사용자:Quotation Marks/documents/합동식

정수론에서 나머지 연산(modular arithmetic) 혹은 법산(法算)은 법에 대해 고정된 정수 집합에 주어진 연산으로 정수를 나눈 나머지와 관계된다. 1801년 카를 프리드리히 가우스가 그의 저서 〈산술연구〉에서 처음으로 사용하였다.

시계는 나머지 연산의 좋은 예이다.

일상적으로 볼 수 있는 나머지 연산의 예로는 하루를 12 시간으로 나누는 12시간 시계를 들 수 있다. 지금이 7시라면 8시간 후에는 3시가 될 것이다. 보통의 덧셈으로 계산하면 7 더하기 8이 되어 15가 되지만, 시계는 항상 12개의 수로 고정되어 있으므로 바른 답이 아니다. 마찬가지로 정오에서 시작해 21시간이 지났다면, 33시가 아닌 그 다음 날 9시가 될 것이다. 시간은 12시에서 출발해서 다시 12시에 이르면 새로 시작하며, 이것을 법 12라고 일컫는다.

합동

나머지 연산은 정수환에서 합동 관계를 다룬다. 정수 a와 b를 자연수 n으로 나눈 나머지가 같으면 a와 b는 법 n에 대해 합동이라 쓰고 다음과 같이 쓴다.

$a\equiv b{\pmod {n}},\,$

한 편 a와 b가 법 n에 대해 합동이면 정의에 의해 a-b는 n에 의해 나누어 떨어진다. 예를 들어, 38과 14는 차가 12로서 12에 의해 나누어 떨어지므로 법 12에 대해 합동이고 다음과 같이 쓴다.

$38\equiv 14{\pmod {12}}\,$

38과 2 역시 법 12에 대해 나머지가 2로 같으므로 합동이고 다음과 같이 쓴다.

$38\equiv 2{\pmod {12}}\,$

비슷하게, 두 수의 비를 분수로 나타내면 소수 부분이 모두 구문 분석 실패 (구문 오류): {\displaystyle 0.1666... (38/12&nbsp;=&nbsp;3.166..., 2/12&nbsp;=&nbsp;0.1666...). From the prior definition we also see that their difference, ''a''&nbsp;&minus;&nbsp;''b'' = 36, is a whole number ([[integer]]) multiple of 12 (''n''&nbsp;=&nbsp;12, 36/12 =&nbsp;3). The same rule holds for negative values of&nbsp;''a'': :<math> -3 \equiv 2 \pmod 5.\,}

A remark on the notation: Because it is common to consider several congruence relations for different moduli at the same time, the modulus is incorporated in the notation. In spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been clearer if the notation a ≡_n b had been used, instead of the common traditional notation.

The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.

If

$a_{1}\equiv b_{1}{\pmod {n}}$

and

$a_{2}\equiv b_{2}{\pmod {n}},$

then:

• $(a_{1}+a_{2})\equiv (b_{1}+b_{2}){\pmod {n}}\,$
• $(a_{1}-a_{2})\equiv (b_{1}-b_{2}){\pmod {n}}\,$
• $(a_{1}a_{2})\equiv (b_{1}b_{2}){\pmod {n}}.\,$

The ring of congruence classes

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by ${\overline {a}}_{n}$, is the set $\left\{\ldots ,a-2n,a-n,a,a+n,a+2n,\ldots \right\}$. This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class of a modulo n. Another notation for this congruence class, which requires that in the context the modulus is known, is $\displaystyle [a]$.

The set of congruence classes modulo n is denoted as $\mathbb {Z} /n\mathbb {Z}$ (or, alternatively, $\mathbb {Z} /n$ or $\mathbb {Z} _{n}$) and defined by:

$\mathbb {Z} /n\mathbb {Z} =\left\{{\overline {a}}_{n}|a\in \mathbb {Z} \right\}.$

When n ≠ 0, $\mathbb {Z} /n\mathbb {Z}$ has n elements, and can be written as:

$\mathbb {Z} /n\mathbb {Z} =\left\{{\overline {0}}_{n},{\overline {1}}_{n},{\overline {2}}_{n},\ldots ,{\overline {n-1}}_{n}\right\}.$

When n = 0, $\mathbb {Z} /n\mathbb {Z}$ does not have zero elements; rather, it is isomorphic to $\mathbb {Z}$, since ${\overline {a}}_{0}=\left\{a\right\}$.

We can define addition, subtraction, and multiplication on $\mathbb {Z} /n\mathbb {Z}$ by the following rules:

• ${\overline {a}}_{n}+{\overline {b}}_{n}={\overline {(a+b)}}_{n}$
• ${\overline {a}}_{n}-{\overline {b}}_{n}={\overline {(a-b)}}_{n}$
• ${\overline {a}}_{n}{\overline {b}}_{n}={\overline {(ab)}}_{n}.$

The verification that this is a proper definition uses the properties given before.

In this way, $\mathbb {Z} /n\mathbb {Z}$ becomes a commutative ring. For example, in the ring $\mathbb {Z} /24\mathbb {Z}$, we have

${\overline {12}}_{24}+{\overline {21}}_{24}={\overline {9}}_{24}$

as in the arithmetic for the 24-hour clock.

The notation $\mathbb {Z} /n\mathbb {Z}$ is used, because it is the factor ring of $\mathbb {Z}$ by the ideal $n\mathbb {Z}$ containing all integers divisible by n, where $0\mathbb {Z}$ is the singleton set $\left\{0\right\}$. Thus $\mathbb {Z} /n\mathbb {Z}$ is a field when $n\mathbb {Z}$ is a maximal ideal, that is, when $n$ is prime.

In terms of groups, the residue class ${\overline {a}}_{n}$ is the coset of a in the quotient group $\mathbb {Z} /n\mathbb {Z}$, a cyclic group.

The set $\mathbb {Z} /n\mathbb {Z}$ has a number of important mathematical properties that are foundational to various branches of mathematics.

Rather than excluding the special case n = 0, it is more useful to include $\mathbb {Z} /0\mathbb {Z}$ (which, as mentioned before, is isomorphic to the ring $\mathbb {Z}$ of integers), for example when discussing the characteristic of a ring.

Remainders

The notion of modular arithmetic is related to that of the remainder in division. The operation of finding the remainder is sometimes referred to as the modulo operation and we may see 2 = 14 (mod 12). The difference is in the use of congruency, indicated by "≡", and equality indicated by "=". Equality implies specifically the "common residue", the least non-negative member of an equivalence class. When working with modular arithmetic, each equivalence class is usually represented by its common residue, for example 38 ≡ 2 (mod 12) which can be found using long division. It follows that, while it is correct to say 38 ≡ 14 (mod 12), and 2 ≡ 14 (mod 12), it is incorrect to say 38 = 14 (mod 12) (with "=" rather than "≡").

The difference is clearest when dividing a negative number, since in that case remainders are negative. Hence to express the remainder we would have to write −5 ≡ −17 (mod 12), rather than 7 = −17 (mod 12), since equivalence can only be said of common residues with the same sign.

In computer science, it is the remainder operator that is usually indicated by either "%" (e.g. in C, Java, Javascript, and Python) or "mod" (e.g. in SQL, Visual Basic), with exceptions (e.g. Excel). These operators are commonly pronounced as "mod", but it is specifically a remainder that is computed (since in C99 negative number will be returned if the first argument is negative, and in Python a negative number will be returned if the second argument is negative). The function modulo instead of mod, like 38 ≡ 14 (modulo 12) is sometimes used to indicate the common residue rather than a remainder (e.g. in Ruby).

Parentheses are sometimes dropped from the expression, e.g. 38 ≡ 14 mod 12 or 2 = 14 mod 12, or placed around the divisor e.g. 38 ≡ 14 mod (12). Notation such as 38(mod 12) has also been observed, but is ambiguous without contextual clarification.

Functional representation of the remainder

Computation with modular arithmetic can be implemented using other functions.

One such functional representation uses the floor function. If a ≡ b (mod n), a ≥ 0, and 0 ≤ b < n, then there exists an integer k ≥ 0 such that a = kn + b. The remainder b can be written

$b=a-\left\lfloor {\frac {a}{n}}\right\rfloor \times n,$

where $\left\lfloor {\frac {a}{n}}\right\rfloor \,$ is the largest integer less than or equal to ${\frac {a}{n}}$. If instead −n ≤ b < 0, then

$b=a-\left\lfloor {\frac {a}{n}}\right\rfloor \times n-n.$

Another functional representation uses sine and arcsine, taking advantage of the fact that arcsine is multivalued. Let

$g(x)={\frac {2n}{\pi }}\arcsin \left(\sin \left({\frac {\pi x}{2n}}\right)\right).$

Then

g(x) = x (mod n)

for 0 ≤ πx/2n < π/2 or π ≤ πx/2n < 3π/2, and

g(x) = −x (mod n)

for π/2 ≤ πx/2n < π or 3π/2 ≤ πx/2n < 2π.

Applications

Modular arithmetic is referenced in number theory, group theory, ring theory, knot theory, abstract algebra, cryptography, computer science, chemistry and the visual and musical arts.

It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra.

In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie-Hellman, as well as providing finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including AES, IDEA, and RC4.

In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context.

In chemistry, the last digit of the CAS registry number (a number which is unique for each chemical compound) is a check digit, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the next digit times 2, the next digit times 3 etc., adding all these up and computing the sum modulo 10.

In music, arithmetic modulo 12 is used in the consideration of the system of twelve-tone equal temperament, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-sharp is considered the same as D-flat).

The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).

More generally, modular arithmetic also has application in disciplines such as law (see e.g., apportionment), economics, (see e.g., game theory) and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis.

Computational complexity

Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see Linear congruence theorem. Algorithms, such as Montgomery reduction, also exist to allow simple arithmetic operations, such as multiplication and exponentiation modulo n, to be performed efficiently on large numbers.

Solving a system of non-linear modular arithmetic equations is NP-complete. For details, see for example M. R. Garey, D. S. Johnson: Computers and Intractability, a Guide to the Theory of NP-Completeness, W. H. Freeman 1979.

See also

• Quadratic residue
• Legendre symbol
• Quadratic reciprocity
• Primitive root
• Finite field
• Topics relating to the group theory behind modular arithmetic:
  • Cyclic group
  • Multiplicative group of integers modulo n
• Other important theorems relating to modular arithmetic:
  • Carmichael's theorem
  • Euler's theorem
  • Fermat's little theorem – a special case of Euler's theorem.
  • Chinese remainder theorem
  • Lagrange's theorem
• Modulo
• Modulo operation
• Division
• Modular exponentiation
• Remainder
• Pisano period – Fibonacci sequences modulo n
• Boolean ring
• Circular buffer circular math memory addressing

References

• 틀:Apostol IANT. See in particular chapters 5 and 6 for a review of basic modular arithmetic.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.3: Modular arithmetic, pp.862–868.
• Anthony Gioia, Number Theory, an Introduction Reprint (2001) Dover. ISBN 0-486-41449-3

External links

• In this modular art article, one can learn more about applications of modular arithmetic in art.
• Weisstein, Eric Wolfgang. “Modular Arithmetic”. 《Wolfram MathWorld》 (영어). Wolfram Research. 
• An article on modular arithmetic on the GIMPS wiki
• Modular Arithmetic and patterns in addition and multiplication tables
• Automated modular arithmetic theorem provers:
  • Spear
  • AAProver - Simple C++ framework easy to use in other applications, supporting (among others) all integer operators present in languages such as C/C++/Java and arbitrary bit-width.