec26e88c55ba83585e6a71f34c3e9a3f25aa) and ![PICTURE](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e39d5470b0ba4e7c0b2c690586aba4c33a66cb)(equality may occur only if one of a and b is a multiple of the other).



Many other theorems in elementary number theory, such as Euclid's lemma or Chinese remainder theorem, result from Bézout's identity.

A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in all these domains.