« 都営バス C-231 ( 自動車 ) | トップページ | 夢をの宅急便 - ありがとう »

2018年9月16日 (日)

factorization[factorisation ]因数分解

In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x ? 2)(x + 2) is a factorization of the polynomial x^2 ? 4.Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any x? can be trivially written as ( x y ) × ( 1 /? y )?? }? whenever y? is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator.Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product
of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography.Polynomial factorization has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up to ordering) factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root
-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials).A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals.?Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function写像を特定の性質を持つ写像の合成の形に分解することが挙げられる. たとえば、任意の写像は全射と単射の合成\xA4
噺ǂ襪海箸❹任④\xEB. Matrices possess many kinds of matrix factorizations[decomposition ]. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, and a permutation matrix P; this is a matrix formulation行列形成[構築?]??行列形成[構築?]??{数値解析において,異なる分解が?効率的な行列アルゴリズムを実装するために用いられる?. <例ーー/wiki/行列の分解 > }?of Gaussian elimination「ガウスの消去法(掃き出し法--こだわりをなくしたけいさんほうほう ?!?, を解くためのであり、通常は問題となる連立一次方程式の係数からなるに対して行われる一連の変形操作を意味する. ) 」. <figure--(線形方程式系の解の計算例 )線型方程式系/wiki/行列の基本変形 ?> by Wikipedia (公開鍵暗号=OS {一般的に、公開鍵暗号は共通鍵暗号よりも暗号化、復号に時間がかかる. そのため、実際の運用では、デー
タの暗号化には「その場限りの共通鍵 」を使用し、その共通鍵の配送のみを公開鍵暗号で行う方式がとられることが多い. また、公開鍵暗号はデジタル署名のために利用されることも多い。署名のような認証機能を提供できることが公開鍵暗号の最大の特長であるとも言える. 公開鍵暗号を初めて実現したのがRSA暗号である。デジタル署名も可能な方式であったことから、RSA暗号は公開鍵暗号として、実際に広く利用されることとなった. <実際の使われ方ーー/wiki/公開鍵暗号 > } ????   <厳密な定義(秘匿性 - 公開鍵暗号方式の厳密な定義)ーー/wiki/公開鍵暗号 > ) ? <figure--public-key cryptography 2-nd > In an asymmetric key encryption scheme, anyone can encrypt messages using the public key, but only the holder of the paired private key can decrypt. Security depends on the secrecy of the private key. (かんれん, ) https://qiita.com/y_miyoshi/items/da814d96e8890224aad3 C#で素因数分解するプログラム p206-? 試行除算

« 都営バス C-231 ( 自動車 ) | トップページ | 夢をの宅急便 - ありがとう »