The notion of modular arithmetic is related to that of the "remainder in "Euclidean division. The operation of finding the remainder is sometimes referred to as the "modulo operation, and denoted with "mod" used as an "infix operator. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12; as this remainder is 2, we have 14 mod 12 = 2.

The congruence, indicated by "≡" followed by "mod" between parentheses, means that the operator "mod", applied to both members, gives the same result. That is

is equivalent to

The fundamental property of multiplication in modular arithmetic may thus be written

or, equivalently,

In "computer science, it is the remainder operator that is usually indicated by either "%" (for example, in "C, "C++, "Java, "JavaScript, "Perl, "Python and "Scala) or "mod" (for example, in "Pascal, "BASIC, "SQL, "Haskell, "ABAP, and "MATLAB), with exceptions (for example, Excel). These operators are commonly pronounced as "mod", but it is specifically a remainder that is computed (since in C++ a 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. For example, the programming language "Ruby has both `modulo`

(which returns a nonnegative result) and `remainder`

(which returns a negative result for some inputs). For details of the specific operations defined in different languages, see the "modulo operation page.

## Congruence classes [edit]

Like any congruence relation, congruence modulo *n* is an "equivalence relation, and the "equivalence class of the integer *a*, denoted by *a*_{n}, is the set {… , *a* − 2*n*, *a* − *n*, *a*, *a* + *n*, *a* + 2*n*, …}. This set, consisting of the integers congruent to *a* modulo *n*, is called the **congruence class** or **residue class** or simply **residue** of the integer *a*, modulo *n*. When the modulus *n* is known from the context, that residue may also be denoted [*a*].

## Residue systems[edit]

Each residue class modulo *n* 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 (since this is the proper remainder which results from division). Any two members of different residue classes modulo *n* are incongruent modulo *n*. Furthermore, every integer belongs to one and only one residue class modulo *n*.^{[1]}

The set of integers {0, 1, 2, …, *n* − 1} is called the **least residue system modulo n**. Any set of

*n*integers, no two of which are congruent modulo

*n*, is called a

**complete residue system modulo**.

*n*It is clear that the least residue system is a complete residue system, and that a complete residue system is simply a set containing precisely one representative of each residue class modulo *n*.^{[2]} The least residue system modulo 4 is {0, 1, 2, 3}. Some other complete residue systems modulo 4 are:

- {1, 2, 3, 4}
- {13, 14, 15, 16}
- {−2, −1, 0, 1}
- {−13, 4, 17, 18}
- {−5, 0, 6, 21}
- {27, 32, 37, 42}

Some sets which are *not* complete residue systems modulo 4 are:

- {−5, 0, 6, 22} since 6 is congruent to 22 modulo 4.
- {5, 15} since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.

### Reduced residue systems[edit]

Any set of φ(*n*) integers that are relatively prime to *n* and that are mutually incongruent modulo *n*, where φ(*n*) denotes "Euler's totient function, is called a **reduced residue system modulo n**.

^{[3]}The example above, {5,15} is an example of a reduced residue system modulo 4.

## Integers modulo *n*[edit]

The set of all "congruence classes of the integers for a modulus *n* is called the **ring of integers modulo n**,

^{[4]}and is denoted , , or . The notation is, however, not recommended because it can be confused with the set of "

*n*-adic integers. The set is defined as follows.

When *n* ≠ 0, has *n* elements, and can be written as:

When *n* = 0, does not have zero elements; rather, it is "isomorphic to , since *a*_{0} = {*a*}.

We can define addition, subtraction, and multiplication on by the following rules:

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

In this way, becomes a "commutative ring. For example, in the ring , we have

as in the arithmetic for the 24-hour clock.

The notation is used, because it is the "quotient ring of by the "ideal containing all integers divisible by *n*, where is the "singleton set {0}. Thus is a "field when is a "maximal ideal, that is, when *n* is prime.

In terms of groups, the residue class *a*_{n} is the "coset of *a* in the "quotient group , a "cyclic group.^{[5]}

The set 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 (which, as mentioned before, is isomorphic to the ring of integers), for example, when discussing the "characteristic of a "ring.

The ring of integers modulo *n* is a "finite field if and only if *n* is "prime. If *n* is a non-prime "prime power, there exists a unique (up to isomorphism) finite field GF(*n*) with *n* elements, which must not be confused with the ring of integers modulo *n*, although they have the same number of elements.

## Modular exponentiation[edit]

It is a very useful fact that the value *y* ≡ *a*^{x} (mod *n*) can be computed very efficiently even when the number *a*^{x} is too large to compute (see "modular exponentiation for details.) *y* can be computed without dealing with numbers bigger than *n*^{2}.

## Applications[edit]

Modular arithmetic is referenced in "number theory, "group theory, "ring theory, "knot theory, "abstract algebra, "computer 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.

Modular arithmetic is often used to calculate checksums that are used within identifiers. "International Bank Account Numbers (IBANs), for example, make use of modulo 97 arithmetic to trap user input errors in bank account numbers.

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.

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. "XOR is the sum of 2 bits, modulo 2.

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 previous digit times 2, the previous 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).

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 (see for example, "apportionment), "economics, (see for example, "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 *n*, 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.^{[6]}

## Example implementations[edit]

Below are two reasonably fast C functions for performing modular multiplication on unsigned integers not larger than 63 bits, without overflow of the transient operations. An algorithmic way to compute *a* × *b* (mod *m*):

```
uint64_t mul_mod(uint64_t a, uint64_t b, uint64_t m)
{
uint64_t d = 0, mp2 = m >> 1;
int i;
if (a >= m) a %= m;
if (b >= m) b %= m;
for (i = 0; i < 64; ++i)
{
d = (d > mp2) ? (d << 1) - m : d << 1;
if (a & 0x8000000000000000ULL)
d += b;
if (d > m) d -= m;
a <<= 1;
}
return d;
}
```

On computer architectures where an "extended precision format with at least 64 bits of mantissa is available (such as the "long double type of most x86 C compilers), the following routine is faster than any algorithmic solution, 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(uint64_t a, uint64_t b, uint64_t m)
{
long double x;
uint64_t c;
int64_t r;
if (a >= m) a %= m;
if (b >= m) b %= m;
x = a;
c = x * b / m;
r = (int64_t)(a * b - c * m) % (int64_t)m;
return r < 0 ? r + m : r;
}
```

However, for both routines to work, *m* must not exceed 63 bits.

## See also[edit]

- "Boolean ring
- "Circular buffer
- "Congruence relation
- "Division (mathematics)
- "Finite field
- "Legendre symbol
- "Modular exponentiation
- "Modular multiplicative inverse
- "Modulo operation
- "Number theory
- "Pisano period (Fibonacci sequences modulo
*n*) - "Primitive root modulo n
- "Quadratic reciprocity
- "Quadratic residue
- "Rational reconstruction (mathematics)
- "Reduced residue system
- "Serial number arithmetic (a special case of modular arithmetic)
- "Two-element Boolean algebra
- Topics relating to the group theory behind modular arithmetic:
- Other important theorems relating to modular arithmetic:
- "Carmichael's theorem
- "Chinese remainder theorem
- "Euler's theorem
- "Fermat's little theorem (a special case of Euler's theorem)
- "Lagrange's theorem
- "Thue's lemma

## Notes[edit]

**^**Pettofrezzo & Byrkit (1970, p. 90)**^**Long (1972, p. 78)**^**Long (1972, p. 85)**^**It is a "ring, as shown below.**^**Sengadir T.,*Discrete Mathematics and Combinatorics*, p. 293, at "Google Books**^**Garey, M. R.; Johnson, D. S. (1979).*Computers and Intractability, a Guide to the Theory of NP-Completeness*. W. H. Freeman. "ISBN "0716710447.

## References[edit]

- John L. Berggren. "modular arithmetic". "Encyclopædia Britannica.
- "Apostol, Tom M. (1976),
*Introduction to analytic number theory*, Undergraduate Texts in Mathematics, New York-Heidelberg: "Springer-Verlag, "ISBN "978-0-387-90163-3, "MR 0434929, "Zbl 0335.10001. See in particular chapters 5 and 6 for a review of basic modular arithmetic. - Maarten Bullynck "Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th-century Germany"
- "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 - Long, Calvin T. (1972),
*Elementary Introduction to Number Theory*(2nd ed.), Lexington: "D. C. Heath and Company, "LCCN 77171950 - Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970),
*Elements of Number Theory*, Englewood Cliffs: "Prentice Hall, "LCCN 71081766 - Sengadir, T. (2009).
*Discrete Mathematics and Combinatorics*. Chennai, India: Pearson Education India. "ISBN "978-81-317-1405-8. "OCLC 778356123.

## External links[edit]

- Hazewinkel, Michiel, ed. (2001), "Congruence",
*"Encyclopedia of Mathematics*, "Springer, "ISBN "978-1-55608-010-4 - In this modular art article, one can learn more about applications of modular arithmetic in art.
- "Weisstein, Eric W. "Modular Arithmetic".
*"MathWorld*.

- An article on modular arithmetic on the GIMPS wiki
- Modular Arithmetic and patterns in addition and multiplication tables
- Whitney Music Box—an audio/video demonstration of integer modular math