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.
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, but clocks "wrap around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.
Congruence
Given an integer, called a modulus, two integers 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 the operations of addition, subtraction, and multiplication. Congruence modulo is denoted:
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 modulo operation: denotes the unique integer such that and
The congruence relation may be rewritten as
explicitly showing its relationship with Euclidean division. However, need not be the remainder of the division of by More precisely, what the statement asserts is that and have the same remainder when divided by. That is,
where is the common remainder. Subtracting these two expressions, we recover the previous relation:
by setting
Examples
For example,because, which is a multiple of 12, or, equivalently, because both 38 and 14 have the same remainder 2 when divided by 12.
The definition of congruence also applies to negative values:
Properties
The congruence relation satisfies all the conditions of an equivalence relation:- Reflexivity:
- Symmetry: if for all,, and.
- Transitivity: If and, then
- for any integer
- for any integer
- for any non-negative integer
- , for any polynomial with integer coefficients
- If where is Euler's totient function, then provided is coprime with
- If for any integer, then
- If and is coprime with, then
- 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, the solution to this linear congruence is given by
In particular, if is a prime number, then is coprime with for every such that ; thus a multiplicative inverse exists for all not congruent to zero modulo.
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.
- 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: The congruence, where is prime, and is a polynomial with integer coefficients such that, has at most roots.
- Primitive root modulo n: 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
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. The least residue system modulo 4 is. Some other complete residue systems modulo 4 are:
Some sets which are not complete residue systems modulo 4 are:
- since 6 is congruent to 22 modulo 4.
- since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.
Reduced residue systems
Integers modulo ''n''
The set of all congruence classes of the integers for a modulus is called the ring of integers modulo , and is denoted,, or. The notation is, however, not recommended because it can be confused with the set of -adic integers. The ring is fundamental to various branches of mathematics.The set is defined for n > 0 as:
We define addition, subtraction, and multiplication on by the following rules:
In this way, becomes a commutative ring. For example, in the ring, we have
as in the arithmetic for the 24-hour clock.
We use the notation because this is the quotient ring of by the ideal containing all integers divisible by, where is the singleton set. Thus is a field when is a maximal ideal, that is, when is prime.
This can also be constructed from the group under the addition operation alone. The residue class is the group coset of in the quotient group, a cyclic group.
Rather than excluding the special case, it is more useful to include , for example, when discussing the characteristic of a ring.
The ring of integers modulo is a finite field if and only if is prime.. If is a prime power with k > 1, there exists a unique finite field with elements, but this is not, which fails to be a field because it has zero-divisors.
We denote the multiplicative subgroup of the modular integers by. This consists of for a coprime to n, which are precisely the classes possessing a multiplicative inverse. This forms a commutative group under multiplication, with order.
Applications
In theoretical 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, 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 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.
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.
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 law, economics, 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 , 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.
Example implementations
Below are three reasonably fast C functions, two for performing modular multiplication and one for modular exponentiation on unsigned integers not larger than 63 bits, without overflow of the transient operations.An algorithmic way to compute :
uint64_t mul_mod
On computer architectures where an extended precision format with at least 64 bits of mantissa is available, the following routine is, by employing the trick that, by hardware, floating-point multiplication results in the most significant bits of the product kept, while integer multiplication results in the least significant bits kept:
uint64_t mul_mod
Below is a C function for performing modular exponentiation, that uses the function implemented above.
An algorithmic way to compute :
uint64_t pow_mod
However, for all above routines to work, must not exceed 63 bits.