Modular arithmetic


In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, 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.
A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in, but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is congruent to 3 modulo 12, written 15 ≡ 3, so that 7 + 8 ≡ 3.
Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This can be written as 2 × 8 ≡ 4. Note that after a wait of exactly 12 hours, the hour hand will always be right where it was before, so 12 acts the same as zero, thus 12 ≡ 0.

Congruence

Given an integer, called a modulus, two integers and are said to be congruent modulo, if is a divisor of their difference; that is, if there is an integer such that
Congruence modulo is a congruence relation, meaning that it is an equivalence relation that is compatible with addition, subtraction, and multiplication. Congruence modulo is denoted by
The parentheses mean that applies to the entire equation, not just to the right-hand side.
This notation is not to be confused with the notation , which refers to the remainder of when divided by, known as the modulo operation: that is, denotes the unique integer such that and. Indeed, the expression does not appear in the statement, since it is parsed as
The congruence relation may be rewritten as
explicitly showing its relationship with Euclidean division. However, the here need not be the remainder in the division of by Rather, asserts that and have the same remainder when divided by. That is,
where is the common remainder. We recover the previous relation by subtracting these two expressions and setting
Because the congruence modulo is defined by the divisibility by and because is a unit in the ring of integers, a number is divisible by exactly if it is divisible by.
This means that every non-zero integer may be taken as a modulus.

Examples

In modulus 12, one can assert that:
because the difference is, a multiple of. Equivalently, and have the same remainder when divided by.
The definition of congruence also applies to negative values. For example:

Basic properties

The congruence relation satisfies all the conditions of an equivalence relation:
If and, or if, then:
  • for any integer
  • for any integer
  • for any integer
  • for any non-negative integer
  • , for any polynomial with integer coefficients
If, then it is generally false that. However, the following is true:
If, then and.
For cancellation of common terms, we have the following rules:
  • If, where is any integer, then.
  • If and is coprime with, then.
  • If and, then.
The last rule can be used to move modular arithmetic into division. If divides, then.
The modular multiplicative inverse is defined by the following rules:
  • Existence: There exists an integer denoted such that if and only if is coprime with. This integer is called a modular multiplicative inverse of modulo.
  • If and exists, then .
  • If and is coprime to, then the solution to this linear congruence is given by.
The multiplicative inverse may be efficiently computed by solving Bézout's equation for,, by using the Extended Euclidean algorithm.
In particular, if is a prime number, then is coprime with for every such that ; thus a multiplicative inverse exists for all that is not congruent to zero modulo.

Advanced properties

Some of the more advanced properties of congruence relations are the following:
  • Fermat's little theorem: If is prime and does not divide, then.
  • Euler's theorem: If and are coprime, then, where is Euler's totient function.
  • A simple consequence of Fermat's little theorem is that if is prime, then is the multiplicative inverse of. More generally, from Euler's theorem, if and are coprime, then. Hence, if, then.
  • Another simple consequence is that if, where is Euler's totient function, then provided is coprime with.
  • Wilson's theorem: is prime if and only if.
  • Chinese remainder theorem: For any, and coprime,, there exists a unique such that and. In fact, where is the inverse of modulo and is the inverse of modulo.
  • Lagrange's theorem: If is prime and is a polynomial with integer coefficients such that is not a divisor of, then the congruence has at most non-congruent solutions.
  • Primitive root modulo : A number is a primitive root modulo if, for every integer coprime to, there is an integer such that. A primitive root modulo exists if and only if is equal to or, where is an odd prime number and is a positive integer. If a primitive root modulo exists, then there are exactly such primitive roots, where is the Euler's totient function.
  • Quadratic residue: An integer is a quadratic residue modulo, if there exists an integer such that. Euler's criterion asserts that, if is an odd prime, and is not a multiple of, then is a quadratic residue modulo if and only if
  • :.

    Congruence classes

The congruence relation is an equivalence relation. The equivalence class modulo of an integer is the set of all integers of the form, where is any integer. It is called the congruence class or residue class of modulo , and may be denoted, or as or when the modulus is known from the context.
Each residue class modulo contains exactly one integer in the range. Thus, these 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, denotes generally the unique integer such that and ; it is called the residue of modulo .
In particular, is equivalent to, and this explains why "" is often used instead of "" in this context.

Residue systems

Each residue class modulo may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class. Any two members of different residue classes modulo are incongruent modulo. Furthermore, every integer belongs to one and only one residue class modulo.
The set of integers is called the least residue system modulo . Any set of integers, no two of which are congruent modulo, is called a complete residue system modulo .
The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one representative of each residue class modulo. For example, the least residue system modulo is. Some other complete residue systems modulo include:
Some sets that are not complete residue systems modulo 4 are:
  • , since is congruent to modulo.
  • , since a complete residue system modulo must have exactly incongruent residue classes.

    Reduced residue systems

Given the Euler's totient function, any set of integers that are relatively prime to and mutually incongruent under modulus is called a reduced residue system modulo . The set from above, for example, is an instance of a reduced residue system modulo 4.

Covering systems

Covering systems represent yet another type of residue system that may contain residues with varying moduli.

Integers modulo ''m''

In the context of this paragraph, the modulus is almost always taken as positive.
The set of all [|congruence classes] modulo is a ring called the ring of integers modulo , and is denoted,, or. The ring is fundamental to various branches of mathematics.

For one has
When, is the zero ring; when, is not an empty set; rather, it is isomorphic to, since.
Addition, subtraction, and multiplication are defined on by the following rules:
The properties given before imply that, with these operations, is a commutative ring. For example, in the ring, one has
as in the arithmetic for the 24-hour clock.
The notation is used because this ring is the quotient ring of by the ideal, the set formed by all multiples of, that is, all numbers with
Under addition, is a cyclic group. All finite cyclic groups are isomorphic with for some.
The ring of integers modulo is a field; that is, every nonzero element has a multiplicative inverse, if and only if is prime. If is a prime power with, there exists a unique finite field with elements, which is not isomorphic to, which fails to be a field because it has zero-divisors.
If, denotes the multiplicative group of the integers modulo that are invertible. It consists of the congruence classes, where is coprime to ; these are precisely the classes possessing a multiplicative inverse. They form an abelian group under multiplication; its order is, where is Euler's totient function.

Applications

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 uses modulo 11 or modulo 10 arithmetic for error detection. Likewise, International Bank Account Numbers use modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the CAS registry number 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, International Data Encryption Algorithm, 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.
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.
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.
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 politics, economics and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis.