Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

This article is about the (mod $m$ ) notation. For the binary operation mod( $a,m$ ), see modulo.

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but 15:00 reads as 3:00 on the clock face because clocks "wrap around" every 12 hours and the hour number starts over at zero when it reaches 12. We say that 15 is congruent to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, 8:00 represents a period of 8 hours, and twice this would give 16:00, which reads as 4:00 on the clock face, written as 2 × 8 ≡ 4 (mod 12).

Reflexivity: $a \equiv a (mod m)$

Symmetry: $a \equiv b (mod m)$ if $b \equiv a (mod m)$ .

Transitivity: If $a \equiv b (mod m)$ and $b \equiv c (mod m)$ , then $a \equiv c (mod m)$

The congruence relation satisfies all the conditions of an equivalence relation:

If $a 1 \equiv b 1 (mod m)$ and $a 2 \equiv b 2 (mod m)$ , or if $a \equiv b (mod m)$ , then:^[1]

If $a \equiv b (mod m)$ , then it is generally false that $k a \equiv k b (mod m)$ . However, the following is true:

For cancellation of common terms, we have the following rules:

The last rule can be used to move modular arithmetic into division. If $b$ divides $a$ , then $(a / b) mod m = (a mod b m) / b$ .

The modular multiplicative inverse is defined by the following rules:

The multiplicative inverse $x \equiv a -1 (mod m)$ may be efficiently computed by solving Bézout's equation $a x + m y = 1$ for $x$ , $y$ , by using the Extended Euclidean algorithm.

In particular, if $p$ is a prime number, then $a$ is coprime with $p$ for every $a$ such that $0 < a < p$ ; thus a multiplicative inverse exists for all $a$ that is not congruent to zero modulo $p$ .

: If $p$ is prime and does not divide $a$ , then $a p -1 \equiv 1 (mod p)$ .

Fermat's little theorem

: If $a$ and $m$ are coprime, then $a φ (m) \equiv 1 (mod m)$ , where $φ$ is Euler's totient function.

Euler's theorem

A simple consequence of Fermat's little theorem is that if $p$ is prime, then $a -1 \equiv a p -2 (mod p)$ is the multiplicative inverse of $0 < a < p$ . More generally, from Euler's theorem, if $a$ and $m$ are coprime, then $a -1 \equiv a φ (m)-1 (mod m)$ .

Another simple consequence is that if $a \equiv b (mod φ (m))$ , where $φ$ is Euler's totient function, then $k a \equiv k b (mod m)$ provided $k$ is with $m$ .

coprime

: $p$ is prime if and only if $(p - 1)! \equiv -1 (mod p)$ .

Wilson's theorem

: For any $a$ , $b$ and coprime $m$ , $n$ , there exists a unique $x (mod m n)$ such that $x \equiv a (mod m)$ and $x \equiv b (mod n)$ . In fact, $x \equiv b m n -1 m + a n m -1 n (mod mn)$ where $m n -1$ is the inverse of $m$ modulo $n$ and $n m -1$ is the inverse of $n$ modulo $m$ .

Chinese remainder theorem

: The congruence $f (x) \equiv 0 (mod p)$ , where $p$ is prime, and $f (x) = a 0 x m + ... + a m$ is a polynomial with integer coefficients such that $a 0 \neq 0 (mod p)$ , has at most $m$ roots.

Lagrange's theorem

: A number $g$ is a primitive root modulo $m$ if, for every integer $a$ coprime to $m$ , there is an integer $k$ such that $g k \equiv a (mod m)$ . A primitive root modulo $m$ exists if and only if $m$ is equal to $2, 4, p k$ or $2 p k$ , where $p$ is an odd prime number and $k$ is a positive integer. If a primitive root modulo $m$ exists, then there are exactly $φ (φ (m))$ such primitive roots, where $φ$ is the Euler's totient function.

Primitive root modulo $m$

: An integer $a$ is a quadratic residue modulo $m$ , if there exists an integer $x$ such that $x 2 \equiv a (mod m)$ . Euler's criterion asserts that, if $p$ is an odd prime, and $a$ is not a multiple of $p$ , then $a$ is a quadratic residue modulo $p$ if and only if
$a (p -1)/2 \equiv 1 (mod p)$ .

Quadratic residue

Some of the more advanced properties of congruence relations are the following:

Congruence classes [edit]

The congruence relation is an equivalence relation. The equivalence class modulo $m$ of an integer $a$ is the set of all integers of the form $a + k m$ , where $k$ is any integer. It is called the congruence class or residue class of $a$ modulo $m$ , and may be denoted as $(a mod m)$ , or as $a$ or $[a]$ when the modulus $m$ is known from the context.

Each residue class modulo $m$ contains exactly one integer in the range $0,...,|m|-1$ . Thus, these $|m|$ integers are representatives of their respective residue classes.

It is generally easier to work with integers than sets of integers; that is, the representatives most often considered, rather than their residue classes.

Consequently, $(a mod m)$ denotes generally the unique integer $k$ such that $0 \leq k < m$ and $k \equiv a (mod m)$ ; it is called the residue of $a$ modulo $m$ .

In particular, $(a mod m) = (b mod m)$ is equivalent to $a \equiv b (mod m)$ , and this explains why " $=$ " is often used instead of " $\equiv$ " in this context.

${1, 2, 3, 4}$

${13, 14, 15, 16}$

${-2, -1, 0, 1}$

${-13, 4, 17, 18}$

${-5, 0, 6, 21}$

${27, 32, 37, 42}$

${\overline {a}}_{m}+{\overline {b}}_{m}={\overline {(a+b)}}_{m}$

${\overline {a}}_{m}-{\overline {b}}_{m}={\overline {(a-b)}}_{m}$

${\overline {a}}_{m}{\overline {b}}_{m}={\overline {(ab)}}_{m}.$

Remark: In the context of this paragraph, the modulus $m$ is almost always taken as positive.

The set of all congruence classes modulo $m$ is called the ring of integers modulo $m$ ,^[6] and is denoted ${\textstyle \mathbb {Z} /m\mathbb {Z} }$ , $\mathbb {Z} /m$ , or $\mathbb {Z} _{m}$ .^[7] The notation $\mathbb {Z} _{m}$ is, however, not recommended because it can be confused with the set of $m$ -adic integers. The ring $\mathbb {Z} /m\mathbb {Z}$ is fundamental to various branches of mathematics (see § Applications below).

For $m > 0$ one has

When $m = 1$ , $\mathbb {Z} /m\mathbb {Z}$ is the zero ring; when $m = 0$ , $\mathbb {Z} /m\mathbb {Z}$ is not an empty set; rather, it is isomorphic to $\mathbb {Z}$ , since $a 0 = {a}$ .

Addition, subtraction, and multiplication are defined on $\mathbb {Z} /m\mathbb {Z}$ by the following rules:

The properties given before imply that, with these operations, $\mathbb {Z} /m\mathbb {Z}$ is a commutative ring. For example, in the ring $\mathbb {Z} /24\mathbb {Z}$ , one has

as in the arithmetic for the 24-hour clock.

The notation $\mathbb {Z} /m\mathbb {Z}$ is used because this ring is the quotient ring of $\mathbb {Z}$ by the ideal $m\mathbb {Z}$ , the set formed by all $k m$ with $k\in \mathbb {Z} .$

Considered as a group under addition, $\mathbb {Z} /m\mathbb {Z}$ is a cyclic group, and all cyclic groups are isomorphic with $\mathbb {Z} /m\mathbb {Z}$ for some $m$ .^[8]

The ring of integers modulo $m$ is a field if and only if $m$ is prime (this ensures that every nonzero element has a multiplicative inverse). If $m = p k$ is a prime power with $k > 1$ , there exists a unique (up to isomorphism) finite field $\mathrm {GF} (m)=\mathbb {F} _{m}$ with $m$ elements, which is not isomorphic to $\mathbb {Z} /m\mathbb {Z}$ , which fails to be a field because it has zero-divisors.

If $m > 1$ , $(\mathbb {Z} /m\mathbb {Z} )^{\times }$ denotes the multiplicative group of the integers modulo $m$ that are invertible. It consists of the congruence classes $a m$ , where $a$ is coprime to $m$ ; these are precisely the classes possessing a multiplicative inverse. They form an abelian group under multiplication; its order is $φ (m)$ , where $φ$ is Euler's totient function

Applications[edit]

In pure mathematics, modular arithmetic is one of the foundations of number theory, touching on almost every aspect of its study, and it is also used extensively in group theory, ring theory, knot theory, and abstract algebra. In applied mathematics, it is used in computer algebra, cryptography, computer science, chemistry and the visual and musical arts.

A very practical application is to calculate checksums within serial number identifiers. For example, International Standard Book Number (ISBN) uses modulo 11 (for 10-digit ISBN) or modulo 10 (for 13-digit ISBN) arithmetic for error detection. Likewise, International Bank Account Numbers (IBANs), for example, make use of modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the CAS registry number (a unique identifying number 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 previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10.

In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie–Hellman, and provides finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including Advanced Encryption Standard (AES), International Data Encryption Algorithm (IDEA), and RC4. RSA and Diffie–Hellman use modular exponentiation.

In computer algebra, modular arithmetic is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic. It is used by the most efficient implementations of polynomial greatest common divisor, exact linear algebra and Gröbner basis algorithms over the integers and the rational numbers. As posted on Fidonet in the 1980s and archived at Rosetta Code, modular arithmetic was used to disprove Euler's sum of powers conjecture on a Sinclair QL microcomputer using just one-fourth of the integer precision used by a CDC 6600 supercomputer to disprove it two decades earlier via a brute force search.^[9]

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. The logical operator XOR sums 2 bits, modulo 2.

The use of long division to turn a fraction into a repeating decimal in any base b is equivalent to modular multiplication of b modulo the denominator. For example, for decimal, b = 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).

Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular, Zeller's congruence and the Doomsday algorithm make heavy use of modulo-7 arithmetic.

More generally, modular arithmetic also has application in disciplines such as law (e.g., apportionment), economics (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[edit]

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 $m$ , to be performed efficiently on large numbers.

Some operations, like finding a discrete logarithm or a quadratic congruence appear to be as hard as integer factorization and thus are a starting point for cryptographic algorithms and encryption. These problems might be NP-intermediate.

Solving a system of non-linear modular arithmetic equations is NP-complete.^[10]

John L. Berggren. . Encyclopædia Britannica.

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Maarten Bullynck ""

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ISBN

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"Congruence"

In this article, one can learn more about applications of modular arithmetic in art.

modular art

An on modular arithmetic on the GIMPS wiki